A046675 -id:A046675 - cp's OEIS frontend

A000607 Number of partitions of n into prime parts.

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 30, 35, 40, 46, 52, 60, 67, 77, 87, 98, 111, 124, 140, 157, 175, 197, 219, 244, 272, 302, 336, 372, 413, 456, 504, 557, 614, 677, 744, 819, 899, 987, 1083, 1186, 1298, 1420, 1552, 1695, 1850, 2018, 2198, 2394, 2605, 2833, 3079, 3344

Offset: 0

N. J. A. Sloane

a(n) gives the number of values of k such that A001414(k) = n. - Howard A. Landman, Sep 25 2001
Let W(n) = {prime p: There is at least one number m whose spf is p, and sopfr(m) = n}. Let V(n,p) = {m: sopfr(m) = n, p belongs to W(n)}. Then a(n) = sigma(|V(n,p)|). E.g.: W(10) = {2,3,5}, V(10,2) = {30,32,36}, V(10,3) = {21}, V(10,5) = {25}, so a(10) = 3+1+1 = 5. - David James Sycamore, Apr 14 2018
From Gus Wiseman, Jan 18 2020: (Start)
Also the number of integer partitions such that the sum of primes indexed by the parts is n. For example, the sum of primes indexed by the parts of the partition (3,2,1,1) is prime(3)+prime(2)+prime(1)+prime(1) = 12, so (3,2,1,1) is counted under a(12). The a(2) = 1 through a(14) = 10 partitions are:
  1  2  11  3   22   4    32    41    33     5      43      6       44
            21  111  31   221   222   42     322    331     51      52
                     211  1111  311   321    411    421     332     431
                                2111  2211   2221   2222    422     3222
                                      11111  3111   3211    3221    3311
                                             21111  22111   4111    4211
                                                    111111  22211   22221
                                                            31111   32111
                                                            211111  221111
                                                                    1111111
(End)

			n = 10 has a(10) = 5 partitions into prime parts: 10 = 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 = 2 + 3 + 5 = 3 + 7 = 5 + 5.
n = 15 has a(15) = 12 partitions into prime parts: 15 = 2 + 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 5 = 2 + 3 + 5 + 5 = 2 + 3 + 3 + 7 = 2 + 2 + 11 = 2 + 13 = 3 + 3 + 3 + 3 + 3 = 3 + 5 + 7 = 5 + 5 + 5.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 203.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
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).

Hans Havermann, Table of n, a(n) for n = 0..50000 (first 1000 terms from T. D. Noe, terms 1001-20000 from Vaclav Kotesovec, terms 20001-50000 extracted from files by Hans Havermann)
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
George E. Andrews, Arnold Knopfmacher, and John Knopfmacher, Engel expansions and the Rogers-Ramanujan identities, J. Number Theory 80 (2000), 273-290. See Eq. 2.1.
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, Journal of Number Theory, Volume 118, Issue 1, May 2006, pp. 15-30.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, Asymptotic prime partitions of integers, Phys. Rev. E, 95 (2017), 052108, arXiv:1609.06497 [math-ph], 2016-2017.
P. T. Bateman and P. Erdős, Partitions into primes, Publ. Math. Debrecen 4 (1956), 198-200.
J. Browkin, Sur les décompositions des nombres naturels en sommes de nombres premiers, Colloquium Mathematicum 5 (1958), 205-207.
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
L. M. Chawla and S. A. Shad, Review of "On a trio-set of partition functions and their tables", Mathematics of Computation, Vol. 24, No. 110 (Apr., 1970), pp. 490-491.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see Section VIII.6, pp. 576ff.
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187.
O. P. Gupta and S. Luthra, Partitions into primes, Proc. Nat. Inst. Sci. India. Part A. 21 (1955), 181-184.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12. (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Havermann, Tables of sum-of-prime-factors sequences (including sequence lengths, i.e., number of prime-parts partitions, for the first 50000).
BongJu Kim, Partition number identities which are true for all set of parts, arXiv:1803.08095 [math.CO], 2018.
Vaclav Kotesovec, Graph - asymptotic ratio for log(a(n)), without minor asymptotic term.
Vaclav Kotesovec, Wrong asymptotics of OEIS A000607? (MathOverflow). Includes discussion of the contradiction between the results for the next-to-leading term in the asymptotic formulas by Vaughan and by Bartel et al.
John F. Loase (splurge(AT)aol.com), David Lansing, Cassie Hryczaniuk and Jamie Cahoon, A Variant of the Partition Function, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), pp. 320-321.
Ljuben Mutafchiev, A Note on Goldbach Partitions of Large Even Integers, arXiv:1407.4688 [math.NT], 2014-2015.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
N. J. A. Sloane, Transforms.
R. C. Vaughan, On the number of partitions into primes, Ramanujan J. vol. 15, no. 1 (2008) 109-121.
Eric Weisstein's World of Mathematics, Prime Partition.
Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.
Index entries for sequences related to Goldbach conjecture

