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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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A379317 Positive integers with a unique even prime index.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Dec 29 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Crossrefs

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],EvenQ]]==1&]

A222656 Number T(n,k) of partitions of n using exactly k primes; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 3, 4, 3, 1, 3, 6, 4, 2, 5, 7, 6, 3, 1, 6, 9, 8, 5, 2, 8, 11, 12, 7, 3, 1, 8, 17, 14, 10, 5, 2, 12, 20, 19, 14, 8, 3, 1, 13, 26, 25, 19, 11, 5, 2, 17, 31, 35, 24, 16, 8, 3, 1, 19, 41, 42, 34, 21, 12, 5, 2, 26, 47, 56, 44, 29
Offset: 0

Views

Author

Alois P. Heinz, May 29 2013

Keywords

Examples

			T(6,0) = 3: [6], [4,1,1], [1,1,1,1,1,1].
T(6,1) = 4: [5,1], [4,2], [3,1,1,1], [2,1,1,1,1].
T(6,2) = 3: [3,3], [3,2,1], [2,2,1,1].
T(6,3) = 1: [2,2,2].
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2;
  2,  2,  1;
  2,  3,  2;
  3,  4,  3,  1;
  3,  6,  4,  2;
  5,  7,  6,  3, 1;
  6,  9,  8,  5, 2;
  8, 11, 12,  7, 3, 1;
  8, 17, 14, 10, 5, 2;
  ...

Links

Crossrefs

Column k=0 gives: A002095.
Row sums give: A000041.
Cf. A000040, A004526, A224344.

Programs

  • Maple
    b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1 then 0
      else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(isprime(i), 0$j, NULL), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n$2):
seq(T(n), n=0..16);
  • Mathematica
    zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[PadRight[x, m, z], PadRight[y, m, z]]]]; b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1}, i<1, {0}, True, pc = {}; For[j = 0, j <= n/i, j++, pc = zip[Plus, pc, Join[If[PrimeQ[i], Array[0&, j], {}], b[n-i*j, i-1]], 0]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Jan 29 2014, after Alois P. Heinz *)

Formula

Sum_{k=1..floor(n/2)} k * T(n,k) = A037032(n).
G.f.: G(t,x) = Product_{i>=1} (1 - x^prime(i))/((1 - x^i)*(1 - t*x^prime(i))). - Emeric Deutsch, Nov 11 2015

A321378 Number of integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 3, 2, 3, 0, 6, 1, 5, 3, 6, 1, 11, 2, 9, 6, 12, 5, 19, 4, 17, 11, 23, 9, 32, 10, 31, 22, 39, 17, 55, 21, 57, 37, 67, 33, 92, 44, 97, 65, 114, 63, 154, 78, 162, 113, 191, 117, 250, 138, 269, 194, 320
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Examples

			The a(30) = 11 integer partitions:
  (30)
  (24,6)
  (15,15)
  (18,12)
  (20,10)
  (18,6,6)
  (12,12,6)
  (14,10,6)
  (10,10,10)
  (12,6,6,6)
  (6,6,6,6,6)

Crossrefs

Cf. A000607, A000688, A000961, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321347, A321665, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1/(1-x^n)],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A321347 Number of strict integer partitions of n containing no prime powers (including 1).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 4, 4, 2, 3, 4, 4, 5, 6, 5, 6, 7, 7, 9, 10, 10, 13, 12, 11, 15, 17, 16, 19, 20, 20, 25, 28, 26, 30, 33, 35, 41, 43, 42, 50, 55, 57, 64, 67, 67, 79, 86, 87, 97, 105, 109, 124, 131, 135, 151, 163, 169
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

First differs from A286221 at a(30) = 6, A286221(30) = 5.

Examples

			The a(36) = 13 strict integer partitions:
  (36),
  (21,15), (22,14), (24,12), (26,10), (30,6), (35,1),
  (14,12,10), (18,12,6), (20,10,6), (20,15,1), (21,14,1),
  (15,14,6,1).

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321378, A321665, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A321665 Number of strict integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 2, 2, 0, 3, 1, 3, 2, 4, 1, 5, 2, 5, 4, 6, 4, 9, 3, 8, 7, 10, 6, 13, 7, 13, 12, 16, 10, 20, 13, 22, 19, 24, 18, 32, 23, 34, 30, 37, 30, 49, 37, 50, 47, 58, 51, 73, 58, 77, 74, 89, 80, 108, 91, 116
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Examples

			The a(36) = 9 strict integer partitions:
  (36)
  (30,6)
  (21,15)
  (22,14)
  (24,12)
  (26,10)
  (18,12,6)
  (20,10,6)
  (14,12,10)

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A321346, A321347, A321378, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

