A248143 -id:A248143

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A248144

Least positive integer m such that m + n divides p(m*n), where p(.) is the partition function given by A000041.

+10
3

4, 5, 3, 1, 2, 14, 2, 10, 1, 4, 17, 5, 9, 1, 1, 6, 4, 10, 16, 357, 1, 197, 14, 1, 3, 9, 6, 1123, 15, 93, 4, 1, 8, 46, 77, 99, 18, 53, 10, 76, 4, 2, 15, 152, 4, 3, 10, 29, 6, 12, 4, 1, 25, 1, 252, 64, 106, 11, 11, 136

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..496

EXAMPLE

a(6) = 14 since 6 + 14 = 20 divides p(6*14) = 26543660.

MATHEMATICA

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

CROSSREFS

Cf. A000041, A248004, A248008, A248143, A248175.

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 02 2014

STATUS

approved

A248175

Least positive integer m such that m + n divides q(m*n), where q(.) is the strict partition function given by A000009.

+10
2

11, 4, 9, 2, 12, 10, 9, 16, 3, 6, 1, 5, 2, 18, 7, 8, 5, 14, 11, 36, 2, 34, 4, 8, 31, 6, 15, 36, 23, 2, 9, 14, 17, 22, 11, 18, 1, 22, 11, 7, 1, 22, 12, 7, 55, 7, 19, 40, 15, 6, 31, 12, 43, 10, 25, 40, 7, 91, 61, 20

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

(ii) For each n > 0, there is a positive integer m such that m + n divides q(m) + q(n).

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..405

EXAMPLE

a(3) = 9 since 9 + 3 = 12 divides q(9*3) = 192 = 12*16.

MATHEMATICA

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