cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A341774 Number of partitions of n into 3 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A063993, A068980, A341775, A341776, A341777, A341778.

A341775 Number of partitions of n into 4 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 0, 2, 0, 1, 2, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 2, 2, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 2, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 0, 2, 1, 1
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319814, A341774, A341776, A341777, A341778.

A341776 Number of partitions of n into 5 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 1, 1, 0, 2, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 1, 1, 2, 1, 0, 3, 1, 1, 3, 2, 0, 2, 1, 0, 2, 2, 0, 3, 2, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 3, 1, 2, 3, 1, 1, 4, 1, 2, 3, 1, 0, 4, 2
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319815, A341774, A341775, A341777, A341778.

A341777 Number of partitions of n into 6 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 2, 1, 0, 2, 1, 0, 2, 2, 0, 2, 2, 0, 1, 3, 0, 2, 3, 1, 1, 3, 1, 1, 3, 1, 1, 4, 2, 1, 3, 2, 0, 3, 2, 1, 4, 3, 1, 3, 2, 1, 3, 3, 1, 4, 3, 2, 3, 3, 1, 3, 3, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 5, 2, 2, 5, 2, 2, 5, 2, 2, 6, 3, 3
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319816, A341774, A341775, A341776, A341778.

A341778 Number of partitions of n into 7 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 2, 1, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 4, 1, 2, 3, 1, 1, 4, 1, 2, 4, 2, 1, 4, 2, 1, 4, 3, 1, 5, 3, 2, 4, 3, 1, 4, 3, 2, 5, 4, 2, 4, 4, 2, 4, 4, 2, 5, 5, 3, 4, 5, 2, 4, 5, 3, 4, 7, 3, 4, 6, 3, 3, 7, 3, 4, 7, 4
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A024692, A068980, A319817, A341774, A341775, A341776, A341777.

A341794 Number of ways to write n as an ordered sum of 3 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 6, 0, 0, 3, 0, 0, 3, 3, 0, 3, 6, 0, 0, 3, 0, 1, 6, 0, 0, 6, 0, 0, 3, 0, 0, 9, 3, 0, 3, 3, 0, 6, 0, 0, 6, 3, 0, 0, 0, 0, 3, 6, 0, 3, 6, 1, 6, 0, 0, 3, 6, 0, 6, 0, 0, 6, 3, 0, 0, 3, 3, 3, 6, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 6, 3, 0, 9, 0, 0, 12
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Crossrefs

Cf. A000292, A023533, A023670, A053604, A282582, A341774, A341795, A341796, A341797.

Programs

  • Mathematica
    nmax = 95; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^3, {x, 0, nmax}], x] // Drop[#, 3] &

Formula

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^3.

A341796 Number of ways to write n as an ordered sum of 5 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 5, 0, 0, 10, 0, 0, 15, 0, 0, 25, 0, 0, 31, 0, 0, 30, 5, 0, 35, 20, 0, 30, 30, 0, 20, 40, 0, 20, 65, 0, 10, 65, 0, 5, 70, 10, 5, 90, 30, 0, 70, 30, 1, 85, 40, 0, 80, 60, 0, 50, 50, 0, 70, 90, 10, 50, 90, 20, 50, 80, 10, 60, 130, 20, 65, 70, 20, 65, 90, 30, 50, 110, 70, 65, 100
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Links

Crossrefs

Cf. A000292, A023533, A023670, A282172, A282582, A340950, A341776, A341794, A341795, A341797, A341806, A341807, A341808, A341809.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 100);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^5 )); // G. C. Greubel, Jul 20 2022
  • Mathematica
    nmax = 82; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &
  • SageMath
    def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(5, x) ).list()
a=A341796_list(120); a[5:100] # G. C. Greubel, Jul 20 2022

Formula

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^5.

A341797 Number of ways to write n as an ordered sum of 6 nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 15, 0, 0, 26, 0, 0, 45, 0, 0, 66, 0, 0, 76, 6, 0, 90, 30, 0, 96, 60, 0, 80, 90, 0, 75, 150, 0, 60, 192, 0, 35, 210, 15, 30, 270, 60, 15, 270, 90, 6, 270, 120, 6, 306, 195, 0, 240, 210, 1, 246, 270, 20, 240, 360, 60, 180, 330, 60, 216, 450, 80, 210, 435, 120, 216, 360
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

Links

Crossrefs

Cf. A000292, A023533, A023670, A282582, A340951, A341777, A341794, A341795, A341796, A341797, A341806, A341807, A341808, A341809.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 80);
Coefficients(R!( (&+[x^Binomial(j+2,3): j in [1..20]])^6 )); // G. C. Greubel, Jul 20 2022
  • Mathematica
    nmax = 77; CoefficientList[Series[Sum[x^Binomial[k + 2, 3], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &
  • SageMath
    def f(m, x): return ( sum( x^(binomial(j+2,3)) for j in (1..20) ) )^m
def A341797_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(6, x) ).list()
a=A341797_list(100); a[6:81] # G. C. Greubel, Jul 20 2022

Formula

G.f.: ( Sum_{k>=1} x^binomial(k+2,3) )^6.

A145397 Numbers not of the form m*(m+1)*(m+2)/6, the non-tetrahedral numbers.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 14 2008

Keywords

Comments

Complement of A000292; A000040 is a subsequence.

Links

Crossrefs

Cf. A000040, A000292, A014306, A023533.

Programs

  • Magma
    [n: n in [1..100] | Binomial(Floor((6*n-1)^(1/3))+2, 3) ne n ]; // G. C. Greubel, Feb 20 2022
  • Mathematica
    Select[Range[100], Binomial[Floor[Surd[6*# -1, 3]] +2, 3] != # &] (* G. C. Greubel, Feb 20 2022 *)
  • PARI
    is(n)=binomial(sqrtnint(6*n,3)+2,3)!=n \\ Charles R Greathouse IV, Feb 22 2017
  • Python
    from itertools import count
from math import comb
from sympy import integer_nthroot
def A145397(n):
    def f(x): return n+next(i for i in count(integer_nthroot(6*x,3)[0],-1) if comb(i+2,3)<=x)
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    return iterfun(f,n) # Chai Wah Wu, Sep 09 2024
  • Python
    from math import comb
from sympy import integer_nthroot
def A145397(n): return n+(m:=integer_nthroot(6*n,3)[0])-(n+m<=comb(m+2,3)) # Chai Wah Wu, Oct 01 2024
  • Sage
    [n for n in (1..100) if binomial( floor( real_nth_root(6*n-1, 3) ) +2, 3) != n ] # G. C. Greubel, Feb 20 2022

Formula

A014306(a(n)) = 1; A023533(a(n)) = 0.
a(n) = n+m if 6(n+m)>m(m+1)(m+2) and a(n)=n+m-1 otherwise where m is floor((6n)^(1/3)). - Chai Wah Wu, Oct 01 2024

Extensions

Definition corrected by Ant King, Sep 20 2012

A279495 Number of tetrahedral numbers dividing n.

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 13 2016

Keywords

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

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
Previous Showing 41-50 of 62 results. Next

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