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A000607 Number of partitions of n into prime parts.
(Formerly M0265 N0093)
53
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

a(n) gives the number of values of k for which A001414(k) = n. - Howard A. Landman, Sep 25 2001

REFERENCES

Andrews, George E.; Knopfmacher, Arnold; and Knopfmacher, John; Engel expansions and the Rogers-Ramanujan identities. J. Number Theory 80 (2000), 273-290. See Eq. 2.1.

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).

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.

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.

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, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

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.

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).

Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580

H. Havermann: Tables of sum-of-prime-factors sequences (including sequence lengths, i.e. number of prime-parts partitions, for the first 50000).

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.

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(n-k). - Vladeta Jovovic, Aug 27 2002

G.f. 1/product(1-x^prime(k),k=1..infty).

See the partition arrays A116864 and A116865.

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

t1:=mul(1/(1-q^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] [From Robert G. Wilson v, Jul 23 2010]

Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&]], {n, 0, 60}] (* Harvey P. Dale, Apr 22 2012 *)

PROG

(PARI) A000607(n, m)={local(p); (m==1 | n<3) & return(1-n%2); if( m, A000607[n, m] & return(A000607[n, m]); m>(p=primepi(n)) & A000607[n, m=p] & return(A000607[n, m]), A000607=matrix(n, m=primepi(n))); A000607[n, m]=sum(i=0, n\p=prime(m), A000607(n-i*p, m-1))} - M. F. Hasler, Jan 22 2008

(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

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.

Row sums of array A116865.

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.

STATUS

approved

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Last modified August 21 23:54 EDT 2014. Contains 245918 sequences.