A024938 -id:A024938 - cp's OEIS frontend

A084993 Total number of parts in all partitions of n into prime parts.

0, 1, 1, 2, 3, 5, 6, 9, 12, 16, 20, 27, 33, 42, 53, 64, 80, 96, 117, 141, 169, 201, 239, 282, 333, 390, 456, 532, 617, 715, 826, 951, 1094, 1253, 1435, 1636, 1864, 2119, 2404, 2723, 3078, 3473, 3915, 4403, 4947, 5549, 6215, 6952, 7767, 8665, 9656, 10748

Offset: 1

Vladeta Jovovic, Jul 17 2003

nonn

			Partitions of 9 into primes are 2+2+2+3=3+3+3=2+2+5=2+7; thus a(9)=4+3+3+2=12.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000607, A024938.

g:=sum(x^ithprime(j)/(1-x^ithprime(j)),j=1..20)/product(1-x^ithprime(j),j=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..57); # Emeric Deutsch, Mar 07 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]
    elif i=1 then `if`(irem(n, 2)=0, [1, n/2], [0, 0])
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i));
         b(n, i-1) +g +[0, g[1]]
      fi
    end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 30 2012

nn=40;a=Product[1/(1-y x^i),{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[Series[D[a,y]/.y->1,{x,0,nn}],x],1]  (* Geoffrey Critzer, Oct 30 2012 *)
b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i == 1, If[EvenQ[n], {1, n/2}, {0, 0}], True, g = If[Prime[i] > n, {0, 0}, b[n - Prime[i], i]]; b[n, i - 1] + g + {0, g[[1]]}]];
a[n_] := b[n, PrimePi[n]][[2]];
Array[a, 52] (* Jean-François Alcover, Dec 30 2017, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n],AllTrue[#,PrimeQ]&]]],{n,60}] (* Harvey P. Dale, Jul 11 2023 *)

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, isprime), -n)} \\ Andrew Howroyd, Dec 28 2017

G.f.: sum(x^p(j)/(1-x^p(j)),j=1..infinity)/product(1-x^p(j), j=1..infinity), where p(j) is the j-th prime. - Emeric Deutsch, Mar 07 2006

A281477 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, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

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

			a(38) = 3 because we have [25, 9, 4].

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Extended graphical example
Index entries for related partition-counting sequences

Cf. A001248, A024938, A048261, A111900, A121518, A281449, A281542, A281668.

Primes:= select(isprime, [$1..20]):
g:= add(x^(p^2)/(1+x^(p^2)),p=Primes)*mul(1+x^(p^2),p=Primes):
S:= series(g, x, 20^2+1):
seq(coeff(S,x,n),n=1..20^2); # Robert Israel, Feb 08 2017

nmax = 125; 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)).

A281545 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, 0, 1, 2, 0, 2, 1, 2, 1, 2, 3, 4, 1, 4, 4, 4, 6, 4, 7, 6, 6, 8, 9, 8, 10, 6, 13, 12, 12, 14, 15, 16, 16, 18, 23, 22, 19, 24, 24, 30, 28, 30, 33, 34, 34, 40, 44, 46, 44, 46, 58, 56, 60, 64, 65, 68, 70, 80, 86, 88, 87, 94, 101, 112, 114, 116, 125, 130, 132, 148, 159, 162, 163, 168, 190, 196

Offset: 1

Ilya Gutkovskiy, Jan 23 2017

nonn

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

			a(23) = 7 because we have [23], [13, 7, 3], [11, 7, 5] and 1 + 3 + 3 = 7.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for related partition-counting sequences

Cf. A024938, A024939, A065091.

nmax = 80; 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/(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, 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)).

