[seqfan] Re: "Prime sums" of underlined neighbors

[franktaw at netscape.net]

OK - but  can you actually prove that every prime does appear? It isn't 
obvious to me.

I think you have a few constraints on your sequence you haven't 
mentioned:

1) The underlined pairs must not overlap; i.e., you can't have 1,2,3.
2) After the first two pairs, every third number must not be part of an 
underlined pair, and the remaining pairs must be underlined.

Even with these constraints, it seems to me that the minimum sequence 
would start:
1,1,3,8,1,3,14,1,5,18,3,1,2,8,...

Franklin T. Adams-Watters

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>

Hello Franklin,
thank you for your comments -- but
note that 2 is not missing. By cons-
truction all primes will show, sooner
or later.
Ok, this sequence is not fractal in
the sense you mention -- let's say
it contains an infinite amount of
copies of itself.
Best,
E.
Envoyé d'un iPhone


Le 21 sept. 2011 à 20:49, "franktaw at netscape.net" 
<franktaw at netscape.net> a
écrit :

> By all means, submit it.
>
> You don't quite get a rearrangement of the primes; 2 is missing.
>
> This is not a fractal sequence by the way that is defined. (You have 
to
> get the sequence back by removing specifically the first instance of
> each number in the sequence.)
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Eric Angelini <Eric.Angelini at kntv.be>
>
> Hello SeqFans,
>
> Submit this sequence to a teenager and ask him to underline all
> couples of neighboring integers which sum up to a prime:
>
> S = 
1,2,4,3,1,3,2,2,6,5,4,2,11,3,3,14,1,7,12,3,1,22,2,4,25,2,2,29,6,...
>
> He will underline 1,2, then 4,3, then 3,2, then 6,5, etc. -- like
> this:
>
> S : 
1,2,4,3,1,3,2,2,6,5,4,2,11,3,3,14,1,7,12,3,1,22,2,4,25,2,2,29,6,...
>    *** ***   ***   ***   ****   ****   ****   ****   ****   ****
>
> Now for the first amazing remark:
> - all "prime sums" are different and they seem to be a derangement
>  of the prime numbers (they are);
> And the second remark:
> - the not underlined integers re-build the sequence S itself.
>
> We have thus a fractal sequence.
>
> Additional remarks:
> - Lars Blomberg has computed this sequence (see below)
> - after the start 1,2,4,3, we always have a single "non underlined"
> integer followed by two underlined ones
> - S is the lexically first sequence obeying those constraints.
>
> Is S worth entering the OEIS? _My_ (doubly) teenagers were stumped!
>
> Best,
> É.
>
> -------------------------------------------------------------------
> Start with:
> 1,2,4,3
> Sequence:
> 
1,2,4,3,1,3,2,2,6,5,4,2,11,3,3,14,1,7,12,3,1,22,2,4,25,2,2,29,6,2,35,5,1,

