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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 (list; graph; refs; listen; history; text; internal format)
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 an 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

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

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

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

Sequence in context: A201446 A220772 A242135 * A135559 A254449 A131478

Adjacent sequences:  A332978 A332979 A332980 * A332982 A332983 A332984

KEYWORD

nonn

AUTHOR

Michel Lagneau, Mar 05 2020

STATUS

approved

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Last modified June 20 13:02 EDT 2021. Contains 345164 sequences. (Running on oeis4.)