A063667 Number of solutions of phi(x) = 12n + 2.

3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Labos Elemer, Aug 22 2001

Keywords

• nonn

Examples

			In the range n=0..10000, only 18 invphi(12n + 2) sets are nonempty, always with 2 terms. E.g., n = 8034, a(8034) = 2 because 12*8034 + 2 = 96410 and invphi(96410) = {96721,193442}. - Original comment corrected by _Antti Karttunen_, Nov 07 2018
In the range n <= 100000, there are 48 nonzero values. - _Antti Karttunen_, Nov 07 2018
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..100000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A002202, A005277, A014197, A017545.

Programs

Maple

with(numtheory): [seq(nops(invphi(2+12*j)),j=0..10000)];

PARI

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197 by M. F. Hasler
A063667(n) = A014197(2+(12*n)); \\ Antti Karttunen, Nov 07 2018

PARI

a(n) = invphiNum(12*n+2); \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = A014197(A017545(n)). - Antti Karttunen, Nov 07 2018

Extensions

Term a(0) = 3 prepended by Antti Karttunen, Nov 07 2018