A237115 Lesser prime factor of the smallest semiprime of the form k^prime(n)+1, or 0 if no such semiprime exists.

2, 3, 3, 3, 3, 3, 3, 3, 3, 691, 3, 17, 313, 3, 7, 11, 7, 3, 11, 47, 19, 3, 1499, 17, 71, 3, 97, 7, 13, 823, 3, 97, 1163, 31, 17, 199, 1907, 53, 3, 17, 1231, 1013, 3, 13, 53, 3, 67, 47, 23, 1013, 787, 127, 347, 17, 37, 97, 683, 631, 73, 4549, 173, 11, 17, 1039, 3, 17, 47, 6389, 3, 461, 23, 673, 37, 29, 331, 7451, 1433, 4561

Offset

1,1

Comments

For n > 1, smallest prime p such that ((p-1)^prime(n)+1)/p is prime; the corresponding primes ((p-1)^prime(n)+1)/p are A237116(n) = 3, 11, 43, 683, 2731, 43691, 174763, 2796203, ... and the corresponding semiprimes (p-1)^prime(n)+1 are A237114(n) = 9, 33, 129, 2049, 8193, 131073, 524289, 8388609, ... .

Links

Table of n, a(n) for n=1..78.

Eric Weisstein's World of Mathematics, Semiprime

Wikipedia, Semiprime

Formula

a(n) = A237114(n)/A237116(n), for n > 0.

(a(n)-1)^prime(n) = A237114(n)-1, for n > 1.

a(n) == A237114(n) (mod prime(n)) (for a proof, see A237114).

a(n) mod prime(n) = A237117(n), if a(n) > 0.

Example

Prime(1)=2 and the smallest semiprime of the form k^2+1 is 3^2+1 = 10 = 2*5, so a(1) = 2.
Prime(2)=3 and the smallest semiprime of the form k^3+1 is 2^3+1 = 9 = 3*3, so a(2) = 3.

Mathematica

L = {2}; Do[p = Prime[k]; n = 1; q = Prime[n] - 1; cp = (q^p + 1)/(q + 1); While[! PrimeQ[cp], n = n + 1; q = Prime[n] - 1; cp = (q^p + 1)/(q + 1)]; L = Append[L, q + 1], {k, 2, 78}]; L

Crossrefs

Cf. A001358, A103795, A123627, A123628, A237040, A237114, A237116, A237117.

Sequence in context: A025791 A358474 A324608 * A362915 A069637 A072292

Adjacent sequences: A237112 A237113 A237114 * A237116 A237117 A237118

Keyword

nonn

Author

Jonathan Sondow, Feb 04 2014

Status

approved