> 
40,4,5,38,2,2,45,11,1,1,3,1,52,3,1,58,14,2,59,1,5,62,7,1,70,12,2,71,3,3,7

> 
6,1,3,80,22,3,86,2,2,95,4,2,99,25,1,102,2,4,103,2,2,107,29,1,112,6,3,124,

> 
2,2,129,35,1,136,5,4,135,1,3,146,40,2,149,4,2,155,5,1,162,38,1,166,2,4,16

> 
9,2,2,177,45,3,178,11,3,188,1,5,188,1,5,192,3,5,194,1,3,208,52,2,221,3,3,

> 
224,1,3,226,58,2,231,14,1,238,2,2,239,59,3,248,1,3,254,5,1,262,62,1,268,7

> 
,2,269,1,3,274,70,2,279,12,3,280,2,2,291,71,1,306,3,5,306,3,1,312,76,2,31

> 
5,1,5,326,3,1,336,80,1,346,22,2,347,3,1,352,86,1,358,2,4,363,2,2,371,95,1

> 
,378,4,4,379,2,2,387,99,5,392,25,2,399,1,3,406,102,3,416,2,2,419,4,2,429,

> 
103,1,432,2,4,435,2,2,441,107,1,448,29,3,454,1,3,458,112,3,460,6,2,465,3,

> 
1,478,124,1,486,2,4,487,2,2,497,129,1,502,35,3,506,1,3,518,136,2,521,5,1,

> 
540,4,2,545,135,3,554,1,5,558,3,1,568,146,1,570,40,4,573,2,2,585,149,1,59

> 
2,4,4,595,2,2,599,155,1,606,5,3,610,1,3,614,162,2,617,38,2,629,1,3,638,16

> 
6,3,640,2,2,645,4,2,651,169,1,658,2,6,655,2,2,671,177,1,676,45,1,682,3,1,

> 
690,178,2,699,11,1,708,3,1,718,188,2,725,1,3,730,5,1,738,188,2,741,1,3,74

> 
8,5,1,756,192,2,759,3,3,766,5,1,772,194,2,785,1,5,792,3,1,808,208,1,810,5

> 
2,3,818,2,2,821,221,4,823,3,1,828,3,1,838,224,2,851,1,5,852,3,1,858,226,4

> 
,859,58,4,873,2,2,879,231,1,882,14,2,885,1,3,904,238,4,907,2,4,915,2,2,92

> 
7,239,1,936,59,1,940,3,1,946,248,2,951,1,5,962,3,1,970,254,1,976,5,3,980,

> 
1,3,988,262,2,995,62,2,1007,1,3,1010,268,2,1017,7,1,1020,2,2,1029,269,4,1

> 
029,1,5,1034,3,1,1048,274,1,1050,70,4,1057,2,2,1061,279,1,1068,12,2,1085,

> 
3,1,1090,280,5,1088,2,4,1093,2,2,1101,291,1,1108,71,3,1114,1,3,1120,306,2

> 
,1127,3,1,1150,5,1,1152,306,2,1161,3,3,1168,1,3,1178,312,2,1185,76,1,1192

> 
,2,2,1199,315,3,1210,1,3,1214,5,1,1222,326,2,1227,3,5,1226,1,3,1234,336,3

> 
,1246,80,2,1257,1,3,1274,346,2,1277,22,3,1280,2,2,1287,347,1,1290,3,3,129

> 
4,1,3,1298,352,2,1301,86,2,1305,1,3,1316,358,3,1318,2,2,1325,4,2,1359,363

> 
,1,1366,2,4,1369,2,2,1379,371,3,1396,95,3,1406,1,3,1420,378,3,1424,4,2,14

> 
27,4,2,1431,379,1,1438,2,4,1443,2,2,1449,387,1,1452,99,1,1458,5,1,1470,39

> 
2,1,1480,25,1,1482,2,2,1485,399,4,1485,1,5,1488,3,1,1498,406,1,1510,102,2

> 
,1521,3,1,1530,416,1,1542,2,4,1545,2,2,1551,419,1,1558,4,4,1563,2,2,1569,

> 
429,1,1578,103,2,1581,1,3,1594,432,3,1598,2,6,1601,4,2,1607,435,1,1612,2,

> 
4,1615,2,2,1619,441,3,1624,107,3,1634,1,3,1654,448,2,1661,29,4,1663,3,1,1

> 
668,454,2,1691,1,5,1692,3,1,1698,458,1,1708,112,2,1719,3,1,1722,460,2,173

> 
1,6,3,1738,2,2,1745,465,1,1752,3,3,1756,1,3,1774,478,2,1781,124,2,1785,1,

> 
3,1786,486,3,1798,2,2,1809,4,2,1821,487,1,1830,2,4,1843,2,2,1859,497,1,18

> 
66,129,3,1868,1,3,1870,502,2,1875,35,1,1878,3,1,1888,506,2,1899,1,5,1902,

> 
3,1,1912,518,1,1930,136,5,1928,2,2,1947,521,1,1950,5,3,1970,1,3,1976,540,

> 
2,1985,4,4,1989,2,2,1995,545,1,1998,135,1,2002,3,1,2010,554,2,2015,1,7,20

> 
20,5,1,2028,558,1,2038,3,3,2050,1,3,2060,568,4,2065,146,2,2079,1,3,2080,5

> 
70,2,2085,40,2,2087,4,2,2097,573,1,2110,2,6,2107,2,2,2127,585,1,2130,149,

> 
3,2134,1,3,2138,592,2,2141,4,2,2151,4,2,2159,595,1,2178,2,4,2199,2,2,2205

> ,599,1,2212,155,3,2218,1,3,2234,606,2,2237,5,1,2242,3
> Primes used:
> 
3,7,5,11,13,17,19,23,29,31,37,41,43,47,2,53,59,61,67,71,73,79,83,89,97,10

> 
1,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193

> 
,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,

> 
307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,4

> 
19,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,52

> 
3,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643

> 
,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,

> 
769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,8

> 
87,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019

> 
,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093,1097,1103,11

> 
09,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,

> 
1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,131

> 
9,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427,1429,1433,1439,1447,1

