A332981 Smallest semiprime m = p*q such that the sum s = p + q can be expressed as an unordered sum of two primes in exactly n ways.

4, 21, 57, 93, 183, 291, 327, 395, 501, 545, 695, 791, 815, 831, 1145, 1205, 1415, 1631, 1461, 1745, 1941, 1865, 2661, 2315, 2615, 2855, 2495, 2285, 3665, 2705, 2721, 3521, 3561, 3351, 3755, 4341, 3545, 4701, 4265, 4881, 3981, 4821, 5601, 5255, 6671, 6041, 4595

Offset: 1

Author

Michel Lagneau, Mar 05 2020

Keywords

• nonn

Comments

The unique square and even term of the sequence is a(1) = 4.

For n = 1, the sequence of semiprimes having a unique decomposition as the sum of two primes begins with 4, 6, 9, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, ... containing the even semiprimes (A100484).

We observe a majority of terms where a(n) == 5 (mod 10).

Examples

			a(11) = 695 because 695 = 5*139 and the sum 5 + 139 = 144 = 5+139 = 7+137 = 13+131 = 17+127 = 31+113 = 37+107 = 41+103 = 43+101 = 47+97 = 61+83 = 71+73. There are exactly 11 decompositions of 144 into an unordered sum of two primes.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..501

Crossrefs

Cf. A001358, A002375, A006881, A023036, A100484, A136244.

Programs

Maple

with(numtheory):
for n from 1 to 50 do:
ii:=0:
for k from 2 to 10^8 while(ii=0) do:
x:=factorset(k):it:=0:
if bigomega(k) = 2
  then
   s:=x[1]+k/x[1]:
    for m from 1 to s/2 do:
     if isprime(m) and isprime(s-m)
      then
       it:=it+1:
       else fi:
     od:
     if it = n
     then
      ii:=1: printf(`%d, `,k):
     else fi:
     fi:
    od:
    od:

PARI

nbp(k) = {my(nb = 0); forprime(p=2, k\2, if (isprime(k-p), nb++););nb;}
a(n) = {forcomposite(k=1, oo, if (bigomega(k)==2, my(x=factor(k)[1,1]); if (nbp(x+k/x)==n, return(k));););} \\ Michel Marcus, Apr 26 2020