

A000607


Number of partitions of n into prime parts.
(Formerly M0265 N0093)


111



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
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OFFSET

0,6


COMMENTS

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


REFERENCES

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. 2428, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 4978 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
D. M. Burton, Elementary Number Theory, 5th ed., McGrawHill, 2002.
L. M. Chawla and S. A. Shad, On a trioset of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 8796.
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).


LINKS

T. D. Noe, Vaclav Kotesovec and H. Havermann, Table of n, a(n) for n = 0..50000 (first 1000 terms from T. D. Noe, terms 100120000 from Vaclav Kotesovec, terms 2000150000 extracted from files by H. Havermann)
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275294.
George E. Andrews, Arnold Knopfmacher, John Knopfmacher, Engel expansions and the RogersRamanujan identities J. Number Theory 80 (2000), 273290. 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, Pages 1530.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 1217. 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 [mathph].
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 trioset of partition functions and their tables", Mathematics of Computation, Vol. 24, No. 110 (Apr., 1970), pp. 490491.
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), 185187.
O. P. Gupta and S. Luthra, Partitions into primes, Proc. Nat. Inst. Sci. India. Part A. 21 (1955), 181184.
R. K. Guy, Letter to N. J. A. Sloane, 19880412 (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]
H. Havermann, Tables of sumofprimefactors sequences (including sequence lengths, i.e. number of primeparts partitions, for the first 50000).
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 nexttoleading 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. 320321.
Ljuben Mutafchiev, A Note on Goldbach Partitions of Large Even Integers, arXiv:1407.4688 [math.NT], 20142015.
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) 109121.
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


FORMULA

Asymptotically a(n) ~ exp(2 Pi sqrt(n/log n) / sqrt(3)) (Ayoub).
a(n) = (1/n)*Sum_{k=1..n} A008472(k)*a(nk).  Vladeta Jovovic, Aug 27 2002
G.f.: 1/Product_{k>=1}(1x^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), p. 109121, 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


EXAMPLE

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.


MAPLE

with(gfun):
t1:=mul(1/(1q^ithprime(n)), n=1..51):
t2:=series(t1, q, 50):
t3:=seriestolist(t2);


MATHEMATICA

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 *)
Comment from Thomas Vogler, Dec 10 2015 (Start): The following code uses the Euler transform. The code caches already computed values and is faster than using the builtin IntegerPartitions[] function.
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}] (End)


PROG

(PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1x^prime(k))) \\ Joerg Arndt, Sep 04 2014
(Haskell)
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
(Sage) [Partitions(n, parts_in=(prime_range(n+1))).cardinality() for n in range(1000)] # Giuseppe Coppoletta, Jul 11 2016
(Python)
from sympy import primefactors
l=[1, 0]
for n in xrange(2, 101):l+=[sum([sum(primefactors(k))*l[n  k] for k in xrange(1, n + 1)])/n, ]
print l # Indranil Ghosh, Jul 13 2017


CROSSREFS

G.f. = 1 / g.f. for A046675. See A046113 for the ordered (compositions) version.
Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586, A000041, A070214, A192541, A112021, A056768, A128515, A000586, A319265, A319266.
Row sums of array A116865 and of triangle A261013.
Sequence in context: A029022 A140953 A112021 * A114372 A046676 A003114
Adjacent sequences: A000604 A000605 A000606 * A000608 A000609 A000610


KEYWORD

easy,nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Maple program fixed by Vaclav Kotesovec, Sep 14 2014


STATUS

approved



