A237117 Remainder mod p of the smallest semiprime of the form k^p+1, where p = prime(n); or -1 if no such semiprime exists.

0, 0, 3, 3, 3, 3, 3, 3, 3, 24, 3, 17, 26, 3, 7, 11, 7, 3, 11, 47, 19, 3, 5, 17, 71, 3, 97, 7, 13, 32, 3, 97, 67, 31, 17, 48, 23, 53, 3, 17, 157, 108, 3, 13, 53, 3, 67, 47, 23, 97, 88, 127, 106, 17, 37, 97, 145, 89, 73, 53, 173, 11, 17, 106, 3, 17, 47, 323, 3, 112, 23, 314, 37, 29, 331, 174, 266, 194, 226, 397, 29, 16, 176, 45, 44, 152, 373, 349, 101, 143, 53, 386, 133, 29, 345, 1

Offset

1,3

Comments

It appears that a(n) > 0 for all n > 2. See the comments in A237114.

Links

Table of n, a(n) for n=1..96.

Eric Weisstein's World of Mathematics, Semiprime

Wikipedia, Semiprime

Formula

a(n) = A237114(n) mod prime(n) = A237115(n) mod prime(n), if A237114(n)>0.

Example

Prime(2)=3 and the smallest semiprime of the form k^3+1 is 2^3+1 = 9 = 3*3, so a(2) = 9 mod 3 = 0.
Prime(3)=5 and the smallest semiprime of the form k^5+1 is 2^5+1 = 33 = 3*11, so a(3) = 33 mod 5 = 3.

Mathematica

L = {0}; 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, Mod[q^p + 1, p]], {k, 2, 87}]; L

Crossrefs

Cf. A001358, A006881, A103795, A123627, A123628, A237040, A237114, A237115, A237116.

Sequence in context: A210745 A187471 A083565 * A247970 A063438 A276870

Adjacent sequences: A237114 A237115 A237116 * A237118 A237119 A237120

Keyword

nonn

Author

Jonathan Sondow, Feb 06 2014

Status

approved