Are there infinitely many primes in sequences defined by $a_{k+1} = a_k + \operatorname{digitsum}(a_k)$?
问题内容
For any positive integer $n$, define $S(n)$ as the sum of the digits of $n$ in decimal (e.g., $S(19) = 1 + 9 = 10$). Define the sequence $a_k$ as: $a_1 = 7$ $a_{k+1} = a_k + S(a_k)$
Edit: To add, the first term ($a_1$) can be any positive integer(except $a_1 \not\equiv 0 \pmod 3$).
I conjecture: In the sequence $\{a_k\}$, there exist infinitely many terms that are prime numbers.
For example:
$a_1 = 7$ (prime number)
$a_2 = 7 + S(7) = 14$
$a_3 = 14 + S(14) = 19$ (prime number)
$a_4 = 19 + S(19) = 29$ (prime number)
$a_5 = 29 + S(29) = 40$
$a_6 = 40 + S(40) = 44$
$a_7 = 44 + S(44) = 52$
$a_8 = 52 + S(52) = 59$ (prime number)
Motivation:
The reason is that I tried to find a pattern between "ordinary addition" and "the sum of the digits in decimal", and unexpectedly this question popped up today.
My question:
I want to know if there are any known relevant documents in history, not for proof (unless it is really possible).
Edit :
The following are the percentages of the first 10,000 items that are different from the first item:
$a_1$ = 2: 1317 primes (13.17%)
$a_1$ = 5: 1320 primes (13.20%)
$a_1$ = 7: 1319 primes (13.19%)
$a_1$ = 11: 1319 primes (13.19%)
$a_1$ = 13: 1318 primes (13.18%)
$a_1$ = 17: 1318 primes (13.18%)
$a_1$ = 19: 1318 primes (13.18%)
$a_1$ = 23: 1316 primes (13.16%)
$a_1$ = 29: 1317 primes (13.17%)
$a_1$ = 31: 1315 primes (13.15%)
$a_1$ = 37: 1318 primes (13.18%)
$a_1$ = 41: 1316 primes (13.16%)
$a_1$ = 43: 1316 primes (13.16%)
$a_1$ = 47: 1317 primes (13.17%)
$a_1$ = 53: 1317 primes (13.17%)
$a_1$ = 59: 1318 primes (13.18%)
$a_1$ = 61: 1317 primes (13.17%)
$a_1$ = 67: 1316 primes (13.16%)
$a_1$ = 71: 1318 primes (13.18%)
$a_1$ = 73: 1317 primes (13.17%)
$a_1$ = 79: 1318 primes (13.18%)
$a_1$ = 83: 1316 primes (13.16%)
$a_1$ = 89: 1317 primes (13.17%)
$a_1$ = 97: 1318 primes (13.18%)
$a_1$ = 101: 1317 primes (13.17%)
$a_1$ = 103: 1317 primes (13.17%)
$a_1$ = 107: 1317 primes (13.17%)
$a_1$ = 109: 1317 primes (13.17%)
$a_1$ = 113: 1317 primes (13.17%)
$a_1$ = 127: 1317 primes (13.17%)
$a_1$ = 131: 1316 primes (13.16%)
$a_1$ = 137: 1316 primes (13.16%)
$a_1$ = 139: 1316 primes (13.16%)
$a_1$ = 149: 1316 primes (13.16%)
$a_1$ = 151: 1314 primes (13.14%)
$a_1$ = 157: 1315 primes (13.15%)
$a_1$ = 163: 1315 primes (13.15%)
$a_1$ = 167: 1315 primes (13.15%)
$a_1$ = 173: 1314 primes (13.14%)
$a_1$ = 179: 1313 primes (13.13%)
$a_1$ = 181: 1314 primes (13.14%)
$a_1$ = 191: 1313 primes (13.13%)
