A094701 Smallest linear combination of phi(n) and sigma(n) with nonnegative coefficients: a(n) = Min_{x>=0,y>=0} (x+y) for which x*phi(n) + y*sigma(n) is a multiple of n.

1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 3, 2, 4, 5, 2, 2, 3, 2, 4, 7, 4, 2, 2, 5, 4, 3, 1, 2, 5, 2, 2, 9, 4, 18, 3, 2, 4, 9, 4, 2, 7, 2, 6, 3, 4, 2, 3, 7, 5, 9, 5, 2, 3, 7, 3, 9, 4, 2, 5, 2, 4, 7, 2, 11, 7, 2, 9, 9, 10, 2, 3, 2, 4, 12, 4, 10, 7, 2, 5, 3, 4, 2, 3, 13, 4, 9, 4, 2, 5, 9, 9, 9, 4, 19, 3, 2, 7, 5, 5, 2, 7

Offset: 1

Author

Walter Nissen, May 20 2004

Keywords

• nonn

Comments

a(n) is a generalization of the multiperfect numbers in A007691.

Examples

			a(6) = 1 as 1*sigma(6) is a multiple of 6.
a(4) = 2 as 2*phi(4) + 0*sigma(4) = 4. - Example added by _Antti Karttunen_, Feb 24 2020
a(14) = 4 as 3*phi(14) + 1*sigma(14) = 3*6 + 24 = 3*14, where 3+1 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12857
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000010, A000203, A000396, A005820, A007691 (positions of ones), A027687, A046060, A046061.

Programs

PARI

A094701(n) = { my(x=eulerphi(n),y=sigma(n)); for(s=1,oo,for(t=0,s,if(!(((t*x)+((s-t)*y))%n),return(s)))); }; \\ Antti Karttunen, Feb 24 2020

Formula

a(multiperfect) = 1.

a(prime) = 2 as 1*phi(prime) + 1*sigma(prime) and 1+1 = 2.

For primes > 5, a(2*prime) = 4.

Extensions

Name clarified by Antti Karttunen, Feb 24 2020