A248052 -id:A248052

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A248054

Least positive integer m such that m + n divides sigma(m^2) + sigma(n^2), where sigma(k) is the sum of all positive divisors of k.

+0
4

1, 3, 2, 7, 24, 34, 3, 81, 209, 16, 63, 25, 7, 20, 140, 10, 3, 10, 22, 2, 39, 4, 35, 5, 4, 2, 28, 27, 75, 41, 16, 78, 44, 6, 23, 14, 207, 59, 21, 84, 17, 78, 7, 3, 11725, 10, 5, 2, 1669, 361, 134, 10, 141, 310, 21, 73, 21, 33, 38, 121

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: a(n) exists for any n > 0.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3190

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

EXAMPLE

a(4) = 7 since 7 + 4 = 11 divides sigma(7^2) + sigma(4^2) = 57 + 31 = 88.

MATHEMATICA

Do[m=1; Label[aa]; If[Mod[DivisorSigma[1, m^2]+DivisorSigma[1, n^2], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

CROSSREFS

Cf. A000203, A248008, A248036, A248052.

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 30 2014

STATUS

approved

A248354

Least positive integer m such that m + n divides prime(m^2) + prime(n^2).

+0
1

1, 1, 2, 1, 3, 8, 2, 6, 6, 45, 9, 4, 15, 2, 13, 17, 4, 12, 9, 8, 11, 6, 101, 20, 2, 15, 7, 50, 4, 183, 48, 15, 9, 5, 4, 4, 157, 1, 123, 4, 13, 112, 76, 4, 7, 13, 44, 2, 16, 28, 83, 202, 114, 50, 85, 31, 14, 62, 19, 25

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1)/2 for all n > 1.