|
%I #18 Feb 23 2024 13:42:21
%S 1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,4,1,1,1,2,1,1,1,2,1,2,1,2,1,1,
%T 2,2,1,1,1,4,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,1,1,4,1,1,1,2,1,1,1,2,
%U 1,3,1,2,1,1,1,2,1,1,1,4,1,1,1,3,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,4,1,1,1,2,2,1,1,2,1,2,1,3,1,1,1,2,1,1,1,5
%N Number of tetrahedral numbers dividing n.
%C Inverse Möbius transform of A023533. - _Antti Karttunen_, Oct 01 2018
%C Records are a(1) = 1, a(4) = 2, a(20) = 4, a(120) = 5, a(280) = 6, a(560) = 7, a(840) = 8, a(1680) = 9, a(9240) = 11, a(18480) = 12, a(55440) = 13, a(120120) = 14, a(240240) = 15, a(314160) = 16, a(628320) = 17, a(1441440) = 18, a(2282280) = 19, a(4564560) = 21, a(9129120) = 22, a(13693680) = 23, a(27387360) = 24, a(54774720) = 25, a(68468400) = 26, a(77597520) = 27, a(136936800) = 28, a(155195040) = 29, a(310390080) = 30, a(465585120) = 31, a(775975200) = 32, a(1163962800) = 33, a(2327925600) = 36, a(4655851200) = 37, a(13967553600) = 38, a(16295479200) = 40. - _Charles R Greathouse IV_, Dec 19 2016
%H Antti Karttunen, <a href="/A279495/b279495.txt">Table of n, a(n) for n = 1..65537</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>.
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F G.f.: Sum_{k>=1} x^(k*(k+1)*(k+2)/6)/(1 - x^(k*(k+1)*(k+2)/6)).
%F a(n) = Sum_{d|n} A023533(d). - _Antti Karttunen_, Oct 01 2018
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - _Amiram Eldar_, Jan 02 2024
%e a(10) = 2 because 10 has 4 divisors {1,2,5,10} among which 2 divisors {1,10} are tetrahedral numbers.
%t Table[SeriesCoefficient[Sum[x^(k (k + 1) (k + 2)/6)/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n}], {n, 1, 120}]
%o (PARI) a(n)=sum(k=1,sqrtnint(6*n,3),n%(k*(k+1)*(k+2)/6)==0) \\ _Charles R Greathouse IV_, Dec 13 2016
%o (PARI) isA000292(n)=my(k=sqrtnint(6*n,3)); k*(k+1)*(k+2)==6*n
%o a(n)=sumdiv(n,d,isA000292(d)) \\ _Charles R Greathouse IV_, Dec 13 2016
%Y Cf. A000292, A007862, A023533, A061704.
%K nonn,easy
%O 1,4
%A _Ilya Gutkovskiy_, Dec 13 2016
|
