A175931 -id:A175931 - cp's OEIS frontend

A129363 Number of partitions of 2n into the sum of two twin primes.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 4, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 2, 0, 0, 0, 2, 4, 3, 2, 2, 2, 4, 6, 3, 3, 5, 3, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 5, 3, 2, 4, 3, 3, 4, 2, 2, 4, 2, 3, 6, 3, 1, 2, 1, 3, 6, 4, 2, 2, 1, 2, 4, 3, 4, 6, 4, 4, 5, 3, 6, 12

Offset: 1

T. D. Noe, Apr 11 2007

nonn

a(n/2)=0 for the n in A007534. The logarithmic plot of this sequence seems very regular after 200000 terms.

			a(11)=3 because 22 = 3+19 = 5+17 = 11+11.

T. D. Noe, Table of n, a(n) for n=1..10000
James Grime and Brady Haran, Goldbach Conjecture (but with TWIN PRIMES), Numberphile video (2024)
T. D. Noe, Logarithmic plot of 10^6 terms

Cf. A175931 (n for which a(n-1), a(n), a(n+1) are equal).

Cf. A001097, A002375, A007534.

a129363 n = sum $ map (a164292 . (2*n -)) $ takeWhile (<= n) a001097_list
-- Reinhard Zumkeller, Feb 03 2014

nn=1000; tw=Select[Prime[Range[PrimePi[nn]]], PrimeQ[ #+2]&]; tw=Union[tw,tw+2]; tc=Table[0,{nn}]; tc[[tw]]=1; Table[cnt=0; k=1; While[tw[[k]]<=n/2, cnt=cnt+tc[[n-tw[[k]]]]; k++ ]; cnt, {n,2,nn,2}]

a(n) = Sum_{i=1..n} ceiling((A010051(i+2) + A010051(i-2))/2) * ceiling((A010051(2n-i+2) + A010051(2n-i-2))/2) * A010051(2n-i) * A010051(i). - Wesley Ivan Hurt, Jan 30 2014

a(n) = sum(A164292(2*n - A001097(k)): A001097(k) <= n). - Reinhard Zumkeller, Feb 03 2014

Comment converted to crossref by Klaus Brockhaus, Oct 27 2010

A175940 Number of ways of writing n=p+f with p a prime and f a factorial.

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2010

nonn

Number of partitions of n into the sum of a prime number and a factorial number. Number of decompositions of n into an unordered sum of a prime number and a factorial number.

			a(29)=2 because 29 has two prime + factorial representations, 5+4! and 23+3!.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000142, A062602, A175931, A175933.

a:= proc(n) local t,k;
       t:= 0;
       for k while k! < n do
         if isprime(n-k!) then t:= t+1 fi
       od;
       t
end proc:
seq(a(n), n=1..100); # Robert Israel, Oct 13 2014

a[n_] := Module[{t = 0, k}, For[k = 1, k! < n, k++, If[PrimeQ[n - k!] , t++]]; t];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023, after Robert Israel *)

a(n) = c=0;for(i=1,n,if(isprime(n-i!),c++));c
vector(100,n,a(n)) \\ Derek Orr, Oct 13 2014

Edited and entries checked by D. S. McNeil, Nov 26 2010

cp's OEIS Frontend

A129363 Number of partitions of 2n into the sum of two twin primes.

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