G.f. = 1 / g.f. for A046675. See A046113 for the ordered (compositions) version.
Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586 (distinct parts), A000041, A070214, A192541, A112021, A056768, A128515, A319265, A319266.
Row sums of array A116865 and of triangle A261013.
Column sums of A331416.
Partitions whose Heinz number is divisible by their sum of primes are A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Cf. A000720, A014689, A331385, A331387, A331415, A331417.

a000607 = p a000040_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012

[1] cat [#RestrictedPartitions(n,{p:p in PrimesUpTo(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019

with(gfun):
t1:=mul(1/(1-q^ithprime(n)),n=1..51):
t2:=series(t1,q,50):
t3:=seriestolist(t2); # fixed by Vaclav Kotesovec, Sep 14 2014

CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
f[n_] := Length@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]; Array[f, 57] (* Robert G. Wilson v, Jul 23 2010 *)
Table[Length[Select[IntegerPartitions[n],And@@PrimeQ/@#&]],{n,0,60}] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := a[n] = If[PrimeQ[n], 1, 0]; c[n_] := c[n] = Plus @@ Map[# a[#] &, Divisors[n]]; b[n_] := b[n] = (c[n] + Sum[c[k] b[n - k], {k, 1, n - 1}])/n; Table[b[n], {n, 1, 20}] (* Thomas Vogler, Dec 10 2015: Uses Euler transform, caches computed values, faster than IntegerPartitions[] function. *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j + 1]] -= poly[[j + 1 - p]], {j, nmax, p, -1}];, {k, 2, pmax}]; s = Sum[poly[[k + 1]]*x^k, {k, 0, Length[poly] - 1}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2021 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,1-x^prime(k))) \\ Joerg Arndt, Sep 04 2014

from sympy import primefactors
l = [1, 0]
for n in range(2, 101):
    l.append(sum(sum(primefactors(k)) * l[n - k] for k in range(1, n + 1)) // n)
l  # Indranil Ghosh, Jul 13 2017

[Partitions(n, parts_in=prime_range(n + 1)).cardinality() for n in range(100)]  # Giuseppe Coppoletta, Jul 11 2016

