A248029 -id:A248029 - 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

A248030 Least positive integer m such that m + n divides sigma(m)*phi(n), where sigma(.) and phi(.) are given by A000203 and A000010.

Original entry on oeis.org

2, 12, 4, 2, 3, 6, 2, 10, 3, 2, 21, 8, 3, 22, 13, 8, 9, 6, 8, 12, 3, 8, 10, 4, 5, 10, 21, 8, 20, 26, 4, 8, 7, 14, 13, 12, 8, 4, 33, 8, 23, 6, 20, 12, 3, 16, 22, 72, 7, 10, 13, 4, 27, 42, 5, 24, 15, 26, 57, 18, 11, 38, 27, 20, 31, 4, 21, 36, 19, 2

Offset: 1

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: a(n) < 2*n for any n > 2.

Numbers n such that a(n) > n: 1, 2, 3, 8, 11, 14, 48, 227, 908, 4478, ... The next number, if it exists, is greater than 10^5. - Derek Orr, Sep 29 2014

			a(2) = 12 since 12 + 2 = 14 divides sigma(12)*phi(2) = 28.

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

Cf. A000010, A000203, A248004, A248007, A248008, A248029.

Do[m=1;Label[aa];If[Mod[DivisorSigma[1,m]*EulerPhi[n],m+n]==0,Print[n," ",m];Goto[bb]];m= m+1; Goto[aa];Label[bb];Continue,{n,1,70}]
lpim[n_]:=Module[{m=1,ephn=EulerPhi[n]},While[Mod[ephn*DivisorSigma[1,m],m+n]!=0, m++]; m]; Array[lpim,70] (* Harvey P. Dale, Feb 14 2024 *)

a(n)=m=1;while((eulerphi(n)*sigma(m))%(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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