

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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



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



KEYWORD

nonn


AUTHOR



STATUS

approved



