A129363 -id:A129363 - cp's OEIS frontend

A172366 Number of partitions of prime(n) into the sum of two semiprimes.

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 3, 2, 2, 4, 4, 3, 5, 6, 4, 6, 5, 5, 7, 4, 7, 6, 7, 7, 5, 5, 8, 8, 8, 8, 9, 10, 9, 7, 13, 10, 10, 11, 13, 10, 10, 10, 11, 15, 18, 13, 10, 13, 14, 16, 12, 18, 12, 15, 12, 11

Offset: 1

Juri-Stepan Gerasimov, Nov 20 2010

nonn

			a(8)=2 because prime(8)=19=4(semiprime)+15(semiprime)=9(semiprime)+10(semiprime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A129363, A175933,

Table[Total[If[PrimeOmega[#]=={2,2},1,0]&/@Table[{x-n,n},{n,x/2}]],{x, Prime[ Range[60]]}] (* Harvey P. Dale, Dec 21 2015 *)

a(n) = A072931(A000040(n)).

Corrected and extended by D. S. McNeil, Nov 20 2010

A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2, 0, 0, 1, 1, 3, 1, 1, 0, 0, 2, 3, 2, 0, 0, 1, 2, 3, 2, 1, 0, 0, 2, 3, 3, 0, 0, 0, 1, 4, 3, 1, 0, 0, 2, 3, 3, 2, 0, 0, 1, 4, 4

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(31) = 2 because we have [31] and [19, 7, 5].

Eric Weisstein's World of Mathematics, Twin Primes
Index entries for related partition-counting sequences

Cf. A000586, A000607, A001097, A006512, A077608, A129363, A283875, A283876, A299196.

nmax = 105; CoefficientList[Series[Product[1 + Boole[PrimeQ[k] && PrimeQ[k - 2]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A006512(k)).

A318202 Prime numbers p such that p-3 cannot be written as sum of two twin prime numbers (not necessarily forming a pair of twin primes).

Original entry on oeis.org

2, 3, 5, 7, 97, 101, 521, 787, 907, 911, 1117, 1151, 1361, 3251, 4211

Offset: 1

Dimitris Valianatos, Aug 21 2018

Conjecture 1: All prime numbers >= 11 can be written as sum of 3 twin prime numbers.
Conjecture 2: All prime numbers aside from the 15 terms given here can be written as a sum of three twin prime numbers with (at least) one of them equal to 3.
If the sequence is finite then there are infinitely many twin primes.
For the terms in this sequence, the lexicographically least partitions into three twin primes are: 97=5+19+73, 101=11+17+73, 521=11+197+313, 787=5+139+643, 907=5+19+883, 911=11+17+883, 1117=5+19+1093, 1151=11+107+1033, 1361=11+29+1321, 3251=11+71+3169, 4211=11+41+4159.
a(16) > 10^6 if it exists. - Amiram Eldar, Dec 06 2018

			a(6) = 101 because 101 - 3 = 98 and (98 - 73 = 25, 98 - 71 = 27), (98 - 61 = 37, 98 - 59 = 39), ..., (98 - 5 = 93, 98 - 3 = 95) aren't twin primes.

David A. Corneth, PARI prog

Cf. A000040, A001097, A007534, A129363, A321221.

p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* Amiram Eldar, Nov 15 2018 *)

{forprime(n=2,10^4,p=n-3;forprime(t1=2,n,forprime(t2=t1,n,t12=t1+t2; if((isprime(t1-2)||isprime(t1+2))&&(isprime(t2-2)||isprime(t2+2)), if(t12==p,break(2)))));if(t12==2*n,print1(n", ")))}

isok(p) = {if (isprime(p), p -= 3; forprime(q = 2, p, if (isprime(r=p-q), if ((isprime(r+2) || isprime(r-2)) && (isprime(q-2) || isprime(q+2)), return (0)););); return (1));} \\ Michel Marcus, Dec 05 2018

\\ See Corneth link \\ David A. Corneth, Dec 05 2018

2,3,5,7 prepended by David A. Corneth, Dec 05 2018

A274001 Even numbers with a unique resolution as the sum of two primes, each of which has a twin.

Original entry on oeis.org

6, 8, 12, 28, 40, 52, 56, 68, 124, 128, 136, 172, 176, 188, 226, 262, 266, 304, 308, 394, 396, 398, 412, 416, 442, 446, 484, 488, 544, 548, 556, 560, 608, 634, 638, 668, 682, 686, 694, 696, 698, 724, 728, 736, 740, 754, 758, 772, 776, 802, 806, 874, 878, 934

Offset: 1

Thomas Curtright, Jun 06 2016

nonn

The sequence is infinite only if the number of twin primes is infinite.

Note that not all even integers can be written as the sum of two twins (e.g. 94, 96, 98,...).

			6 = 3 + 3 is an element since (3,5) are twins, as is 8 = 5 + 3.
10 = 7 + 3 = 5 + 5 is not an element, since it is not uniquely resolved, even though the two resolutions both involve primes with twins.

Cf. A129363, A007534.

ok[n_] := 1 == Length@ IntegerPartitions[n, {2}, Select[Prime@ Range@ PrimePi@ n, Or @@ PrimeQ[# + {-2, 2}] &]]; Select[2 Range[500], ok] (* Giovanni Resta, Jun 06 2016 *)

a(7)-a(54) from Giovanni Resta, Jun 06 2016

A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime.

Original entry on oeis.org

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

Offset: 1

Harry E. Neel, Feb 18 2020

nonn

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).

			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.

Cf. A000040, A001097, A002375, A129363.

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

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A172366 Number of partitions of prime(n) into the sum of two semiprimes.

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A299197 Number of partitions of n into distinct parts that are greater of twin primes (A006512).

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A318202 Prime numbers p such that p-3 cannot be written as sum of two twin prime numbers (not necessarily forming a pair of twin primes).

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A274001 Even numbers with a unique resolution as the sum of two primes, each of which has a twin.

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A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime.

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PARI