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 A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime. 0
 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 3, 5, 4, 3, 5, 3, 4, 5, 3, 5, 6, 2, 4, 6, 4, 4, 7, 4, 5, 7, 5, 4, 7, 4, 4, 7, 3, 5, 7, 4, 4, 8, 6, 6, 9, 5, 6, 9, 4, 5, 8, 3, 6, 8, 4, 2, 8, 7, 7, 10, 5, 5, 8, 4, 7, 10, 4, 7, 9, 3, 4, 11, 9, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is the number of ways that a twin prime may be summed with another prime to provide the even number 2n.  It is not known if summing primes with twin primes will provide every even number greater than 4 (or greater than or equal to 6). LINKS EXAMPLE a(6) = 1 because the only way to express 2*6 = 12 as the sum of two primes, one of which is a twin prime, is 5+7. (Since the sequence counts unordered sums, 7+5 is not counted as distinct from 5+7.) Also, 37+61 = 98 is a valid sum, 61 being a part of a twin prime pair; while 37+47 = 84 is not a valid sum because neither 37 nor 47 is a part of a twin prime pair. PROG (PARI) istwin(p) = isprime(p-2) || isprime(p+2); a(n) = {n *= 2; my(nb = 0, q, v=[]); forprime(p=2, n, q = n-p; if ((q>=p) && isprime(q) && (istwin(p) || istwin(q)), nb++; v= concat(v, p)); ); nb; } \\ Michel Marcus, Feb 28 2020 CROSSREFS Cf. A000040, A001097, A002375, A129363. Sequence in context: A230197 A094570 A225638 * A230443 A254610 A002375 Adjacent sequences:  A332653 A332654 A332655 * A332657 A332658 A332659 KEYWORD nonn AUTHOR Harry E. Neel, Feb 18 2020 STATUS approved

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Last modified April 20 11:43 EDT 2021. Contains 343135 sequences. (Running on oeis4.)