A248030 -id:A248030 - cp's OEIS frontend

A248588 Least positive integer m such that m + n divides sigma(m), where sigma(m) is the sum of all positive divisors of m.

2, 12, 4, 9, 40, 6, 8, 10, 15, 14, 21, 112, 27, 22, 16, 12, 39, 289, 65, 34, 18, 20, 57, 60, 95, 46, 69, 28, 115, 96, 32, 58, 45, 62, 93, 24, 155, 340, 217, 44, 63, 30, 50, 82, 123, 52, 129, 204, 75, 40, 141, 228, 235, 42, 36, 106, 99, 68, 265, 120

Offset: 1

Zhi-Wei Sun, Oct 09 2014

nonn

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

			a(5) = 40 since 40 + 5 = 45 divides sigma(40) =  90.
a(1162) = 24031232 since 24031232 + 1162 = 24032394 divides sigma(24031232) =  48064788 = 2*24032394.

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

Cf. A000203, A248007, A248008, A248029, A248030, A248568, A248590.

Do[m=1;Label[aa];If[Mod[DivisorSigma[1,m],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
lpi[n_]:=Module[{m=1},While[!Divisible[DivisorSigma[1,m],m+n],m++];m]; Array[lpi,60] (* Harvey P. Dale, Feb 21 2020 *)

a(n) = my(m = 1); while(sigma(m) % (m+n), m++); m; \\ Michel Marcus, Aug 08 2017

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

Original entry on oeis.org

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

Zhi-Wei Sun, Sep 29 2014

nonn

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

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

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.

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

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}]

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

cp's OEIS Frontend

A248588 Least positive integer m such that m + n divides sigma(m), where sigma(m) is the sum of all positive divisors of m.

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