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A237115 Lesser prime factor of the smallest semiprime of the form k^prime(n)+1, or 0 if no such semiprime exists. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A102680 A025791 A324608 * A069637 A072292 A243282

Adjacent sequences:  A237112 A237113 A237114 * A237116 A237117 A237118

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Feb 04 2014

STATUS

approved

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Last modified September 20 22:20 EDT 2019. Contains 327252 sequences. (Running on oeis4.)