A279495 Number of tetrahedral numbers dividing n.

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, 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, 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

Offset

1,4

Comments

Inverse Möbius transform of A023533. - Antti Karttunen, Oct 01 2018

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Eric Weisstein's World of Mathematics, Tetrahedral Number.

Index to sequences related to pyramidal numbers.

Formula

G.f.: Sum_{k>=1} x^(k*(k+1)*(k+2)/6)/(1 - x^(k*(k+1)*(k+2)/6)).

a(n) = Sum_{d|n} A023533(d). - Antti Karttunen, Oct 01 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Jan 02 2024

Example

a(10) = 2 because 10 has 4 divisors {1,2,5,10} among which 2 divisors {1,10} are tetrahedral numbers.

Mathematica

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}]

Prog

(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
(PARI) isA000292(n)=my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n
a(n)=sumdiv(n, d, isA000292(d)) \\ Charles R Greathouse IV, Dec 13 2016

Crossrefs

Cf. A000292, A007862, A023533, A061704.

Sequence in context: A056832 A105931 A349163 * A300409 A361631 A247462

Adjacent sequences: A279492 A279493 A279494 * A279496 A279497 A279498

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 13 2016

Status

approved