a(a(n) + n) is prime

[corrected sequence below]

Hello SeqFans,

am I wrong or this seq is not in the OEIS yet?

S = 1,2,3,5,4,7,8,6,11,9,10,12,13,17,19,14,15,16,23,29,31,18,20,37,21,41,22,24,25,43,47,53,26,59,...

S is build with those rules:

. a(1)=1

. a(a(n)+n) is prime

. S is extended with the smallest integer not yet in S and not leading to a contradiction.

We see here, that:

a(1)+1 -> 1+1=2 -> a(2) which is prime [2]

a(2)+2 -> 2+2=4 -> a(4) which is prime [5]

a(3)+3 -> 3+3=6 -> a(6) which is prime [7]

a(4)+4 -> 5+4=9 -> a(9) which is prime [11]

a(5)+5 -> 4+5=9 -> a(9) which is prime [11]

a(6)+6 -> 7+6=13 -> a(13) which is prime [13]

a(7)+7 -> 8+7=15 -> a(15) which is prime [19]

a(8)+8 -> 6+8=14 -> a(14) which is prime [17]

a(9)+9 -> 11+9=20 -> a(20) which is prime [29] ...

n = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

S = 1,2,3,5,4,7,8,6,11,9,10,12,13,17,19,14,15,16,23,29,31,18,20,37,21,41,22,24,25,43,47,53,26,59,...

sum 2 4 6 9 9 1 1 1  2 1 21 24 26 31 34 30 32 ...

3 5 4  0 9

I guess S is a permutation of the Naturals.

Primes appear in S in their natural order.

Best,

É.

[Sent: Tuesday, October 08, 2013 2:23 PM

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[Lars Blomberg]:

Hello Eric!

My results are below, they differ a little from yours but I have checked that the prime property is correct.

However, I cannot figure out what the difference in our methods might be.

Apart from a(3) = 3, all entries that do not require a prime have a composite number.

S=1, 2, 3, 5, 4, 7, 6, 8, 11, 9, 10, 12, 13, 14, 15, 17, 16, 18, 19, 23, 29, 20, 21, 31, 22, 37, 24, 41, 25, 43, 26, 27, 47, 28, 30, 53, 32, 59, 33, 34, 35, 61, 67, 71, 36, 38, 73, 39, 40, 79, 83, 42, 44, 89, 97, 45, 101, 46, 103, 48, 49, 107, 109, 50, 113, 51, 52, 54, 127, 55, 56, 131, 137, 139, 57, 149, 58, 60, 62, 151, 157, 63, 64, 163, 65, 66, 167, 68, 173, 69, 70, 72, 74, 179, 75, 76, 181, 77, 78, 80, 191, 81, 193, 197, 82, 84, 85, 199, 86, 211, 87, 88, 90, 223, 227, 91, 229, 92, 233, 239, 93, 241, 94, 95, 251, 96, 257, 98, 263, 99, 100, 269, 102, 271, 277, 104, 105, 281, 106, 108, 283, 110, 293, 111, 307, 112, 311, 114, 115, 313, 116, 317, 117, 118, 119, 331, 120, 337, 347, 121, 349, 353, 122, 359, 123, 124, 367, 125, 373, 379, 126, 383, 128, 129, 389, 130, 397, 401, 132, 409, 133, 134, 419, 135, 136, 138, 421, 140, 141, 431, 142, 433, 143, 144, 439, 443, 145, 449, 146, 457, 147, 148, 461, 150, 152, 153, 463, 154, 155, 467, 156, 158, 479, 487, 159, 160, 491, 161, 499, 162, 164, 503, 165, 166, 509, 521, 168, 169, 523, 170, 541, 171, 172, 174, 547, 175, 176, 557, 177, 563, 178, 569, 180, 182, 571, 183, 577, 587, 184, 185, 186, 593, 187, 599, 601, 188, 189, 607, 190, 192, 194, 613, 195, 617, 196, 198, 619, 200, 201, 631, 202, 641, 643, 647, 203, 204, 653, 659, 205, 206, 661, 207, 208, 209, 673, 210, 212, 677, 213, 683, 214, 691, 701, 215, 216, 709, 719, 217, 218, 219, 727, 220, 733, 221, 222, 739, 743, 224, 225, 226, 751, 228, 230, 757, 231, 761, 232, 234, 769, 235, 773, 236, 237, 787, 238, 240, 242, 797, 243, 809, 244, 245, 811, 246, 247, 821, 823, 827, 248, 249, 250, 829, 252, 253, 839, 853, 254, 857, 255, 859, 256, 863, 258, 877, 259, 260, 881, 261, 883, 262, 264, 887, 907, 911, 265, 266, 919, 267, 268, 929, 270, 272, 273, 937, 274, 941, 275, 276, 947, 278, 279, 953, 280, 967, 971, 282, 284, 977, 285, 983, 286, 991, 287, 288, 997, 289, 1009, 290, 291, 1013, 1019, 292, 1021, 294, 1031, 295, 296, 1033, 297, 298, 1039, 1049, 1051, 299, 300, 1061, 301, 302, 1063, 303, 304, 305, 1069, 1087, 306, 1091, 308, 309, 1093, 310, 312, 314, 1097, 315, 1103, 1109, 1117, 316, 318, 1123, 319, 320, 321, 1129, 322, 1151, 323, 324, 1153, 325, 326, 1163, 327, 328, 1171, 329, 330, 1181, 332, 333, 1187, 334, 1193, 1201, 335, 336, 338, 1213, 1217, 1223, 339, 340, 1229, 341, 342, 343, 344, 1231, 345, 1237, 346, 348, 350, 1249, 351, 1259, 1277, 352, 1279, 354, 1283, 355, 1289,...

(more values are available if you need them)

/Lars B

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Many thanks, Lars! And yes, your version is correct, indeed (my apologizes to everyone)!

This has been checked by Maximilian Hasler and is now http://oeis.org/A230086

Best,

É.