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

Author

Ilya Gutkovskiy, Dec 13 2016

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

Cf. A000292, A007862, A023533, A061704.

Programs

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}]

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

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