A084993 -id:A084993 - cp's OEIS frontend

A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 3, 2, 3, 2, 1, 0, 2, 2, 3, 3, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 1, 0, 2, 2, 4, 3, 4, 2, 1, 1, 1, 1, 3, 4, 5, 3, 3, 2, 1, 0, 2, 2, 6, 4, 4, 4, 2, 1, 1, 0, 1, 5, 3, 6

Offset: 2

Emeric Deutsch, Mar 07 2006

Row n has floor(n/2) terms. Row sums yield A000607. T(n,1) = A010051(n) (the characteristic function of the primes). T(n,2) = A061358(n). Sum(k*T(n,k), k>=1) = A084993(n).

			T(12,3) = 2 because we have [7,3,2] and [5,5,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  0, 1, 1;
  1, 1, 1;
  0, 1, 1, 1;
  0, 1, 2, 1;
  ...

Alois P. Heinz, Rows n = 2..200, flattened

Columns k=1-10 give: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, A259200, A259201.
Row sums give A000607.
T(A000040(n),n) gives A259254(n).
Cf. A084993, A219180.

g:=1/product(1-t*x^(ithprime(j)),j=1..30): gser:=simplify(series(g,x=0,30)): for n from 2 to 22 do P[n]:=sort(coeff(gser,x^n)) od: for n from 2 to 22 do seq(coeff(P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i))[]], 0)))
    end:
T:= n-> subsop(1=NULL, b(n, numtheory[pi](n)))[]:
seq(T(n), n=2..25);  # Alois P. Heinz, Nov 16 2012

(* As triangle: *) nn=20;a=Product[1/(1-y x^i),{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[Series[a,{x,0,nn}],{x,y}],2,1]//Grid (* Geoffrey Critzer, Oct 30 2012 *)

parts(n, pred)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=15); apply(p->Vecrev(p/y), Vec(parts(n, isprime)-1))} \\ Andrew Howroyd, Dec 28 2017

G.f.: G(t,x) = -1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.

A024938 Total number of parts in all partitions of n into distinct prime parts.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 3, 2, 2, 5, 1, 5, 3, 5, 5, 7, 5, 10, 6, 10, 12, 10, 15, 12, 16, 17, 17, 19, 22, 17, 27, 21, 30, 30, 31, 35, 36, 40, 45, 45, 49, 53, 50, 62, 60, 69, 69, 73, 78, 85, 88, 98, 100, 105, 116, 116, 134, 135, 141, 149, 154, 168, 176, 188, 195, 206, 211, 232, 242, 255, 267, 276

Offset: 1

Clark Kimberling

			a(16) = 7 because the partitions of 16 into distinct prime parts are [13,3], [11,5] and [11,3,2].

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A084993.

g:=sum(x^ithprime(j)/(1+x^ithprime(j)),j=1..30)*product(1+x^ithprime(j),j=1..30): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # Emeric Deutsch, Apr 01 2006
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; local g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i-1));
         b(n, i-1) +g +[0, g[1]]
      fi
    end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 30 2012

Rest@ CoefficientList[ Series[ Sum[x^Prime@j/(1 + x^Prime@j), {j, 20}]* Product[1 + x^Prime@j, {j, 20}], {x, 0, 70}], x] (* Robert G. Wilson v *)
b[n_, i_] := b[n, i] = Module[{g}, If[n==0, {1, 0}, If[i < 1, {0, 0}, g = If[ Prime[i] > n, {0, 0}, b[n - Prime[i], i-1]]; b[n, i-1] + g + {0, g[[1]]}]]]; a[n_] := b[n, PrimePi[n]][[2]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)

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

G.f.: sum(x^p(j)/(1+x^p(j)),j>=1)*product(1+x^p(j), j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Jul 17 2003

More terms from Vladeta Jovovic, Jul 17 2003

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)).

A281617 Expansion of Sum_{i = pq, p prime, q prime} x^i/(1 - x^i) / Product_{j = pq, p prime, q prime} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 3, 0, 5, 2, 6, 3, 9, 3, 14, 7, 16, 10, 23, 12, 32, 20, 37, 28, 52, 35, 69, 49, 80, 68, 110, 83, 137, 112, 166, 150, 215, 178, 268, 239, 324, 303, 406, 365, 504, 472, 604, 584, 747, 708, 917, 888, 1089, 1085, 1337, 1311, 1618, 1606, 1916, 1954, 2332, 2334, 2782, 2829, 3300, 3407, 3963

Offset: 0

Ilya Gutkovskiy, Jan 25 2017

nonn

Total number of parts in all partitions of n into semiprimes (A001358).

Convolution of A086971 and A101048.

			a(12) = 5 because we have [6, 6], [4, 4, 4] and 2 + 3 = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A001358, A084993, A086971, A101048, A344447.

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0$2],
     `if`(i>n, 0, (p-> p+[0, p[1]])(b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=0..70);  # Alois P. Heinz, May 19 2021

nmax = 70; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - Floor[PrimeOmega[j]/2] Floor[2/PrimeOmega[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

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

a(n) = Sum_{k>0} k * A344447(n,k). - Alois P. Heinz, May 19 2021

A284827 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 4, 6, 9, 11, 13, 18, 20, 26, 34, 37, 47, 55, 66, 80, 96, 111, 130, 150, 180, 206, 240, 278, 318, 366, 419, 483, 549, 626, 716, 803, 913, 1034, 1167, 1314, 1477, 1659, 1861, 2085, 2332, 2605, 2902, 3232, 3602, 3999, 4442, 4930, 5454, 6034, 6675, 7375, 8133, 8967, 9870, 10855

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of smallest parts in all partitions of n into prime parts.

			a(10) = 11 because we have [7, 3], [5, 5], [5, 3, 2], [3, 3, 2, 2], [2, 2, 2, 2, 2] and 1 + 2 + 1 + 2 + 5 = 11.