Asymptotically a(n) ~ exp(2 Pi sqrt(n/log n) / sqrt(3)) (Ayoub).
a(n) = (1/n)*Sum_{k=1..n} A008472(k)*a(n-k). - Vladeta Jovovic, Aug 27 2002
G.f.: 1/Product_{k>=1} (1-x^prime(k)).
See the partition arrays A116864 and A116865.
From Vaclav Kotesovec, Sep 15 2014 [Corrected by Andrey Zabolotskiy, May 26 2017]: (Start)
It is surprising that the ratio of the formula for log(a(n)) to the approximation 2 * Pi * sqrt(n/(3*log(n))) exceeds 1. For n=20000 the ratio is 1.00953, and for n=50000 (using the value from Havermann's tables) the ratio is 1.02458, so the ratio is increasing. See graph above.
A more refined asymptotic formula is found by Vaughan in Ramanujan J. 15 (2008), pp. 109-121, and corrected by Bartel et al. (2017): log(a(n)) = 2*Pi*sqrt(n/(3*log(n))) * (1 - log(log(n))/(2*log(n)) + O(1/log(n))).
See Bartel, Bhaduri, Brack, Murthy (2017) for a more complete asymptotic expansion. (End)
G.f.: 1 + Sum_{i>=1} x^prime(i) / Product_{j=1..i} (1 - x^prime(j)). - Ilya Gutkovskiy, May 07 2017
a(n) = A184198(n) + A184199(n). - Vaclav Kotesovec, Jan 11 2021

A000586 Number of partitions of n into distinct primes.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 5, 6, 7, 6, 9, 7, 9, 9, 9, 11, 11, 11, 13, 12, 14, 15, 15, 17, 16, 18, 19, 20, 21, 23, 22, 25, 26, 27, 30, 29, 32, 32, 35, 37, 39, 40, 42, 44, 45, 50, 50, 53, 55, 57, 61, 64, 67, 70, 71, 76, 78, 83, 87, 89, 93, 96

Offset: 0

N. J. A. Sloane

			n=16 has a(16) = 3 partitions into distinct prime parts: 16 = 2+3+11 = 3+13 = 5+11.

H. Gupta, Certain averages connected with partitions. Res. Bull. Panjab Univ. no. 124 1957 427-430.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0004 and N0039).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187. [broken link]
BongJu Kim, Partition number identities which are true for all set of parts, arXiv:1803.08095 [math.CO], 2018.
Vaclav Kotesovec, Plot of log(a(n)) / log(Qas(n)) for n = 2..10^8, for Qas see the formula (25) from the article by Murthy, Brack and Bhaduri, p. 7.
M. V. N. Murthy, M. Brack, R. K. Bhaduri, On the asymptotic distinct prime partitions of integers, arXiv:1904.02776 [math.NT], Mar 22 2019.
K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Quart. J. Math., Oxford Ser. (2) 5 (1954), 241-259.

Cf. A000041, A070215, A000607 (parts may repeat), A112022, A000009, A046675, A319264, A319267.

a000586 = p a000040_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

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(ithprime(i)>n, 0, b(n-ithprime(i), i-1))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 15 2012

CoefficientList[Series[Product[(1+x^Prime[k]), {k, 24}], {x, 0, Prime[24]}], x]
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[Prime[i] > n, 0, b[n - Prime[i], i-1]]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = 1; Do[p = Prime[k]; Do[poly[[j]] += poly[[j - p]], {j, nmax + 1, p + 1, -1}];, {k, 2, pmax}]; poly (* Vaclav Kotesovec, Sep 20 2018 *)

a(n,k=n)=if(n<1, !n, my(s);forprime(p=2,k,s+=a(n-p,p-1));s) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import isprime, primerange
from functools import cache
@cache
def a(n, k=None):
    if k == None: k = n
    if n < 1: return int(n == 0)
    return sum(a(n-p, p-1) for p in primerange(1, k+1))
print([a(n) for n in range(83)]) # Michael S. Branicky, Sep 03 2021 after Charles R Greathouse IV

G.f.: Product_{k>=1} (1+x^prime(k)).
a(n) = A184171(n) + A184172(n). - R. J. Mathar, Jan 10 2011
a(n) = Sum_{k=0..A024936(n)} A219180(n,k). - Alois P. Heinz, Nov 13 2012
log(a(n)) ~ Pi*sqrt(2*n/(3*log(n))) [Roth and Szekeres, 1954]. - Vaclav Kotesovec, Sep 13 2018