A281668 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, 1, 3, 2, 5, 3, 8, 7, 10, 12, 13, 20, 18, 26, 25, 36, 34, 45, 47, 59, 62, 71, 82, 91, 105, 112, 132, 143, 163, 174, 201, 220, 244, 266, 298, 327, 362, 388, 437, 470, 521, 558, 621, 671, 733, 788, 864, 938, 1011, 1100, 1182, 1295, 1379, 1501, 1606, 1753, 1861, 2017, 2158, 2335, 2493, 2672, 2871, 3078

Offset: 1

Ilya Gutkovskiy, Jan 26 2017

nonn

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

			a(10) = 7 because we have [8, 2], [7, 3], [5, 3, 2] and 2 + 2 + 3 = 7.

Index entries for related partition-counting sequences

Cf. A015723, A024938, A054685, A246655, A281616.

nmax = 66; 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)).

A279412 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 + x^prime(k)) Product_{k>=1} (1 + x^prime(k)).

Original entry on oeis.org

0, 2, 3, 0, 10, 0, 14, 8, 9, 20, 11, 24, 26, 28, 30, 48, 34, 72, 57, 80, 84, 88, 115, 120, 125, 156, 135, 168, 203, 180, 279, 224, 297, 306, 315, 396, 407, 418, 507, 480, 574, 630, 645, 748, 720, 828, 893, 960, 1029, 1150, 1122, 1300, 1378, 1458, 1650, 1624, 1824, 1856, 2065, 2220, 2379, 2480, 2646, 2816, 2925

Offset: 1

Ilya Gutkovskiy, Apr 11 2017

nonn

Sum of all parts of all partitions of n into distinct primes.

			a(12) = 24 because we have [7, 5], [7, 3, 2] and 2*12 = 24.

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

Cf. A000586, A024938, A066189, A276560.

nmax = 65; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}] Product[1 + x^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 65; Rest[CoefficientList[Series[x D[Product[1 + x^Prime[k], {k, 1, nmax}], x], {x, 0, nmax}], x]]

G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^prime(k)).
a(n) = n*A000586(n).

A281667 Expansion of Sum_{i>=1} mu(i)^2x^i/(1 + x^i) Product_{j>=1} (1 + mu(j)^2*x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 3, 2, 3, 6, 5, 9, 10, 12, 15, 16, 20, 24, 27, 38, 41, 48, 56, 62, 78, 88, 101, 120, 131, 149, 174, 189, 221, 243, 278, 318, 349, 394, 444, 491, 556, 622, 693, 773, 849, 953, 1048, 1158, 1292, 1422, 1568, 1735, 1901, 2101, 2307, 2534, 2795, 3060, 3357, 3681, 4024, 4404, 4809, 5245, 5734, 6242, 6805, 7418

Offset: 1

Ilya Gutkovskiy, Jan 26 2017

nonn

Total number of parts in all partitions of n into distinct squarefree parts (A005117).

			a(8) = 9 because we have [7, 1], [6, 2], [5, 3], [5, 2, 1] and 2 + 2 + 2 + 3 = 9.

Index entries for related partition-counting sequences

Cf. A005117, A008683, A015723, A024938, A087188, A281572.

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

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

cp's OEIS Frontend

A084993 Total number of parts in all partitions of n into prime parts.

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A281477 Expansion of Sum_{k>=1} x^(prime(k)^2)/(1 + x^(prime(k)^2)) * Product_{k>=1} (1 + x^(prime(k)^2)).

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A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).

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A281668 Expansion of Sum_{p prime, i>=1} x^(p^i)/(1 + x^(p^i)) * Product_{p prime, j>=1} (1 + x^(p^j)).

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

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A279412 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).

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A281667 Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 + x^i) * Product_{j>=1} (1 + mu(j)^2*x^j), where mu() is the Moebius function (A008683).

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Mathematica

Formula

A279412 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 + x^prime(k)) Product_{k>=1} (1 + x^prime(k)).

A281667 Expansion of Sum_{i>=1} mu(i)^2x^i/(1 + x^i) Product_{j>=1} (1 + mu(j)^2*x^j), where mu() is the Moebius function (A008683).