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A352834
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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.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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, ...
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LINKS
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FORMULA
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Conjecture: a(n) = O(n^(1/3)).
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PROG
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(PARI) apply( {A352834(n, L=n^2*2)=forstep(k=n, L, n, denominator(numdiv(k)/k)==n&&return(k/n))}, [1..99])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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