Formula

G.f.: Product_{k>=2, k not a prime power} 1 + x^k. - Joerg Arndt, Dec 22 2020

A002096 Mixed partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 9, 14, 20, 29, 42, 58, 79, 108, 145, 191, 252, 329, 427, 549, 704, 894, 1136, 1427, 1793, 2237, 2789, 3450, 4268, 5248, 6447, 7880, 9619, 11691, 14199, 17166, 20739, 24966, 30020, 35976, 43076, 51420, 61320, 72927, 86642, 102682
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The number of partitions of n with at least one prime part and at least one nonprime part. - Sean A. Irvine, Mar 23 2016

References

  • L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Formula

a(n) = Sum_{k=1..n-1} A000607(n - k) * A002095(k). - Sean A. Irvine, Mar 23 2016

Extensions

More terms from Sean A. Irvine, Mar 23 2016

A155515 Number of partitions of n into as many primes as nonprimes.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 3, 2, 3, 5, 4, 8, 12, 10, 15, 23, 22, 33, 42, 47, 64, 79, 90, 122, 147, 169, 219, 264, 312, 387, 465, 546, 679, 799, 950, 1151, 1365, 1599, 1937, 2270, 2678, 3181, 3735, 4374, 5192, 6046, 7082, 8318, 9684, 11281, 13208, 15313, 17798, 20702, 23951
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 23 2009

Keywords

Examples

			a(9) = #{6+3, 5+4, 5+2+1+1, 4+2+2+1, 2+2+2+1+1+1} = 5;
a(10) = #{8+2, 5+3+1+1, 4+3+2+1, 3+2+2+1+1+1} = 4.

Links

Crossrefs

Cf. A000041, A000607, A002095, A027187, A355306.

Programs

  • Maple
    b:= proc(n, i, t) local m; m:= n- `if`(t>0, t, -2*t); if m<0 then 0 elif n=0 then 1 elif i<3 then `if`(irem(m,3)=0, 1, 0) else b(n, i, t):= b(n-i, i, t+ `if`(isprime(i), 1, -1)) +b(n, i-1, t) fi end: a:= n-> b(n, n, 0): seq(a(n), n=0..60);  # Alois P. Heinz, Apr 30 2009
  • Mathematica
    pnpQ[n_]:=Count[n,?PrimeQ]==Length[n]/2; Table[Count[ IntegerPartitions[ n], ?pnpQ],{n,60}] (* Harvey P. Dale, Feb 02 2014 *)
b[n_, i_, t_] := b[n, i, t] = Module[{m}, m = n - If[t > 0, t, -2t]; Which[m < 0, 0, n == 0, 1, i < 3, If[Mod[m, 3] == 0, 1, 0], True, b[n, i, t] = b[n-i, i, t + If[PrimeQ[i], 1, -1]] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)
  • PARI
    parts(n)={1/(prod(k=1, n, 1 - if(isprime(k), y, 1/y)*x^k + O(x*x^n)))}
{my(n=60); apply(p->polcoeff(p,0), Vec(parts(n)))} \\ Andrew Howroyd, Dec 29 2017
  • Python
    from sympy import isprime
from sympy.utilities.iterables import partitions
def c(p): return 2*sum(p[i] for i in p if isprime(i)) == sum(p.values())
def a(n): return sum(1 for p in partitions(n) if c(p))
print([a(n) for n in range(55)]) # Michael S. Branicky, Jun 30 2022

Formula

a(n) = A000041(n) - A355306(n). - Omar E. Pol, Jun 30 2022

A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985
Offset: 0

Views

Author

Gus Wiseman, May 16 2022

Keywords

Examples

			The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Links

Crossrefs

The first condition only is A023360, partitions A000607.
For partitions we have A351982, only run-lens A100405, only parts A008483.
The second condition only is A353401, partitions A055923.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A052284 counts compositions into nonprimes, partitions A002095.
A106356 counts compositions by number of adjacent equal parts.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A329738 counts uniform compositions, partitions A047966.
Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

Programs

  • Maple
    b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,15}]

Extensions

a(26)-a(56) from Alois P. Heinz, May 18 2022

A302236 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 + x^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

The difference between the number of partitions of n into an even number of nonprime parts and the number of partitions of n into an odd number of nonprime parts.
Convolution of the sequences A000586 and A081362.

Links

Crossrefs

Cf. A000586, A002095, A018252, A048165, A081362, A096258, A302234.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[1/(1 + Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 + x^A018252(k)).

A347662 Number of partitions of n into at most 4 nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 3, 4, 5, 4, 7, 6, 8, 8, 12, 11, 16, 13, 19, 18, 24, 21, 31, 28, 37, 34, 46, 40, 55, 49, 64, 60, 77, 69, 92, 83, 104, 96, 122, 111, 141, 129, 160, 150, 183, 166, 208, 194, 233, 220, 265, 242, 296, 277, 327, 311, 367, 340, 409, 383, 445
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2021

Keywords

Crossrefs

Cf. A002095, A018252, A307857, A341451, A347663, A347664.
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