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 1-10 of 12 results. Next

A282062 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^2.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 6, 8, 6, 7, 6, 7, 6, 9, 6, 10, 8, 7, 4, 10, 6, 9, 8, 10, 6, 12, 6, 13, 10, 13, 8, 14, 4, 11, 8, 12, 6, 12, 6, 12, 10, 11, 4, 16, 6, 15, 8, 12, 4, 17, 6, 14, 8, 11, 4, 16, 6, 13, 8, 13, 6, 18, 4, 16, 10, 14, 4, 20, 6, 15, 12, 14, 6, 18, 4, 18, 8, 13, 8, 22, 6, 17, 8, 14, 6, 24, 8, 16, 6, 13, 4
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 05 2017

Keywords

Comments

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

Examples

			a(8) = 5 because we have  [7, 1], [5, 3], [4, 4], [3, 5] and [1, 7].

Links

Crossrefs

Cf. A000961, A071330, A071331, A073610, A095840, A280242, A282064.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
P:= select(isprime, [$2..N]):
g:= x + add(add(x^(p^k),k=1..floor(log[p](N))),p=P):
S:= series(g^2,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Feb 10 2017
  • Mathematica
    nmax = 95; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]

Formula

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

A341112 Number of partitions of n into 3 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 8, 8, 10, 9, 12, 10, 13, 12, 15, 13, 17, 15, 18, 15, 19, 16, 21, 17, 23, 18, 24, 19, 27, 23, 30, 24, 32, 25, 32, 26, 34, 26, 36, 26, 36, 28, 38, 28, 40, 30, 42, 32, 43, 30, 45, 32, 47, 35, 49, 30, 50, 35, 51, 36, 53, 35, 55, 37, 54, 40, 57, 36, 61, 40, 61
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A071330, A282064, A307727, A307825, A341122, A341123, A341124, A341125, A341126, A341127.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=3..75);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 3];
Table[a[n], {n, 3, 75}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341140 Number of partitions of n into 3 distinct prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 6, 8, 7, 8, 8, 10, 10, 12, 11, 12, 12, 13, 12, 16, 15, 15, 16, 18, 17, 19, 20, 21, 24, 22, 22, 23, 25, 22, 27, 26, 25, 26, 29, 25, 31, 27, 30, 31, 34, 26, 34, 31, 35, 32, 38, 29, 40, 32, 36, 34, 41, 29, 44, 35, 41, 36, 47, 34, 51, 38, 45, 41, 54
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282064, A307727, A307825, A341112, A341132, A341141, A341142, A341143, A341144, A341145.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=6..77);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    q[n_] := q[n] = PrimeNu[n] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 3];
Table[a[n], {n, 6, 77}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)

A341133 Number of ways to write n as an ordered sum of 4 prime powers (including 1).

Original entry on oeis.org

1, 4, 10, 20, 35, 52, 72, 96, 125, 156, 196, 236, 277, 316, 362, 400, 451, 496, 554, 604, 668, 704, 770, 808, 871, 920, 1014, 1040, 1131, 1172, 1266, 1308, 1449, 1484, 1638, 1672, 1802, 1820, 1992, 1964, 2167, 2172, 2332, 2296, 2534, 2444, 2698, 2648, 2889, 2820, 3140
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341122, A341134, A341135, A341136, A341137, A341138, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..54);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 54; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

A341134 Number of ways to write n as an ordered sum of 5 prime powers (including 1).

Original entry on oeis.org

1, 5, 15, 35, 70, 121, 190, 280, 395, 535, 711, 920, 1160, 1425, 1725, 2041, 2395, 2775, 3200, 3645, 4146, 4640, 5190, 5730, 6325, 6915, 7625, 8270, 9030, 9745, 10576, 11320, 12320, 13185, 14305, 15281, 16510, 17480, 18855, 19835, 21306, 22435, 24010, 25025, 26810, 27790, 29590
Offset: 5

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341123, A341133, A341135, A341136, A341137, A341138, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..51);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 51; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

A341135 Number of ways to write n as an ordered sum of 6 prime powers (including 1).

Original entry on oeis.org

1, 6, 21, 56, 126, 246, 432, 702, 1077, 1576, 2232, 3072, 4118, 5382, 6891, 8638, 10653, 12948, 15563, 18486, 21783, 25398, 29394, 33708, 38422, 43452, 49008, 54888, 61308, 68076, 75434, 83034, 91473, 100248, 109947, 120018, 131191, 142458, 155049, 167622, 181629, 195660
Offset: 6

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341124, A341133, A341134, A341136, A341137, A341138, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..47);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 47; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

A341138 Number of ways to write n as an ordered sum of 9 prime powers (including 1).

Original entry on oeis.org

1, 9, 45, 165, 495, 1278, 2931, 6111, 11790, 21331, 36594, 60057, 94938, 145296, 216153, 313524, 444483, 617229, 841225, 1127187, 1487322, 1935252, 2486124, 3156408, 3964218, 4928841, 6071472, 7414669, 8983179, 10802970, 12902661, 15311277, 18061092, 21185103, 24720552
Offset: 9

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Links

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341127, A341133, A341134, A341135, A341136, A341137, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 9):
seq(a(n), n=9..43);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 43; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

A341136 Number of ways to write n as an ordered sum of 7 prime powers (including 1).

Original entry on oeis.org

1, 7, 28, 84, 210, 455, 882, 1569, 2611, 4116, 6223, 9093, 12901, 17822, 24053, 31759, 41132, 52367, 65702, 81326, 99526, 120471, 144417, 171493, 201985, 235963, 273889, 315805, 362181, 413021, 468888, 529466, 595770, 667541, 745899, 830473, 922866, 1021832, 1129499
Offset: 7

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341125, A341133, A341134, A341135, A341137, A341138, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..45);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

A341137 Number of ways to write n as an ordered sum of 8 prime powers (including 1).

Original entry on oeis.org

1, 8, 36, 120, 330, 784, 1660, 3208, 5763, 9752, 15724, 24368, 36520, 53152, 75392, 104464, 141717, 188624, 246836, 318088, 404356, 507656, 630172, 774048, 941685, 1135304, 1357652, 1611240, 1899138, 2224016, 2589352, 2997544, 3452619, 3957480, 4516912, 5134096, 5815338
Offset: 8

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341126, A341133, A341134, A341135, A341136, A341138, A341139.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..44);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 44; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

A341139 Number of ways to write n as an ordered sum of 10 prime powers (including 1).

Original entry on oeis.org

1, 10, 55, 220, 715, 1992, 4915, 10990, 22660, 43660, 79463, 137830, 229460, 368710, 574410, 870644, 1287545, 1862110, 2639135, 3672050, 5024035, 6768950, 8992340, 11792070, 15279450, 19579514, 24832415, 31193900, 38837085, 47952400, 58750125, 71458860, 86328885
Offset: 10

Views

Author

Ilya Gutkovskiy, Feb 05 2021

Keywords

Crossrefs

Cf. A000961, A010055, A282062, A282064, A341131, A341133, A341134, A341135, A341136, A341137, A341138.

Programs

  • Maple
    q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 10):
seq(a(n), n=10..42);  # Alois P. Heinz, Feb 05 2021
  • Mathematica
    nmax = 42; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^10, {x, 0, nmax}], x] // Drop[#, 10] &
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