Search: id:A157931
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%I A157931
%S A157931 4,6,9,10,14,15,21,22,25,26,33,34,38,39,46,49,55,58,62,69,74,82,85,86,
%T A157931 91,94,106,111,115,118,122,129,133,134,141,142,146,158,159,166,169,178,
%U A157931 183,194,201,202,206,213,214,218,226,235,253,254,259,262,265,274,278
%N A157931 Numbers that are both the sum and the product of two primes.
%C A157931 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
%C A157931 The first few terms of A001358: Semiprimes, not members of A157931 are: 
               35, 51, 57, 65, 77, 87, 93, 95, ..., . [From Robert G. Wilson, v 
               (rgwv(AT)rgwv.com), Mar 15 2009]
%H A157931 Robert G. Wilson, v, <a href="b157931.txt">Table of n, a(n) for n=1..1096</
               a>.<br>
%F A157931 A014091 INTERSECT A001358. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 15 2009]
%e A157931 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).
%p A157931 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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 15 2009]
%t A157931 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@# &] [From Robert G. 
               Wilson, v (rgwv(AT)rgwv.com), Mar 15 2009]
%Y A157931 Cf. A001358, A005843, A052147, A062721.
%Y A157931 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. [From Zak Seidov (zakseidov(AT)yahoo.com), 
               Mar 15 2009]
%Y A157931 Sequence in context: A129336 A103607 A108574 this_sequence A046368 A113433 
               A115654
%Y A157931 Adjacent sequences: A157928 A157929 A157930 this_sequence A157932 A157933 
               A157934
%K A157931 easy,nonn,nice
%O A157931 1,1
%A A157931 William Weeks (dach(AT)kuci.org), Mar 09 2009
%E A157931 Edited by N. J. A. Sloane, Mar 14 2009
%E A157931 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Robert G. 
               Wilson, v (rgwv(AT)rgwv.com), Mar 15 2009

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