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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A326981 Total number of composite parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 22, 31, 51, 70, 105, 145, 210, 283, 398, 530, 726, 958, 1283, 1673, 2212, 2854, 3714, 4756, 6119, 7764, 9893, 12457, 15728, 19674, 24636, 30615, 38079, 47034, 58109, 71396, 87692, 107179, 130943, 159278, 193619, 234486, 283720
Offset: 0

Views

Author

Omar E. Pol, Aug 09 2019

Keywords

Examples

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions               composite
of 6                       parts
--------------------------------------
6 .......................... 1
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 0
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 1
2 + 2 + 1 + 1 .............. 0
3 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 ...... 0
------------------------------------
Total ...................... 3
So a(6) = 3.

Crossrefs

Cf. A000041, A002808, A006128, A037032, A144115, A144116, A144119, A326957, A326982.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), 0, p[1])])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0 || i==1, {1, 0}, b[n, i-1] + # + {0, If[PrimeQ[i], 0, #[[1]]]}&[b[n-i, Min[n-i, i]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Formula

a(n) = A144119(n) - A000070(n-1), n >= 1.
a(n) = A006128(n) - A326957(n).

A363241 Number of partitions of n with prime rank.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 3, 6, 6, 10, 12, 19, 22, 33, 38, 54, 65, 91, 106, 145, 173, 228, 274, 356, 424, 545, 652, 823, 986, 1232, 1468, 1822, 2172, 2665, 3173, 3869, 4590, 5568, 6591, 7938, 9386, 11249, 13256, 15821, 18608, 22100, 25941, 30695, 35933, 42373, 49501, 58160, 67814, 79434, 92396, 107932
Offset: 1

Views

Author

Seiichi Manyama, May 23 2023

Keywords

Examples

			a(6) = 3 counts these partitions: 6, 5+1, 4+2.

Links

Crossrefs

Cf. A000041, A010051, A037032, A063995.

Programs

  • Maple
    b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(isprime(i-c), 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..56);  # Alois P. Heinz, May 23 2023
  • Mathematica
    b[n_, i_, c_] := b[n, i, c] = If[i > n, 0, If[i == n, If[i-c > 0 && PrimeQ[i-c], 1, 0], b[n-i, i, c+1] + b[n, i+1, c]]];
a[n_] := b[n, 1, 1];
Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Dec 20 2024, after Alois P. Heinz *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*sum(j=1, N, isprime(j)*x^(k*j)))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * Sum_{p prime} x^(k*p).
a(n) = Sum_{p prime} A063995(n,p). - Alois P. Heinz, Dec 20 2024

A281611 Expansion of Sum_{p prime, i>=2} x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 7, 10, 16, 23, 36, 50, 73, 100, 144, 193, 267, 355, 481, 631, 838, 1088, 1426, 1833, 2368, 3019, 3861, 4879, 6178, 7751, 9737, 12131, 15120, 18721, 23181, 28535, 35110, 42991, 52606, 64090, 78015, 94609, 114621, 138398, 166927, 200737, 241131, 288864, 345649, 412592, 491931
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

Comments

Total number of proper prime power parts (A246547) in all partitions of n.

Examples

			a(6) = 2 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 0 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 2.

Links

Crossrefs

Cf. A000041, A037032, A073335, A246547.

Programs

  • Mathematica
    nmax = 52; Rest[CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{p prime, i>=2} x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).
a(n) = A073335(n) - A037032(n).

A281612 Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 8, 12, 20, 28, 45, 62, 92, 127, 181, 244, 340, 452, 614, 809, 1077, 1401, 1841, 2371, 3071, 3923, 5026, 6363, 8078, 10149, 12769, 15939, 19899, 24676, 30604, 37726, 46489, 57007, 69849, 85211, 103871, 126119, 152987, 184955, 223349, 268898, 323384, 387830, 464587, 555168, 662619, 789084
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 25 2017

Keywords

Comments

Total number of semiprime parts (A001358) in all partitions of n.
Convolution of A000041 and A086971.

Examples

			a(6) = 3 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 1 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 3.

Links

Crossrefs

Cf. A000041, A001358, A037032, A073335, A086971, A281573.

Programs

  • Mathematica
    nmax = 53; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]

Formula

G.f.: Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).
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