A370687 a(n) is the first number that is the sum of k successive semiprimes for 1 <= k <= n.

4, 10, 134, 2045, 2705, 16626281

Offset: 1

Author

Robert Israel, Feb 27 2024

Keywords

• nonn
• more

Comments

a(7) > 5 * 10^8 if it exists.

Examples

			a(3) = 134 because 134 = 2 * 67 is a semiprime, the sum of two successive semiprimes 65 = 5 * 13 and 69 = 3 * 23, and the sum of three successive semiprimes 39 = 3 * 13, 46 = 2 * 23, 49 = 7 * 7, and is the least such number.
		

Crossrefs

Cf. A001358, A092192, A370162, A370685.

Programs

Maple

N:= 2*10^7: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
nP:= nops(P):
SP:= 0:
for i from 1 while P[i]^2 <= N do
  m:= ListTools:-BinaryPlace(P, N/P[i]);
  SP:= SP, op(P[i]*P[i..m]);
od:
SP:= sort([SP]):
SS:= ListTools:-PartialSums(SP):
V:= Vector(6):
SI:= Vector(6):
II:= Vector(6,1):
for i from 1 to 6 do SI[i]:= SS[i+1]-SS[1] od:
count:= 1: V[1]:= 4: m:= 4: im:= {1}:
while count < 6 do
  for j in im do
    II[j]:= II[j]+1;
    SI[j]:= SS[II[j]+j] - SS[II[j]];
  od;
  m:= min(SI);
  im:= select(j -> SI[j] = m, {$1..20});
  for k from 1 to 20 while {$1..k} subset im do
    if V[k] = 0 then V[k]:= m; count:= count+1 fi
  od;
od:
convert(V,list);