Entry revised by N. J. A. Sloane, Jun 10 2012

A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, -1, -1, 0, 1, 0, -1, 1, 2, 0, -3, 0, 2, 0, -3, 0, 5, 0, -4, -2, 4, 0, -5, 0, 7, 3, -8, -1, 5, 1, -10, 0, 13, 2, -10, -3, 14, -2, -17, -3, 21, 5, -22, 0, 22, 4, -34, -5, 33, 9, -33, -10, 43, 6, -43, -19, 52, 16, -51, -13, 56, 24, -71, -20, 64, 26, -78, -24, 90, 24, -90, -39, 112, 26, -115, -37

Offset: 0

Ilya Gutkovskiy, Sep 19 2017

Convolution inverse of A073576.

The difference between the number of partitions of n into an even number of distinct squarefree parts and the number of partitions of n into an odd number of distinct squarefree parts.

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

Cf. A005117, A008683, A046675, A073576, A087188.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      abs(mobius(d)), d=divisors(j)) *b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 20 2017

nmax = 75; CoefficientList[Series[Product[1 - MoebiusMu[k]^2 x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A005117(k)).

A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

Original entry on oeis.org

1, -1, -1, 0, 1, 0, 0, 0, 1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 1, 0, 0, -1, 2, 0, -1, -1, 1, -1, 1, -1, 2, 0, 0, -2, 2, -2, 0, 0, 3, -2, 1, -2, 2, -1, 0, -3, 5, -1, 0, -3, 3, -3, 3, -3, 3, -1, 2, -5, 6, -4, 1, -2, 6, -5, 3, -6, 5, -2, 4, -8, 9, -5, 3, -5, 7, -8, 7, -8, 8, -4, 5

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of distinct prime parts (including 1) and the number of partitions of n into an odd number of distinct prime parts (including 1).

Convolution inverse of A034891.

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

Cf. A000586, A000607, A034891, A036497, A046675 (partial sums).

nmax = 82; CoefficientList[Series[(1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

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

A350514 Maximal coefficient of Product_{j=1..n} (1 - x^prime(j)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 8, 12, 17, 30, 41, 70, 107, 186, 307, 531, 887, 1561, 2701, 4817, 8514, 15030, 26490, 47200, 84622, 151809, 273912, 496807, 900595, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Alois P. Heinz, Jan 02 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000040, A007504, A046675, A086376, A350457.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1-x^ithprime(n))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..60);

a[n_] := Times@@(1-x^Prime[Range[n]])//Expand//CoefficientList[#, x]&//Max;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 02 2022 *)

a(n) = vecmax(Vec(prod(j=1, n, 1-'x^prime(j)))); \\ Michel Marcus, Jan 04 2022

from sympy.abc import x
from sympy import prime, prod
def A350514(n): return 1 if n == 0 else max(prod(1-x**prime(i) for i in range(1,n+1)).as_poly().coeffs()) # Chai Wah Wu, Jan 04 2022

A298159 G.f.: Product_{k>=1} (1-prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, -2, -3, 0, 1, 0, 3, 15, 14, -9, -11, -7, 9, -37, -79, 28, 193, -46, -150, -142, -58, -142, 171, 73, 652, 643, 349, -1047, 404, -2743, 873, 3082, 2498, -3039, 411, -2188, 4534, -4571, -9997, -5917, 32879, -14830, -11926, -16333, 7208, -22517, 59885, -27601

Offset: 0

Seiichi Manyama, Jan 14 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A002098, A046675, A298160.

Convolution inverse of A002098.

A184082 Decimal expansion of product_{p=primes} (1-1/2^p).

