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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.
1
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
1,1
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).
LINKS
EXAMPLE
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.
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:
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 05 2020
STATUS
approved