A352834 Least k > 0 such that denominator( d(k*n)/(k*n) ) = n, or 0 if no such k exists, where d = A000005 is the number-of-divisors function.

1, 4, 1, 1, 1, 12, 1, 10, 12, 2, 1, 20, 1, 2, 1, 1, 1, 0, 1, 24, 1, 2, 1, 2, 1, 2, 1, 20, 1, 0, 1, 14, 1, 2, 1, 24, 1, 2, 1, 4, 1, 40, 1, 20, 12, 2, 1, 3, 1, 4, 1, 20, 1, 18, 1, 2, 1, 2, 1, 32, 1, 2, 12, 1, 1, 40, 1, 20, 1, 48, 1, 0, 1, 2, 36, 20, 1, 40, 1, 5, 1, 2, 1, 4, 1, 2, 1, 2, 1, 48

Offset: 1

Author

M. F. Hasler, Apr 04 2022

Keywords

• nonn

Comments

This sequence is motivated by the fact that A091895(n) is always a multiple of n, so we list here the ratio A091895(n)/n.

Record values are a(1) = 1, a(2) = 4, a(6) = a(9) = 12, a(12) = 20,

a(20) = a(36) = 24, a(42) = a(66) = 40, a(70) = a(90) = a(110) = a(120) =

a(126) = a(130) = a(170) = a(190) = a(198) = 48, a(210) = a(330) = a(390) = 64,

a(420) = a(660) = a(780) = a(900) = a(1020) = 96,

a(1050) = a(1134) = 120, a(1470) = a(1680) = a(1890) = 144,

a(2310) = a(2730) = a(3150) = a(3570) = a(3990) = a(4290) = 192,

a(4320) = 210, a(6300) = 216, a(7560) = 240, a(9240) = a(10920) = 288,

a(13860) = a(16380) = a(17820) = a(20020) = 336, a(20790) = 360,

a(23760) = a(28080) = 420, a(34650) = a(40950) = 432,

a(41580) = a(49140) = 480, a(60060) = a(78540) = a(80850) = a(87780) = 576,

a(90090) = 672, ...

Up to n = 10^6, the terms are bounded by a(n) < 16*n^(1/3). The largest ratios r(n) := a(n)/n^(1/3) are r(2310) ~ 14.5, r(23760) ~ 14.6, r(60060) ~ 14.7, r(90090) ~ 14.99, r(154440) ~ 15.66, r(201960) = 14.3, r(270270) = 14.85, r(420420) = 14.4, r(510510) = 14.4, r(720720) = 14.05, ...

Crossrefs

Cf. A000005 (number-of-divisors function), A090395 (denominator of d(n)/n), A091895 (a(n)*n), A091896 (indices of zeros of a(n)).

Programs

PARI

apply( {A352834(n,L=n^2*2)=forstep(k=n,L,n,denominator(numdiv(k)/k)==n&&return(k/n))}, [1..99])

Formula

a(n) = A091895(n)/n; a(n) = 0 iff n is in A091896.

Conjecture: a(n) = O(n^(1/3)).