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

Author

Jonathan Sondow, Feb 06 2014

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Semiprime
• Wikipedia, Semiprime

Crossrefs

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

Programs

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

Formula

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