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A248036 Least positive integer m such that m + n divides sigma(m)^2 + sigma(n)^2, where sigma(k) denotes the number of positive divisors of k. 5
1, 3, 2, 1, 10, 6, 3, 50, 1, 5, 34, 28, 7, 6, 10, 18, 3, 16, 33, 5, 20, 14, 83, 24, 1, 10, 10, 12, 56, 6, 33, 2, 15, 11, 93, 13, 204, 27, 52, 38, 17, 6, 7, 6, 15, 14, 5, 944, 1, 8, 17, 39, 32, 33, 5, 24, 7, 59, 58, 15 (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..10000

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

EXAMPLE

a(5) = 10 since 10 + 5 = 15 divides sigma(10)^2 + sigma(5)^2 = 18^2 + 6^2 = 360.

a(1024) = 2098177 since 2098177 + 1024 = 2099201 divides sigma(2098177)^2 + sigma(1024)^2 = 2103300^2 + 2047^2 = 4423875080209 = 2099201*2107409.

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}]

lpi[n_]:=Module[{m=1, dsn=DivisorSigma[1, n]^2}, While[ !Divisible[ DivisorSigma[ 1, m]^2+ dsn, m+n], m++]; m]; Array[lpi, 60] (* Harvey P. Dale, May 07 2016 *)

CROSSREFS

Cf. A000203, A247975, A248008, A248035.

Sequence in context: A267278 A267019 A100100 * A307214 A185967 A188111

Adjacent sequences:  A248033 A248034 A248035 * A248037 A248038 A248039

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 29 2014

STATUS

approved

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)