Original entry on oeis.org

6, 3, 0, 3, 8, 4, 4, 0, 5, 9, 9, 1, 3, 6, 6, 1, 5, 4, 6, 0, 2, 2, 2, 5, 8, 2, 2, 9, 3, 1, 3, 2, 9, 2, 4, 6, 6, 9, 4, 9, 4, 1, 4, 4, 8, 4, 6, 7, 2, 9, 9, 2, 9, 5, 6, 2, 0, 1, 8, 1, 0, 7, 1, 4, 3, 0, 0, 8, 6, 8, 0, 1, 5, 2, 2, 2, 2, 5, 2, 3, 9, 7, 2, 2, 3, 4, 0, 9, 9

Offset: 0

R. J. Mathar, Jan 09 2011

			(1-1/2^2) *(1-1/2^3) *(1-1/2^5) *(1-1/2^7) *(1-1/2^11) *.... = 0.63038440599136615460222582...

RealDigits[Times@@(1-1/2^Prime[Range[100]]),10,120][[1]] (* Harvey P. Dale, Jun 26 2011 *)

Equals product_{p in A000040} (1-2^(-p)) = sum_{n>=0} A046675(n)/2^n.

A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

sign

The difference between the number of partitions of n into an even number of distinct nonprime parts and the number of partitions of n into an odd number of distinct nonprime parts.

Convolution of the sequences A000607 and A010815.

Index entries for sequences related to partitions

Cf. A000607, A002095, A010815, A018252, A046675, A096258, A302236.

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

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

A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, -1, 0, 1, 1, 0, 0, 1, 1, 1, 0, -1, -1, -2, -1, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, -3, -3, -1, 1, 1, 0, -1, -1, 2, 2, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, 0, -2, -3, -1, -1, 0, 2, 0, 1, 3, 0, 1, 3, 1, -3, -2, -3, -2, 3, 2, -1, 0, -2, 1, 1, -2, -1, 1, 2, 2, 3, -1, -2, 4

Offset: 0

Ilya Gutkovskiy, Nov 01 2019

Convolution inverse of A023894.
The difference between the number of partitions of n into an even number of distinct prime power parts and the number of partitions of n into an odd number of distinct prime power parts (1 excluded).
Conjecture: the last zero (38th) occurs at n = 340.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A023894, A046675, A054685, A246655, A292561.

N:= 100: # for a(0)..a(N)
R:= 1:
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  for k from 1 to floor(log[p](N)) do
    R:= series(R*(1-x^(p^k)),x,N+1)
  od;
od:
seq(coeff(R,x,j),j=0..N); # Robert Israel, Nov 03 2019

nmax = 90; CoefficientList[Series[Product[(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 90}]

G.f.: Product_{k>=1} (1 - x^A246655(k)).

A338826 G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, -1, 0, -1, 2, -2, 2, -3, 4, -4, 5, -7, 8, -9, 11, -13, 15, -18, 21, -24, 28, -32, 37, -43, 49, -55, 63, -72, 81, -92, 104, -117, 131, -147, 166, -185, 206, -231, 257, -285, 317, -353, 391, -432, 478, -528, 583, -643, 708, -778, 855, -940, 1031, -1130, 1238, -1354

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

sign

The difference between the number of partitions of n into an even number of prime parts (including 1) and the number of partitions of n into an odd number of prime parts (including 1).

Convolution inverse of A036497.

Cf. A000586, A000607, A008578, A036497, A046675, A048165, A298602.

nmax = 55; CoefficientList[Series[(1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[k, (-1)^(k/#) # &, PrimeQ[#] || # == 1 &] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 55}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * A048165(k).

cp's OEIS Frontend

A000607 Number of partitions of n into prime parts.

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A000586 Number of partitions of n into distinct primes.

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A292561 Expansion of Product_{k>=1} (1 - mu(k)^2*x^k), where mu() is the Moebius function (A008683).

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A298602 Expansion of (1 - x)*Product_{k>=1} (1 - x^prime(k)).

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A350514 Maximal coefficient of Product_{j=1..n} (1 - x^prime(j)).

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A298159 G.f.: Product_{k>=1} (1-prime(k)*x^prime(k)).

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A184082 Decimal expansion of product_{p=primes} (1-1/2^p).

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