$a_1$ = 193: 1313 primes (13.13%)
$a_1$ = 197: 1313 primes (13.13%)
$a_1$ = 199: 1315 primes (13.15%)
$a_1$ = 211: 1316 primes (13.16%)
$a_1$ = 223: 1316 primes (13.16%)
$a_1$ = 227: 1312 primes (13.12%)
$a_1$ = 229: 1314 primes (13.14%)
$a_1$ = 233: 1313 primes (13.13%)
$a_1$ = 239: 1314 primes (13.14%)
$a_1$ = 241: 1313 primes (13.13%)
$a_1$ = 251: 1311 primes (13.11%)
$a_1$ = 257: 1314 primes (13.14%)
$a_1$ = 263: 1313 primes (13.13%)
$a_1$ = 269: 1315 primes (13.15%)
$a_1$ = 271: 1313 primes (13.13%)
$a_1$ = 277: 1316 primes (13.16%)
$a_1$ = 281: 1312 primes (13.12%)
$a_1$ = 283: 1312 primes (13.12%)
$a_1$ = 293: 1315 primes (13.15%)
$a_1$ = 307: 1314 primes (13.14%)
$a_1$ = 311: 1312 primes (13.12%)
$a_1$ = 313: 1311 primes (13.11%)
$a_1$ = 317: 1314 primes (13.14%)
$a_1$ = 331: 1309 primes (13.09%)
$a_1$ = 337: 1312 primes (13.12%)
$a_1$ = 347: 1311 primes (13.11%)
$a_1$ = 349: 1313 primes (13.13%)
$a_1$ = 353: 1309 primes (13.09%)
$a_1$ = 359: 1311 primes (13.11%)
$a_1$ = 367: 1313 primes (13.13%)
$a_1$ = 373: 1309 primes (13.09%)
$a_1$ = 379: 1313 primes (13.13%)
$a_1$ = 383: 1312 primes (13.12%)
$a_1$ = 389: 1313 primes (13.13%)
$a_1$ = 397: 1312 primes (13.12%)
$a_1$ = 401: 1311 primes (13.11%)
$a_1$ = 409: 1312 primes (13.12%)
$a_1$ = 419: 1311 primes (13.11%)
$a_1$ = 421: 1312 primes (13.12%)
$a_1$ = 431: 1312 primes (13.12%)
$a_1$ = 433: 1311 primes (13.11%)
$a_1$ = 439: 1311 primes (13.11%)
$a_1$ = 443: 1310 primes (13.10%)
$a_1$ = 449: 1309 primes (13.09%)
$a_1$ = 457: 1313 primes (13.13%)
$a_1$ = 461: 1312 primes (13.12%)
$a_1$ = 463: 1312 primes (13.12%)
$a_1$ = 467: 1309 primes (13.09%)
$a_1$ = 479: 1313 primes (13.13%)
$a_1$ = 487: 1312 primes (13.12%)
$a_1$ = 491: 1309 primes (13.09%)
$a_1$ = 499: 1312 primes (13.12%)
$a_1$ = 503: 1314 primes (13.14%)
$a_1$ = 509: 1311 primes (13.11%)
$a_1$ = 521: 1311 primes (13.11%)
$a_1$ = 523: 1310 primes (13.10%)
$a_1$ = 541: 1313 primes (13.13%)
$a_1$ = 547: 1312 primes (13.12%)
$a_1$ = 557: 1310 primes (13.10%)
$a_1$ = 563: 1311 primes (13.11%)
$a_1$ = 569: 1312 primes (13.12%)
$a_1$ = 571: 1315 primes (13.15%)
$a_1$ = 577: 1310 primes (13.10%)
$a_1$ = 587: 1312 primes (13.12%)
$a_1$ = 593: 1314 primes (13.14%)
$a_1$ = 599: 1314 primes (13.14%)
$a_1$ = 601: 1314 primes (13.14%)
$a_1$ = 607: 1311 primes (13.11%)
$a_1$ = 613: 1310 primes (13.10%)
$a_1$ = 617: 1313 primes (13.13%)
$a_1$ = 619: 1313 primes (13.13%)
$a_1$ = 631: 1312 primes (13.12%)
$a_1$ = 641: 1311 primes (13.11%)
$a_1$ = 643: 1313 primes (13.13%)
$a_1$ = 647: 1310 primes (13.10%)
$a_1$ = 653: 1312 primes (13.12%)