> 
451,1453,1459,1471,1481,1483,1487,1489,1493,1499,1511,1523,1531,1543,1549

> 
,1553,1559,1567,1571,1579,1583,1597,1601,1607,1609,1613,1619,1621,1627,16

> 
37,1657,1663,1667,1669,1693,1697,1699,1709,1721,1723,1733,1741,1747,1753,

> 
1759,1777,1783,1787,1789,1801,1811,1823,1831,1847,1861,1867,1871,1873,187

> 
7,1879,1889,1901,1907,1913,1931,1933,1949,1951,1973,1979,1987,1993,1997,1

> 
999,2003,2011,2017,2027,2029,2039,2053,2063,2069,2081,2083,2087,2089,2099

> 
,2111,2113,2129,2131,2137,2141,2143,2153,2161,2179,2203,2207,2213,2221,22

> 37,2239,2243,2251
> Primes not used:
> 
2267,2269,2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,235

> 
7,2371,2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2

> 
467,2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609

> 
,2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699,27

> 
07,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797,2801,

> 
2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909,2917,292

> 
7,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037,3041,3049,3

> 
061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169,3181,3187,3191

> 
,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299,3301,3307,3313,33

> 
19,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389,3391,3407,3413,3433,

> 
3449,3457,3461,3463,3467,3469,3491,3499,3511,3517,3527,3529,3533,3539,354

> 
1,3547,3557,3559,3571,3581,3583,3593,3607,3613,3617,3623,3631,3637,3643,3

> 
659,3671,3673,3677,3691,3697,3701,3709,3719,3727,3733,3739,3761,3767,3769

> 
,3779,3793,3797,3803,3821,3823,3833,3847,3851,3853,3863,3877,3881,3889,39

> 
07,3911,3917,3919,3923,3929,3931,3943,3947,3967,3989,4001,4003,4007,4013,

> 
4019,4021,4027,4049,4051,4057,4073,4079,4091,4093,4099,4111,4127,4129,413

> 
3,4139,4153,4157,4159,4177,4201,4211,4217,4219,4229,4231,4241,4243,4253,4

> 
259,4261,4271,4273,4283,4289,4297,4327,4337,4339,4349,4357,4363,4373,4391

> 
,4397,4409,4421,4423,4441,4447,4451,4457,4463,4481,4483,4493,4507,4513,45

> 
17,4519,4523,4547,4549,4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,

> 
4649,4651,4657,4663,4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,478

> 
3,4787,4789,4793,4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4

> 
919,4931,4933,4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009

> 
,5011,5021,5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,51

> 
47,5153,5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,

> 
5281,5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,541

> 
7,5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519,5

> 
521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651,5653

> 
,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749,5779,57

> 
83,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861,5867,5869,

> 
5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011,6029,6037,604

> 
3,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131,6133,6143,6151,6

> 
163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257,6263,6269,6271,6277

> 
,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353,6359,6361,6367,6373,63

> 
79,6389,6397,6421,6427,6449,6451,6469,6473,6481,6491,6521,6529,6547,6551,

> 
6553,6563,6569,6571,6577,6581,6599,6607,6619,6637,6653,6659,6661,6673,667

> 
9,6689,6691,6701,6703,6709,6719,6733,6737,6761,6763,6779,6781,6791,6793,6

> 
803,6823,6827,6829,6833,6841,6857,6863,6869,6871,6883,6899,6907,6911,6917

> 
,6947,6949,6959,6961,6967,6971,6977,6983,6991,6997,7001,7013,7019,7027,70

> 
39,7043,7057,7069,7079,7103,7109,7121,7127,7129,7151,7159,7177,7187,7193,

> 
7207,7211,7213,7219,7229,7237,7243,7247,7253,7283,7297,7307,7309,7321,733

> 
1,7333,7349,7351,7369,7393,7411,7417,7433,7451,7457,7459,7477,7481,7487,7

> 
489,7499,7507,7517,7523,7529,7537,7541,7547,7549,7559,7561,7573,7577,7583

> 
,7589,7591,7603,7607,7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,77

> 
03,7717,7723,7727,7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,

> 
7867,7873,7877,7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,799

> 
3,8009,8011,8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8

> 
123,8147,8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263

> 
,8269,8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,83

> 
89,8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539,

> 
8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669,867

> 
7,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761,8779,8

> 
783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887,8893,8923

> 
,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013,9029,9041,90

> 
43,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157,9161,9173,9181,

> 
9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281,9283,9293,9311,931

> 
9,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403,9413,9419,9421,9431,9

> 
433,9437,9439,9461,9463,9467,9473,9479,9491,9497,9511,9521,9533,9539,9547

> 
,9551,9587,9601,9613,9619,9623,9629,9631,9643,9649,9661,9677,9679,9689,96

> 
97,9719,9721,9733,9739,9743,9749,9767,9769,9781,9787,9791,9803,9811,9817,

> 
9829,9833,9839,9851,9857,9859,9871,9883,9887,9901,9907,9923,9929,9931,994

> 1,9949,9967,9973
>
>
>
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>
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>
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>
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