A099773 -id:A099773 - cp's OEIS frontend

A024939 Number of partitions of n into distinct odd primes.

1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3, 5, 4, 4, 5, 5, 6, 6, 5, 7, 7, 7, 8, 8, 9, 8, 9, 11, 11, 10, 12, 12, 13, 14, 14, 16, 15, 16, 17, 19, 20, 20, 20, 22, 24, 23, 26, 27, 27, 28, 30, 33, 34, 34, 36, 37, 40, 41, 43, 46, 46, 47, 50, 55, 56, 56

Offset: 0

Clark Kimberling

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A024937.

Cf. A099773, A000586.

a024939 = p a065091_list where
   p _  0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012

G.f.: Product_{k>1} (1+x^prime(k)).

Corrected and extended by Vladeta Jovovic, Jul 20 2003

A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 0, 0, 1, 5, 1, 1, 6, 2, 0, 0, 0, 1, 7, 2, 0, 1, 9, 2, 1, 10, 3, 1, 12, 3, 1, 0, 0, 0, 1, 14, 3, 1, 1, 17, 4, 1, 0, 0, 0, 0, 1, 19, 5, 1, 1, 23, 5, 1, 1, 26, 6, 2, 0, 1, 30, 7, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Mar 16 2020

			In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
Triangle T(n,k) begins:
   0    ;
   0    ;
   1    ;
   0, 1    ;
   1       ;
   1, 0, 1    ;
   1, 1       ;
   2, 0, 0, 1    ;
   2, 1          ;
   3, 1          ;
   3, 1, 1       ;
   4, 1, 0, 0, 1    ;
   5, 1, 1          ;
   6, 2, 0, 0, 0, 1    ;
   7, 2, 0, 1          ;
   9, 2, 1             ;
  10, 3, 1             ;
  12, 3, 1, 0, 0, 0, 1    ;
  14, 3, 1, 1             ;
  17, 4, 1, 0, 0, 0, 0, 1    ;
  19, 5, 1, 1                ;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.
Row sums give A000607 for n>0.
Length of n-th row is A000720(A331634(n)) for n>1.
Indices of rows without 1's: A330433.
Cf. A000040, A000720, A010051, A333129, A333238, A333259.

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
      coeff(p, x, i), [][]), i=2..max(2,degree(p))))(b(n, 2, x))
    end:
seq(T(n), n=0..23);

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
T /@ Range[0, 23] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

T(n,pi(n)) = A010051(n) for n > 1.
T(p,pi(p)) = 1 if p is prime.
T(prime(k),k) = 1 for k >= 1.
Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - David James Sycamore, Mar 28 2020

A331981 Number of compositions (ordered partitions) of n into distinct odd primes.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 6, 4, 1, 4, 7, 4, 12, 4, 13, 6, 12, 28, 18, 28, 19, 6, 25, 52, 24, 54, 30, 56, 31, 98, 156, 102, 37, 104, 157, 150, 276, 150, 175, 154, 288, 200, 528, 246, 307, 226, 666, 990, 780, 1038, 679, 348, 799, 1828, 1272, 1162, 1164

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(16) = 4 because we have [13, 3], [11, 5], [5, 11] and [3, 13].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A002124, A023360, A024939, A065091, A099773, A219107, A331926, A331982.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n+1)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i+1))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..72);  # Alois P. Heinz, Feb 03 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n + 1] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, If[# > n, 0, b[n - #, i - 1, t + 1]]&[Prime[i + 1]] + b[n, i - 1, t]]];
a[n_] := b[n, PrimePi[n], 0];
a /@ Range[0, 72] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

A281544 Expansion of Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 2, 3, 4, 4, 6, 7, 8, 11, 12, 15, 18, 20, 26, 29, 34, 40, 46, 54, 62, 71, 82, 94, 106, 122, 138, 157, 178, 201, 226, 254, 286, 321, 360, 402, 448, 501, 558, 619, 690, 764, 846, 938, 1036, 1145, 1264, 1392, 1532, 1687, 1854, 2036, 2234, 2448, 2680, 2934, 3210, 3507, 3828, 4178, 4554, 4961, 5404

Offset: 1

Ilya Gutkovskiy, Jan 23 2017

nonn

Total number of parts in all partitions of n into odd primes.

Convolution of A005087 and A099773.

			a(14) = 8 because we have [11, 3], [7, 7], [5, 3, 3, 3] and 2 + 2 + 4 = 8.

Index entries for related partition-counting sequences

Cf. A005087, A065091, A084993, A099773.

nmax = 68; Rest[CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}]/Product[1 - x^Prime[k], {k, 2, nmax}], {x, 0, nmax}], x]]

sumparts(n, pred)={sum(k=1, n, 1/(1-pred(k)*x^k) - 1 + O(x*x^n))/prod(k=1, n, 1-pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ Andrew Howroyd, Dec 28 2017

G.f.: Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

A284828 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 5, 4, 6, 9, 7, 10, 11, 12, 17, 19, 22, 23, 26, 33, 36, 41, 48, 52, 59, 66, 78, 85, 97, 112, 117, 134, 151, 169, 187, 207, 230, 255, 284, 313, 348, 379, 418, 465, 508, 561, 620, 674, 737, 812, 892, 972, 1064, 1157, 1257, 1379, 1503, 1639, 1776, 1935, 2101, 2279, 2483

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of smallest parts in all partitions of n into odd prime parts (A065091).

			a(16) = 7 because we have [13, 3], [11, 5], [7, 3, 3, 3], [5, 5, 3, 3] and 1 + 1 + 3 + 2 = 7.

