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Search: id:A000607
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| A000607 |
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Number of partitions of n into prime parts.
(Formerly M0265 N0093)
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+0
38
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| 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; listen)
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OFFSET
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0,6
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COMMENT
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a(n) gives the number of values of k for which A001414(k) = n. - Howard A. Landman (howard(AT)polyamory.org), Sep 25 2001
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REFERENCES
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R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 203.
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, 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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580
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, Link to a section of The World of Mathematics.
Index entries for sequences related to Goldbach conjecture
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FORMULA
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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 (vladeta(AT)eunet.rs), Aug 27 2002
G.f. 1/product(1-x^prime(k),k=1..infty).
See the partition arrays A116864 and A116865.
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EXAMPLE
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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.
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MATHEMATICA
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CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
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PROGRAM
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(PARI) A000607(n, m)={local(p); (m==1 | n<3) & return(1-n%2); if( m, A607[n, m] & return(A607[n, m]); m>(p=primepi(n)) & A607[n, m=p] & return(A607[n, m]), A607=matrix(n, m=primepi(n))); A607[n, m]=sum(i=0, n\p=prime(m), A000607(n-i*p, m-1))} - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jan 22 2008
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CROSSREFS
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G.f. = 1 / G.f. for A046675. Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586, A000041, A070214.
Cf. A046113 for the ordered (compositions) version.
Cf. A112021, A056768.
Row sums of array A116865.
Cf. A128515.
Sequence in context: A029022 A140953 A112021 this_sequence A114372 A046676 A003114
Adjacent sequences: A000604 A000605 A000606 this_sequence A000608 A000609 A000610
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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