$a_1$ = 659: 1315 primes (13.15%)
$a_1$ = 661: 1312 primes (13.12%)
$a_1$ = 673: 1314 primes (13.14%)
$a_1$ = 677: 1312 primes (13.12%)
$a_1$ = 683: 1312 primes (13.12%)
$a_1$ = 691: 1313 primes (13.13%)
$a_1$ = 701: 1314 primes (13.14%)
$a_1$ = 709: 1313 primes (13.13%)
$a_1$ = 719: 1311 primes (13.11%)
$a_1$ = 727: 1313 primes (13.13%)
$a_1$ = 733: 1313 primes (13.13%)
$a_1$ = 739: 1312 primes (13.12%)
$a_1$ = 743: 1312 primes (13.12%)
$a_1$ = 751: 1313 primes (13.13%)
$a_1$ = 757: 1311 primes (13.11%)
$a_1$ = 761: 1311 primes (13.11%)
$a_1$ = 769: 1312 primes (13.12%)
$a_1$ = 773: 1312 primes (13.12%)
$a_1$ = 787: 1312 primes (13.12%)
$a_1$ = 797: 1312 primes (13.12%)
$a_1$ = 809: 1311 primes (13.11%)
$a_1$ = 811: 1312 primes (13.12%)
$a_1$ = 821: 1311 primes (13.11%)
$a_1$ = 823: 1312 primes (13.12%)
$a_1$ = 827: 1310 primes (13.10%)
$a_1$ = 829: 1312 primes (13.12%)
$a_1$ = 839: 1313 primes (13.13%)
$a_1$ = 853: 1311 primes (13.11%)
$a_1$ = 857: 1312 primes (13.12%)
$a_1$ = 859: 1313 primes (13.13%)
$a_1$ = 863: 1314 primes (13.14%)
$a_1$ = 877: 1312 primes (13.12%)
$a_1$ = 881: 1312 primes (13.12%)
$a_1$ = 883: 1311 primes (13.11%)
$a_1$ = 887: 1312 primes (13.12%)
$a_1$ = 907: 1312 primes (13.12%)
$a_1$ = 911: 1311 primes (13.11%)
$a_1$ = 919: 1313 primes (13.13%)
$a_1$ = 929: 1312 primes (13.12%)
$a_1$ = 937: 1311 primes (13.11%)
$a_1$ = 941: 1311 primes (13.11%)
$a_1$ = 947: 1313 primes (13.13%)
$a_1$ = 953: 1314 primes (13.14%)
$a_1$ = 967: 1313 primes (13.13%)
$a_1$ = 971: 1312 primes (13.12%)
$a_1$ = 977: 1312 primes (13.12%)
$a_1$ = 983: 1312 primes (13.12%)
$a_1$ = 991: 1311 primes (13.11%)
$a_1$ = 997: 1313 primes (13.13%)
Currently, we know that every positive integer within 1000 will converge into a sequence, which I call the "main sequence." This sequence is: 1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440. 448, 464, 478, 497, 517, 530, 538, 554, 568, 587, 607, 620, 628, 644, 658, 677, 697, 719, 736, 752, 766, 785, 805, 818, 835, 851, 865, 884, 904, 917, 934, 950, 964, 983, 1003, 1007, 1015, 1022, 1027, 1037, 1048, 1061, 1069, 1085, 1099, 1118, 1129, 1142, 1150, 1157, 1171, 1181, 1192, 1205.......
回答 (1)
COMMENT.-The problem could be reduced considerably if you note that it is enough to prove that for arbitrary $a_1=N$ the sequence $a_{k+1}=a_k+S(a_k)$ has at least a prime. With the powerfull of means of computation in hands of the O.P. that can be verified empirically up to a certain point, it's true, unfortunately. However prove this, I think, it must be easier than the original problem.