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a(n) = 0 if n is already prime. a(n) = -1 for n = any multiple of 3 other than 3 itself. The first 5 record holders in this sequence are 1, 14, 20, 25, 40 with the values 1, 2, 3, 8, 483 respectively. 410 may be the next record holder as no solution has been found for it yet. 410 was tested out to 1250 threes with no prime formed.

From Toshitaka Suzuki, May 19 2024: (Start)

The first 6 record holders in this sequence are 1, 14, 20, 25, 40, 410 with the values 1, 2, 3, 8, 483, 37398 respectively. 817 may be the next record holder as no solution has been found for it yet. 817 was tested out to 300000 threes with no prime formed.

a(n) = -1 when n = 37037*k + 2808, 3666, 4070, 9287, 18799, 21574, 28083, 30558, 33300, 33740, 36663 or 36707, because n followed by any positive number, m say, of 3's is divisible by at least one of the primes {7,11,13,37}. (End)

a(n) = 0 if n is already prime. a(n) = -1 for n = any multiple of 7 other than 7 itself. Each multiple of 7 has been tested out to 2000 7's with no result found. The first eight values of n which are not multiples of 7 for which no answer has yet been found are 95, 480, 851, 891, 957, 1184, 1261, 1881. 95 has been tested out to 2100 7's, 1881 has been tested out to 1750 7's, the others have been tested out to 2000 7's. Pending solutions for these values of n, the first 10 record holders are currently 1, 8, 20, 40, 120, 128, 225, 260, 296, 711 with the values 1, 2, 6, 12, 16, 18, 56, 182, 434, 1648 respectively.

From Toshitaka Suzuki, Sep 26 2023: (Start)

a(95) = 2904, a(480) = 11330, a(851) = 28895, a(891) = -1, a(957) = 2903, a(1184) = 4646, a(1261) = -1 and a(1881) = 47927.

a(n) = -1 when n = 15873*k + 891, 1261, 2889, 3263, 3300, 7810, 8917, 9812, 12617, 13024, 14615 or 15066, because n followed by any positive number, m say, of 7's is divisible by at least one of the primes {3,11,13,37}.

Similarly,

a(n) = -1 when n = 11111111*k + 964146, 1207525, 2342974, 3567630, 7525789, 8134540, 8591231 or 9641467 by primes {11,73,101,137};

a(n) = -1 when n = 429000429*k + 23928593, 27079312, 36492115, 41207969, 52285750, 80569929, 89920882, 93857078, 133928703, 217208145, 223492302, 236849444, 239285937, 247857232, 259793116, 270793127, 323985244, 332698824, 333570182, 334985255, 346849554, 364921157, 376698868 or 412079697 by primes {3,11,13,101,9901};

a(n) = -1 when n = 1221001221*k + 14569863, 28792885, 145698637, 167698659, 225079510, 235985156, 247079532, 287928857, 331921124, 399492478, 415286113, 421492500, 437286135, 455985376, 489857474, 529929099, 551921344, 635208563, 709857694, 877208805, 896850104, 993570842, 1029793886 or 1138850346 by primes {3,11,37,101,9901};

a(n) = -1 when n = 1443001443*k + 85928655, 167698659, 176928746, 218921011, 233985154, 247079532, 310492389, 326286024, 376857361, 585793442, 655208583, 700699192, 746208674, 780080065, 791570640, 805850013, 843492922, 859286557, 882570731, 896850104, 1027793884, 1219922012, 1234986155 or 1377858362 by primes {3,13,37,101,9901}.

The first 10 record holders are 1, 8, 20, 40, 95, 480, 851, 1881, 2038, 2174 with the values 1, 2, 6, 12, 2904, 11330, 28895, 47927, 76206, 94146 respectively.

a(4444) > 300000 or a(4444) = -1.

(End)

For multiples of 7 it is confusing that the author writes in the first comment "has been tested out to 2000": If we denote n{m} = n*10^m + (10^m-1)/9*7 the number n with m '7's appended, then it is easy to see that (7k){m} / 7 = k*10^m + (10^m-1)/9 is an integer for all m >= 0. - M. F. Hasler, Jun 05 2024

a(n) = -1 when n = 3*k because no matter how many 9's are appended to n, the resulting number is always divisible by 3 and therefore cannot be prime.

a(n) = -1 when n = 37037*k + 8424, 10175, 10998, 12210, 17600, 19360, 25826, 27146, 27685, 27861, 35915 or 36047, because n followed by any positive number, m say, of 9's is divisible by at least one of the primes {7,11,13,37}.

a(4420) > 300000 or a(4420) = -1.