A248052 -id:A248052 - cp's OEIS frontend

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.

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

Offset: 1

Zhi-Wei Sun, Sep 30 2014

nonn

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

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

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.

Cf. A000203, A248008, A248036, A248052.

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}]

A248354 Least positive integer m such that m + n divides prime(m^2) + prime(n^2).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 05 2014

nonn

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

See also the comments in A248052.

			a(3) = 2 since 2 + 3 = 5 divides prime(2^2) + prime(3^2) = 7 + 23 = 30.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A011757, A247824, A247975, A248052.

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

a(n)=my(N=prime(n^2),m); while((prime(m++^2)+N)%(m+n), ); m \\ Charles R Greathouse IV, Oct 05 2014

cp's OEIS Frontend

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.

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