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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.

%I A352834 #15 May 06 2022 07:34:00
%S A352834 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,
%T A352834 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,
%U A352834 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
%N 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.
%C A352834 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.
%C A352834 Record values are a(1) = 1, a(2) = 4, a(6) = a(9) = 12, a(12) = 20,
%C A352834   a(20) = a(36) = 24, a(42) = a(66) = 40, a(70) = a(90) = a(110) = a(120) =
%C A352834   a(126) = a(130) = a(170) = a(190) = a(198) = 48, a(210) = a(330) = a(390) = 64,
%C A352834   a(420) = a(660) = a(780) = a(900) = a(1020) = 96,
%C A352834   a(1050) = a(1134) = 120, a(1470) = a(1680) = a(1890) = 144,
%C A352834   a(2310) = a(2730) = a(3150) = a(3570) = a(3990) = a(4290) = 192,
%C A352834   a(4320) = 210, a(6300) = 216, a(7560) = 240, a(9240) = a(10920) = 288,
%C A352834   a(13860) = a(16380) = a(17820) = a(20020) = 336, a(20790) = 360,
%C A352834   a(23760) = a(28080) = 420, a(34650) = a(40950) = 432,
%C A352834   a(41580) = a(49140) = 480, a(60060) = a(78540) = a(80850) = a(87780) = 576,
%C A352834   a(90090) = 672, ...
%C A352834 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, ...
%F A352834 a(n) = A091895(n)/n; a(n) = 0 iff n is in A091896.
%F A352834 Conjecture: a(n) = O(n^(1/3)).
%o A352834 (PARI) apply( {A352834(n,L=n^2*2)=forstep(k=n,L,n,denominator(numdiv(k)/k)==n&&return(k/n))}, [1..99])
%Y A352834 Cf. A000005 (number-of-divisors function), A090395 (denominator of d(n)/n), A091895 (a(n)*n), A091896 (indices of zeros of a(n)).
%K A352834 nonn
%O A352834 1,2
%A A352834 _M. F. Hasler_, Apr 04 2022