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

1,2

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, ...

Links

Table of n, a(n) for n=1..90.

Formula

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

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

Prog

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

Crossrefs

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

Sequence in context: A222317 A063851 A124777 * A203639 A358319 A265679

Adjacent sequences: A352831 A352832 A352833 * A352835 A352836 A352837

Keyword

nonn

Author

M. F. Hasler, Apr 04 2022

Status

approved