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

Wed Sep 21 21:58:21 CEST 2011

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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