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Search: id:A073610
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| A073610 |
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Number of primes of the form n-p where p is a prime. |
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10
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| 0, 0, 0, 1, 2, 1, 2, 2, 2, 3, 0, 2, 2, 3, 2, 4, 0, 4, 2, 4, 2, 5, 0, 6, 2, 5, 0, 4, 0, 6, 2, 4, 2, 7, 0, 8, 0, 3, 2, 6, 0, 8, 2, 6, 2, 7, 0, 10, 2, 8, 0, 6, 0, 10, 2, 6, 0, 7, 0, 12, 2, 5, 2, 10, 0, 12, 0, 4, 2, 10, 0, 12, 2, 9, 2, 10, 0, 14, 0, 8, 2, 9, 0, 16, 2, 9, 0, 8, 0, 18, 2, 8, 0, 9, 0, 14, 0, 6
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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a(p) = 2 if p-2 is a prime else a(p) = 0. If n = 2p, p is a prime then a(n) is odd else a(n) is even. As p is counted only once and if q and n-q both are prime then the count is increased by 2. ( Analogous to the fact that perfect squares have odd number of divisors).
a(2k+1) = 2 if (2k-1) is prime, else a(2k+1)=0 (for any k). This sequence can be used to re-describe a couple of conjectures: the Goldbach conjecture == a(2n) > 0 for all n>=2; twin primes conjecture == for any n, there is a prime p>n s.t. a(p)>0.
Number of ordered ways of writing n as the sum of two primes.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
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G.f.: (Sum_{k>0} x^prime(k))^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 12 2005
Convolution of "a(n)=1 if n prime, 0 otherwise" (A010051) with itself. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 18 2006
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EXAMPLE
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a(16) = 4 as there are 4 primes 3,5,11 and 13 such that 16-3,16-5,16-11and 16-13 are primes.
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MAPLE
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for i from 1 to 500 do a[i] := 0:j := 1:while(ithprime(j)<i) do if(isprime(i-ithprime(j))=true) then a[i] := a[i]+1:fi:j := j+1:od:od:seq(a[k], k=1..500);
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CROSSREFS
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Cf. A061358, A065577, A107318.
Cf. A098238
Sequence in context: A029252 A094876 A144159 this_sequence A085693 A067995 A135551
Adjacent sequences: A073607 A073608 A073609 this_sequence A073611 A073612 A073613
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 05 2002
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EXTENSIONS
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Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs) and Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Aug 06 2002
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