Index entries for related partition-counting sequences

Cf. A000607, A084993, A092268, A092269, A195820, A284828.

nmax = 60; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

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

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)).

A276560 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 - x^prime(k)) Product_{k>=1} 1/(1 - x^prime(k)).

Original entry on oeis.org

0, 2, 3, 4, 10, 12, 21, 24, 36, 50, 66, 84, 117, 140, 180, 224, 289, 342, 437, 520, 630, 770, 920, 1104, 1300, 1560, 1809, 2156, 2523, 2940, 3441, 3968, 4620, 5338, 6125, 7092, 8103, 9272, 10608, 12080, 13776, 15624, 17759, 20064, 22680, 25622, 28858, 32496, 36456, 40950, 45849, 51324, 57399, 64044, 71390

Offset: 1

Ilya Gutkovskiy, Apr 10 2017

nonn

Sum of all parts of all partitions of n into prime parts.

Convolution of the sequences A000607 and A008472.

			a(6) = 12 because we have [3, 3], [2, 2, 2] and 2*6 = 12.

Eric Weisstein's World of Mathematics, Prime Partition
Index entries for related partition-counting sequences

Cf. A000607, A008472, A066186, A084993.

nmax = 55; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}] Product[1/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 55; Rest[CoefficientList[Series[x D[Product[1/(1 - x^Prime[k]), {k, 1, nmax}], x], {x, 0, nmax}], x]]
Table[Total@Flatten[IntegerPartitions[n,All,Prime@Range@PrimePi@n]],{n,52}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).
G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^prime(k)).
a(n) = n*A000607(n).
a(n) ~ n*exp(2*Pi*sqrt(n/log(n))/sqrt(3)).

A281449 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 - x^(prime(k)^2)) / Product_{k>=1} (1 - x^(prime(k)^2)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 4, 3, 2, 0, 5, 4, 3, 0, 6, 6, 4, 3, 7, 8, 5, 4, 8, 10, 8, 5, 13, 12, 10, 6, 15, 14, 12, 10, 17, 21, 14, 12, 19, 25, 18, 14, 25, 29, 27, 16, 28, 33, 33, 21, 31, 42, 38, 31, 34, 47, 43, 38, 41, 52, 54, 43, 53, 57, 62, 51, 62, 67, 69, 64, 68, 82, 76, 74, 78, 94, 89, 82, 93

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into squares of primes (A001248).

Convolution of A056170 and A090677.

			a(25) = 6 because we have [25], [9, 4, 4, 4, 4] and 1 + 5 = 6.

Index entries for related partition-counting sequences

Cf. A001248, A056170, A084993, A090677, A281541.

nmax = 88; Rest[CoefficientList[Series[Sum[x^Prime[k]^2/(1 - x^Prime[k]^2), {k, 1, nmax}]/Product[1 - x^Prime[k]^2, {k, 1, nmax}], {x, 0, nmax}], x]]

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

A281616 Expansion of Sum_{p prime, i>=1} x^(p^i)/(1 - x^(p^i)) / Product_{p prime, j>=1} (1 - x^(p^j)).

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 15, 18, 28, 36, 53, 66, 91, 117, 156, 195, 254, 318, 407, 503, 630, 777, 965, 1176, 1439, 1750, 2124, 2559, 3078, 3692, 4417, 5257, 6246, 7405, 8753, 10314, 12127, 14233, 16668, 19464, 22687, 26406, 30662, 35539, 41109, 47495, 54767, 63044, 72454, 83167, 95305, 109054, 124607, 142209, 162076, 184464

Offset: 1

Ilya Gutkovskiy, Jan 25 2017

nonn

Total number of parts in all partitions of n into prime power parts (1 excluded).

Convolution of A001222 and A023894.

			a(9) = 18 because we have [9], [7, 2], [5, 4], [5, 2, 2], [4, 3, 2], [3, 3, 3], [3, 2, 2, 2] and 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.

Index entries for related partition-counting sequences

Cf. A001222, A023894, A084993, A246655.

nmax = 57; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - Floor[1/PrimeNu[j]] x^j, {j, 2, nmax}], {x, 0, nmax}], x]]

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

A186409 Total number of parts in all partitions of prime(n).

Original entry on oeis.org

3, 6, 20, 54, 275, 556, 1965, 3498, 10206, 43453, 68135, 242812, 536104, 785437, 1644136, 4712040, 12760906, 17591088, 44736332, 81493581, 109311863, 257863391, 448980978, 1007135164, 2840344772, 4695605081, 6015397025, 9803584533, 12473509636, 20063812526

Offset: 1

Omar E. Pol, Aug 11 2011

nonn

			For n = 3 the third prime number is 5; the partitions of 5 are [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]; there are 20 parts, so a(3) = 20.

Cf. A024938, A084993, A085410, A182723.

a(n) = A006128(A000040(n)) = A006128(prime(n)).

cp's OEIS Frontend

A117278 Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)).

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A024938 Total number of parts in all partitions of n into distinct prime parts.

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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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A281617 Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j = p*q, p prime, q prime} (1 - x^j).

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A284827 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

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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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A276560 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).

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A281617 Expansion of Sum_{i = pq, p prime, q prime} x^i/(1 - x^i) / Product_{j = pq, p prime, q prime} (1 - x^j).

A276560 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 - x^prime(k)) Product_{k>=1} 1/(1 - x^prime(k)).