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

Charles Greathouse charles.greathouse at case.edu
Wed Sep 21 21:28:55 CEST 2011

```I also think this is worth submitting (and I'm not usually one for
puzzle sequences).  It looks like 1, 1, 2, 3, 1, 6, 5, 1, 12, 2, 15,
3, 1, 18, 6, ... works for the primes rather than the odd primes,
though it has them in order (so far and, I think, always) and lacks
the same structure as the other.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Wed, Sep 21, 2011 at 2:49 PM,  <franktaw at netscape.net> wrote:
> 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.)
>
>
> -----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.
>
> - 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.
>
> -------------------------------------------------------------------
> 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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```