A248590 -id:A248590 - 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

A248593 Least positive integer m such that m + n divides F(m), where F(m) is the m-th Fibonacci number given by A000045.

Original entry on oeis.org

10, 6, 84, 12, 16, 7, 27, 9, 144, 30, 28, 12, 8, 30, 14, 18, 57, 19, 342, 18, 20, 24, 66, 12, 9, 27, 144, 60, 112, 35, 16, 24, 60, 55, 20, 12, 40, 111, 24, 36, 88, 72, 80, 48, 10, 15, 72, 24, 224, 18, 50, 54, 270, 72, 54, 33, 224, 18, 28, 12

Offset: 1

Zhi-Wei Sun, Oct 09 2014

nonn

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n*(n-1) except for n = 1, 2, 3, 9.
In contrast, it is easy to show that for any integer n > 0, there is a positive integer m such that m + n divides 2^m - 1.
a(n) exists for any n > 0. See Bloom (1998). - Amiram Eldar, Jan 15 2022

			a(1) = 10 since 10 + 1 = 11 divides F(10) = 55.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
David M. Bloom, Offset Entries, Solution to Problem B-830, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 36, No. 1 (1998), pp. 89-90.

Cf. A000045, A247937, A247940, A248588, A248590.

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

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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