A048165 -id:A048165 - 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

A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 06 2018

sign

a(n) is the number of prime divisors p|n such that n/p is odd, minus the number of prime divisors p|n such that n/p is even.

			The prime divisors of 12 are 2, 3. We see that 12/2 = 6, 12/3 = 4. None of those are odd, but both of them are even, so a(12) = -2.
The prime divisors of 30 are {2,3,5} with quotients {15,10,6}. One of these is odd and two are even, so a(30) = 1 - 2 = -1.

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

Cf. A000005, A000607, A001221, A008683, A048165, A083399, A088705, A106404, A205745, A280954.

a:= n-> -add((-1)^(n/i[1]), i=ifactors(n)[2]):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 07 2018
# Alternative
N:= 1000: # to get a(0)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  R:= [seq(i,i=p..N,p)];
  W:= ;
  V[R]:= V[R]+W;
od:
[0,seq(V[i],i=1..N)]; # Robert Israel, Jun 07 2018

Table[Sum[If[PrimeQ[d], (-1)^(n/d - 1), 0], {d, Divisors[n]}], {n, 30}]

a(n) = -Sum_{p|n prime} (-1)^(n/p).
From Robert Israel, Jun 07 2018: (Start)
If n is odd, a(n) = A001221(n).
If n == 2 (mod 4), a(n) = 2 - A001221(n).
If n == 0 (mod 4) and n > 0, a(n) = -A001221(n). (End)
L.g.f.: log(Product_{k>=1} (1 + x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018

A184198 Number of partitions of n into an even number of primes.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 5, 8, 7, 11, 10, 15, 13, 20, 17, 26, 23, 34, 29, 43, 38, 55, 49, 69, 62, 88, 78, 109, 97, 135, 122, 167, 150, 205, 186, 251, 227, 306, 277, 371, 337, 448, 407, 539, 492, 647, 591, 773, 707, 922, 845, 1096, 1005, 1298, 1193, 1535, 1412, 1809, 1667, 2127

Offset: 0

R. J. Mathar, Jan 10 2011

			n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=11 are even, namely 11+7, 13+5, 5+5+5+3, 7+5+3+3, 3+3+3+3+3+3, 7+7+2+2, 11+3+2+2, 5+3+3+3+2+2, 5+5+2+2+2+2, 7+3+2+2+2+2, 3+3+2+2+2+2+2+2.
The remaining A184199(18)=8 are odd.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000607, A048165, A184199, A184171.

Table[Count[IntegerPartitions[n],?(AllTrue[#,PrimeQ]&&EvenQ[Length[ #]]&)], {n,0,70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Feb 16 2018 *)

parts(n, pred, y)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ Andrew Howroyd, Dec 28 2017

a(n) = (A000607(n)+A048165(n))/2.

a(31)-a(69) corrected by Andrew Howroyd, Dec 28 2017

A184199 Number of partitions of n into an odd number of primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 4, 3, 5, 4, 7, 6, 10, 8, 13, 11, 17, 15, 23, 20, 29, 26, 38, 34, 49, 43, 62, 55, 78, 69, 97, 88, 122, 109, 150, 135, 186, 167, 227, 205, 277, 251, 337, 306, 407, 371, 492, 448, 591, 539, 707, 647, 845, 773, 1005, 922, 1193, 1096, 1412, 1298, 1667, 1535, 1963, 1809, 2305, 2127

Offset: 0

R. J. Mathar, Jan 10 2011

			n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=8 are odd, namely  11+5+2, 13+3+2, 5+5+3+3+2, 7+3+3+3+2, 7+5+2+2+2, 3+3+3+3+2+2+2, 5+3+2+2+2+2+2, 2+2+2+2+2+2+2+2+2.
The remaining A184198(18)=11 are even.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..200 from Jean-François Alcover)

Cf. A000607, A048165, A184198, A184172.

a[n_] := a[n] = Count[IntegerPartitions[n, All, Prime[Range[PrimePi[n]]]], p_ /; OddQ[Length[p]]];
Reap[Do[Print[n, " ", a[n]]; Sow[a[n]], {n, 0, 200}]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)

parts(n, pred, y)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
{my(n=80); (Vec(parts(n, isprime, 1)) - Vec(parts(n, isprime, -1)))/2} \\ Andrew Howroyd, Dec 28 2017

a(n) = (A000607(n)-A048165(n))/2.

a(31)-a(70) corrected by Andrew Howroyd, Dec 28 2017

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A298160 G.f.: Product_{k>=1} 1/(1+prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, -2, -3, 4, 1, 1, -9, 13, -9, 20, -38, 76, -75, 65, -323, 378, -197, 805, -1394, 1635, -2513, 3175, -5442, 11135, -12570, 12526, -33357, 51563, -46460, 93551, -155750, 186650, -313241, 421641, -620393, 1131820, -1321220, 1663951, -3559915, 5011036, -5207116

Offset: 0

Seiichi Manyama, Jan 14 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..6278

Cf. A002099, A048165, A298159.

CoefficientList[Series[Product[1/(1+Prime[k]x^Prime[k]),{k,50}],{x,0,50}],x] (* Harvey P. Dale, Mar 24 2020 *)

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

Ilya Gutkovskiy, Apr 03 2018

sign

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.

Index entries for sequences related to partitions

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

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]

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

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 4, 6, 5, 4, 7, 10, 8, 7, 11, 15, 13, 12, 17, 22, 19, 18, 25, 30, 28, 26, 35, 42, 39, 38, 49, 59, 56, 54, 69, 81, 77, 76, 94, 110, 105, 105, 127, 147, 141, 142, 171, 195, 189, 190, 227, 257, 250, 254, 299, 335, 328, 334, 390, 432

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

			O.g.f.: 1/((1 - x)(1 + x^2)(1 - x^3)(1 - x^5)(1 + x^6)(1 - x^7)(1 + x^10)...).

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1+(-x)^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 07 2018

sign

Cf. A000607, A001597, A007916, A048165, A081362, A091050, A303707, A304779, A304817, A305614, A305630-A305635.

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Sum[x^p/(1+x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

cp's OEIS Frontend

A000607 Number of partitions of n into prime parts.

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A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

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A184198 Number of partitions of n into an even number of primes.

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A184199 Number of partitions of n into an odd number of primes.

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A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

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A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

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A298160 G.f.: Product_{k>=1} 1/(1+prime(k)*x^prime(k)).

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