A089268 -id:A089268 - cp's OEIS frontend

A062721 Numbers k such that k is a product of two primes and k-2 is prime.

4, 9, 15, 21, 25, 33, 39, 49, 55, 69, 85, 91, 111, 115, 129, 133, 141, 159, 169, 183, 201, 213, 235, 253, 259, 265, 295, 309, 319, 339, 355, 361, 381, 391, 403, 411, 445, 451, 469, 481, 489, 493, 501, 505, 511, 543, 559, 565, 573, 579, 589, 633, 649, 655, 679

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001

This sequence is a subsequence of A107986, which only requires k to be composite. The first term in that sequence which is not in this sequence is 45, a number with three prime factors. - Alonso del Arte, May 03 2014

Zak Seidov, Table of n, a(n) for n = 1..10000.

Cf. A006512, A001358, A043326.

Cf. A063637, A046315, A089268.

Select[ Range[ 2, 1500 ], Plus @@ Last@Transpose@FactorInteger[ # ] == 2 && PrimeQ[ # - 2 ] & ]
Select[Range[700], PrimeOmega[#] == 2 && PrimeQ[# - 2]&] (* Harvey P. Dale, Mar 25 2013 *)

{ n=0; for (m=1, 10^9, a=prime(m) + 2; f=factor(a)~; if ((length(f)==1 && f[2, 1]==2) || (length(f)==2 && f[2, 1]==1 && f[2, 2]==1), write("b062721.txt", n++, " ", a); if (n==10000, break)) ) } \\ Harry J. Smith, Aug 09 2009

A087942 Number of partitions of n into as many primes as n has prime factors.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 7, 1, 3, 7, 3, 1, 2, 1, 11, 1, 4, 0, 15, 1, 2, 1, 21, 1, 3, 1, 4, 12, 4, 1, 26, 1, 5, 0, 4, 1, 33, 1, 38, 0, 4, 1, 41, 1, 3, 19, 137, 0, 5, 1, 6, 1, 2, 1, 61, 1, 5, 22, 5, 0, 5, 1, 67, 24, 5, 1, 81, 1, 5, 0, 96, 1, 93, 1, 9, 0

Offset: 1

Reinhard Zumkeller, Oct 27 2003

nonn

Conjecture, for m>1: a(m)=0 iff n is an odd semiprime such that m-2 is not prime, i.e. m=A089268(k) for some k. - Reinhard Zumkeller, Oct 28 2003

			n=20 = 2*2*5 = 13+5+2 = 11+7+2, all other partitions into 3 primes have fewer than or more than 3 parts, therefore a(20)=2.

Eric Weisstein's World of Mathematics, Prime Partition.
Index entries for sequences related to Goldbach conjecture

Cf. A001222, A000607, A051034, A004526.

Table[Count[IntegerPartitions[n,{PrimeOmega[n]}],?(AllTrue[#,PrimeQ]&)],{n,100}] (* _Harvey P. Dale, Jul 26 2023 *)

A173664 Sums of 2 primes that are not product of 2 primes.

Original entry on oeis.org

5, 7, 8, 12, 13, 16, 18, 19, 20, 24, 28, 30, 31, 32, 36, 40, 42, 43, 44, 45, 48, 50, 52, 54, 56, 60, 61, 63, 64, 66, 68, 70, 72, 73, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 103, 104, 105, 108, 109, 110, 112, 114, 116, 120, 124, 126

Offset: 1

Juri-Stepan Gerasimov, Nov 24 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A157931, A089268.

a:= proc(n) option remember; local k;
      if n=1 then 5
      else for k from a(n-1)+1 do
             if add (i[2], i=ifactors(k)[2])=2 then next fi;
             if irem (k, 2)=0 or isprime (k-2) then break fi
           od; k
      fi
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Nov 24 2010

Select[Union[Flatten[Table[Prime[i] + Prime[j], {i, 25}, {j, 25}]]], PrimeOmega[#] != 2 &] (* Alonso del Arte, Feb 08 2013 *)

is(n)=if(n%2,isprime(n-2)&&bigomega(n)!=2,n>2&&!isprime(n/2)) \\ above 4 * 10^18, conditional on the Goldbach conjecture Charles R Greathouse IV, Feb 09 2013

A014091 \ A001358. - R. J. Mathar, Nov 24 2010

More terms from Alois P. Heinz, Nov 24 2010

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A062721 Numbers k such that k is a product of two primes and k-2 is prime.

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A087942 Number of partitions of n into as many primes as n has prime factors.

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A173664 Sums of 2 primes that are not product of 2 primes.

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