A157931 Numbers that are both the sum and the product of two primes.

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 55, 58, 62, 69, 74, 82, 85, 86, 91, 94, 106, 111, 115, 118, 122, 129, 133, 134, 141, 142, 146, 158, 159, 166, 169, 178, 183, 194, 201, 202, 206, 213, 214, 218, 226, 235, 253, 254, 259, 262, 265, 274, 278

Offset: 1

Author

William Weeks (dach(AT)kuci.org), Mar 09 2009

Keywords

• easy
• nonn
• nice

Comments

Assuming the Goldbach conjecture, this is A001358 intersect (A005843 union A052147), since an odd number n is the sum of two primes iff n-2 is prime. - N. J. A. Sloane, Mar 14 2009

The first few terms of A001358: Semiprimes, not members of A157931 are: 35, 51, 57, 65, 77, 87, 93, 95, ..., . - Robert G. Wilson v, Mar 15 2009

Examples

			For the numbers up to 100, the solutions are 4 = (2+2) = (2*2); 6 = (3+3) = (2*3); 9 = (2+7) = (3*3); 10 = (3+7) = (2*5); 14 = (3+11) = (2*7); 15 = (2+13) = (3*5); 21 = (2+19) = (3*7); 22 = (3+19) = (2*11); 25 = (2+23) = (5*5); 26 = (3+23) = (2*13); 33 = (2+31) = (3*11); 34 = (3+31) = (2*17); 38 = (7+31) = (2*19); 39 = (2+37) = (3*13); 46 = (3+43) = (2*23); 49 = (2+47) = (7*7); 55 = (2+53) = (5*11); 58 = (5+53) = (2*29); 62 = (3+59) = (2*31); 69 = (2+67) = (3*23); 74 = (3+71) = (2*37); 82 = (3+79) = (2*41); 85 = (2+83) = (5*17); 86 = (3+83) = (2*43); 91 = (2+89) = (7*13); 94 = (5+89) = (2*47).
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1096 terms from Robert G. Wilson v)

Crossrefs

Cf. A001358, A005843, A052147, A062721.

Cf. A043326 Numbers n such that n is a product of two different primes and n - 2 is prime, A062721 Numbers n such that n is a product of two primes and n - 2 is prime. - Zak Seidov, Mar 15 2009

Cf. A014091, A064911, A100962.

Programs

Haskell

a157931 n = a157931_list !! (n-1)
a157931_list = filter ((== 1) . a064911) a014091_list
-- Reinhard Zumkeller, Oct 15 2014

Maple

isA014091 := proc(n) for i from 1 do p := ithprime(i) ; if p > n/2 then RETURN(false); fi; if isprime(n-p) then RETURN(true) ; fi; od: end: isA001358 := proc(n) RETURN(numtheory[bigomega](n) = 2) ; end: for n from 4 to 500 do if isA001358(n) and isA014091(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Mar 15 2009

Mathematica

fQ[n_] := Block[{k = 2}, While[k < n, If[ PrimeQ[n - k], Break[]]; k = NextPrime@k]; k + 1 < n]; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; Select[ Range@ 295, fQ@# && semiPrimeQ@# &] (* Robert G. Wilson v, Mar 15 2009 *)
Select[Union[Flatten[Table[Prime[i] + Prime[j], {i, 50}, {j, 50}]]], PrimeOmega[#] == 2 &] (* Alonso del Arte, Feb 08 2013 *)
Union[Select[Total/@Tuples[Prime[Range[60]],2],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jul 27 2015 *)

Formula

A014091 INTERSECT A001358. - R. J. Mathar, Mar 15 2009

Extensions

Edited by N. J. A. Sloane, Mar 14 2009

Extended by R. J. Mathar and Robert G. Wilson v, Mar 15 2009