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.

Showing 1-3 of 3 results.

A282064 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^3.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 15, 18, 22, 27, 33, 37, 45, 48, 52, 54, 60, 60, 69, 69, 79, 81, 87, 79, 93, 87, 97, 99, 114, 99, 120, 111, 130, 126, 150, 135, 168, 141, 160, 147, 177, 144, 189, 156, 183, 162, 201, 157, 213, 171, 214, 189, 231, 168, 237, 189, 244, 201, 261, 177, 270, 201, 261, 210, 282, 192, 297, 216, 283, 228
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2017

Keywords

Comments

Number of ways to write n as an ordered sum of three prime powers (1 included).

Examples

			a(6) = 10 because we have  [4, 1, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 2, 2], [2, 1, 3], [1, 4, 1], [1, 3, 2], [1, 2, 3] and [1, 1, 4].

Links

Crossrefs

Cf. A000961, A098238, A280243, A282062.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^3, {x, 0, nmax}], x]

Formula

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

A307727 Number of partitions of n into 3 prime powers (not including 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 8, 7, 9, 9, 10, 10, 12, 11, 14, 13, 14, 13, 16, 13, 18, 15, 18, 16, 20, 18, 23, 20, 25, 23, 26, 22, 28, 23, 30, 23, 30, 23, 32, 26, 32, 27, 34, 28, 37, 28, 36, 29, 40, 31, 43, 28, 42, 32, 44, 32, 46, 32, 46, 35, 46, 35, 50, 34, 51, 37, 53, 36, 59, 36, 57, 41
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2019

Keywords

Examples

			a(11) = 4 because we have [7, 2, 2], [5, 4, 2], [5, 3, 3] and [4, 4, 3].

Links

Crossrefs

Cf. A000961, A023894, A068307, A246655, A280243, A307726.

Programs

  • Maple
    f:= proc(n,k,pmax) option remember;
  local t,p,j;
  if n = 0 then return `if`(k=0, 1, 0) fi;
  if k = 0 then return 0 fi;
  if n > k*pmax then return 0 fi;
  t:= 0:
  for p in A246655 do
    if p > pmax then return t fi;
    t:= t + add(procname(n-j*p, k-j, min(p-1,n-j*p)),j=1..min(k,floor(n/p)))
  od;
  t
end proc:
seq(f(n,3,n),n=0..80) # Robert Israel, Apr 25 2019
  • Mathematica
    Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, PrimePowerQ] &)] &, 81, 0]

Formula

a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^A246655(k)).
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [omega(i) * omega(j) * omega(n-i-j) == 1], where omega(n) is the number of distinct prime factors of n and [==] is the Iverson bracket. - Wesley Ivan Hurt, Apr 25 2019

A282289 Expansion of (Sum_{p prime, k>=2} x^(p^k))^4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 4, 0, 0, 6, 12, 6, 0, 8, 12, 12, 4, 13, 16, 6, 4, 13, 28, 12, 4, 10, 24, 24, 16, 28, 24, 24, 24, 42, 52, 18, 28, 32, 60, 40, 24, 44, 28, 42, 28, 60, 52, 18, 24, 37, 84, 54, 48, 42, 60, 78, 48, 72, 44, 60, 52, 68, 96, 36, 40, 22, 72, 72, 52, 76, 52, 66, 36, 88, 88, 64, 56
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 11 2017

Keywords

Comments

Number of ways to write n as an ordered sum of 4 proper prime powers (A246547).
Conjecture: a(n) > 0 for all n > 27.

Examples

			a(28) = 8 because we have [16, 4, 4, 4], [8, 8, 8, 4], [8, 8, 4, 8], [8, 4, 8, 8], [4, 16, 4, 4], [4, 8, 8, 8], [4, 4, 16, 4] and [4, 4, 4, 16].

Links

Crossrefs

Cf. A246547, A280242, A280243, A282062, A282064.

Programs

  • Mathematica
    nmax = 91; CoefficientList[Series[Sum[Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}]^4, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{p prime, k>=2} x^(p^k))^4.
Showing 1-3 of 3 results.

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