A248029 Least positive integer m such that m + n divides phi(m)*sigma(n), where phi(.) and sigma(.) are given by A000010 and A000203.

1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 2, 1, 10, 9, 15, 1, 8, 1, 1, 11, 14, 1, 6, 6, 16, 5, 14, 1, 6, 1, 10, 15, 11, 13, 16, 1, 7, 9, 5, 1, 6, 1, 12, 7, 26, 1, 14, 8, 12, 21, 46, 1, 6, 17, 4, 23, 32, 1, 24, 1, 34, 41, 63, 7, 6, 1, 16, 11, 2

Offset

2,3

Comments

Conjecture: For any n > 1, we have a(n) <= n.

The existence of a(n) is easy; in fact, for m = sigma(n) - n, obviously m + n divides phi(m)*sigma(n). - Zhi-Wei Sun, Oct 02 2014

Links

Zhi-Wei Sun, Table of n, a(n) for n = 2..10000

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Example

a(8) = 7 since 7 + 8 = 15 divides phi(7)*sigma(8) = 6*15 = 90.

Mathematica

Do[m=1; Label[aa]; If[Mod[EulerPhi[m]*DivisorSigma[1, n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 2, 70}]

Prog

(PARI)
a(n)=m=1; while((eulerphi(m)*sigma(n))%(m+n), m++); m
vector(100, n, a(n)) \\ Derek Orr, Sep 29 2014

Crossrefs

Cf. A000010, A000203, A248004, A248007, A248008, A248030.

Sequence in context: A193279 A076889 A134689 * A117552 A294886 A357698

Adjacent sequences: A248026 A248027 A248028 * A248030 A248031 A248032

Keyword

nonn

Author

Zhi-Wei Sun, Sep 29 2014

Status

approved