Index entries for related partition-counting sequences

Cf. A065091, A084993, A092268, A092269, A099773, A195820, A281544, A284827.

nmax = 68; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, i, nmax}], {i, 2, nmax}], {x, 0, nmax}], x]]

x = 'x + O('x ^ 70); concat([0, 0], Vec(sum(i=2, 70, x^prime(i)/(1 - x^prime(i)) * prod(j=i, 70, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 05 2017

G.f.: Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

A298603 Number of partitions of n into odd prime parts (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 22, 26, 31, 36, 42, 49, 56, 65, 75, 86, 98, 112, 127, 144, 164, 185, 209, 235, 264, 297, 332, 372, 416, 463, 516, 574, 638, 708, 785, 869, 960, 1061, 1171, 1291, 1421, 1563, 1718, 1886, 2070, 2269, 2484, 2718, 2972, 3247, 3545, 3868, 4216, 4592

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

Partial sums of A099773.

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

Index entries for related partition-counting sequences

Cf. A000607, A006005, A008578, A034891, A099773, A298604.

nmax = 62; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=2} 1/(1 - x^prime(k)).

A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 0

Gus Wiseman, Nov 01 2023

To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

			Triangle begins:
  1
  1
  1 0 0 1
  1 0 0 1 0 1
  1 0 0 1 0 1 1 1
  1 0 0 1 0 1 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1
The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

Row lengths are A055670.
Columns appear to converge to A099773.
A bisected even version is A116598 (counts partitions by number of 1's).
Counting all parts (not just > 1) gives A331416, shifted A331385.
A000041 counts integer partitions, strict A000009 (also into odds).
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A330953 counts partitions with Heinz number divisible by sum of primes.
A331381 counts partitions with (product)|(sum of primes), equality A331383.
Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]

A280912 Number of partitions of n into odd semiprimes (A046315).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 0, 3, 1, 1, 3, 1, 0, 4, 2, 2, 5, 1, 1, 5, 3, 1, 6, 3, 2, 8, 2, 1, 7, 5, 4, 9, 4, 3, 11, 6, 3, 11, 6, 6, 14, 7, 5, 15, 9, 7, 16, 9, 8, 20, 14, 9, 21, 13, 11, 26, 16, 12, 28, 19, 17, 29, 19, 17, 37, 27

Offset: 0

Ilya Gutkovskiy, Jan 10 2017

nonn

			a(39) = 3 because we have [39], [21, 9, 9] and [15, 15, 9].

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Semiprime
Index entries for related partition-counting sequences

Cf. A001222, A001358, A046315, A099773, A101048, A112020.

nmax = 100; CoefficientList[Series[Product[1/(1 - Floor[PrimeOmega[2 k + 1]/2] Floor[2/PrimeOmega[2 k + 1]] x^(2 k + 1)), {k, 1, nmax}], {x, 0, nmax}], x]
Join[{1},Table[Count[IntegerPartitions[n],?(AllTrue[#,OddQ]&&Union[PrimeOmega[#]]=={2}&)],{n,110}]] (* _Harvey P. Dale, Nov 11 2024 *)

G.f.: Product_{k>=1} 1/(1 - floor(bigomega(2*k+1)/2)*floor(2/bigomega(2*k+ 1))*x^(2*k+1)), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

A352165 Number of partitions of n into odd prime powers (1 included).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 71, 83, 97, 112, 130, 150, 173, 199, 228, 261, 298, 340, 386, 439, 497, 563, 637, 718, 809, 910, 1023, 1147, 1286, 1439, 1608, 1796, 2003, 2231, 2483, 2761, 3065, 3401, 3770, 4175, 4619

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A023893, A023894, A061345, A099773, A134345, A280151, A352166.

nmax = 55; CoefficientList[Series[Product[1/(1 - Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1 / (1 - x^A061345(k)).

A284834 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j=2..i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 2, 5, 4, 4, 9, 5, 6, 12, 8, 11, 17, 12, 14, 23, 19, 21, 29, 27, 29, 41, 37, 36, 56, 49, 55, 72, 62, 74, 91, 90, 96, 116, 117, 125, 155, 149, 162, 195, 194, 215, 246, 248, 270, 311, 324, 344, 389, 406, 435, 494, 509, 546, 615, 636, 694, 763, 787, 861, 942, 994, 1063

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of largest parts in all partitions of n into odd prime parts (A065091).

			a(16) = 5 because we have [13, 3], [11, 5], [7, 3, 3, 3], [5, 5, 3, 3] and 1 + 1 + 1 + 2 = 5.

Index entries for related partition-counting sequences

Cf. A046746, A065091, A084993, A092311, A099773, A284828.

nmax = 64; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, 2, i}], {i, 2, nmax}], {x, 0, nmax}], x]]

x='x+O('x^70); concat([0, 0], Vec(sum(i=2, 70, x^prime(i)/(1 - x^prime(i)) * prod(j=2,i, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 04 2017

G.f.: Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j=2..i} 1/(1 - x^prime(j)).

cp's OEIS Frontend

A024939 Number of partitions of n into distinct odd primes.

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A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

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A331981 Number of compositions (ordered partitions) of n into distinct odd primes.

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A281544 Expansion of Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

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A284828 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

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A298603 Number of partitions of n into odd prime parts (including 1).

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A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

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A280912 Number of partitions of n into odd semiprimes (A046315).

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