A306342 a(n) is the smallest divisor > 1 of prime(n)# = A002110(n) that is congruent to 1 modulo prime(n+1), or 0 if no such divisor exists.

0, 0, 6, 15, 210, 14, 35, 39, 70, 30, 187, 38, 42, 87, 95, 266, 119, 62, 403, 143, 74, 159, 665, 357, 195, 102, 310, 215, 110, 114, 255, 1311, 138, 418, 299, 303, 158, 327, 335, 174, 1254, 182, 574, 194, 395, 399, 1267, 447, 455, 230, 1166, 957, 483, 754, 258, 527, 1615, 543, 278, 282, 1133, 1173, 615, 623, 314, 318, 663, 1349, 695, 699, 354

Offset

0,3

Comments

Also, a(n) is the smallest number congruent 1 modulo prime(n+1) that can be written as the product of distinct primes <= prime(n).

a(n) > 0 for 2 <= n <= 10^4. It seems that a(n) > 0 for all n >= 2.

It is surprising that a(n) is usually small compared with n. Let b(n) = (a(n) - 1)/prime(n+1) if a(n) != 0. Except for b(4) = 19, b(n) <= 14 for 2 <= n <= 10^4, and there are only 21 n's <= 10^4 such that b(n) >= 10.

Links

Table of n, a(n) for n=0..70.

Formula

a(2) = 6 = 2*3 = 1*5 + 1.

a(3) = 15 = 3*5 = 2*7 + 1.

a(4) = 210 = 2*3*5*7 = 19*11 + 1.

a(5) = 14 = 2*7 = 1*13 + 1.

a(6) = 35 = 5*7 = 2*17 + 1.

Prog

(PARI) a(n)=my(N=prod(i=1, n, prime(i))\prime(n+1)); for(k=1, N+1, if(k==N+1, return(0)); my(m=k*prime(n+1)+1, p=vecmax(factor(m)[, 1])); if(issquarefree(m)&&p<=prime(n), return(m)))

Crossrefs

Cf. A002110.

Sequence in context: A199224 A013224 A013230 * A117062 A003155 A335578

Adjacent sequences: A306339 A306340 A306341 * A306343 A306344 A306345

Keyword

nonn

Author

Jianing Song, Feb 08 2019

Status

approved