A127269 Suppose the sum of the prime factors of the composites between prime(n) and prime(n+1) is prime. Sequence gives prime(n).

5, 7, 11, 67, 97, 103, 107, 109, 113, 163, 173, 197, 263, 283, 331, 359, 389, 409, 419, 431, 461, 463, 521, 569, 599, 607, 659, 761, 787, 797, 809, 811, 829, 857, 877, 911, 1019, 1039, 1061, 1087, 1093, 1277, 1283, 1289, 1301, 1409, 1427, 1451, 1481, 1627

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

Prime(4) = 7, prime(5) = 11. Sum of prime factors of 8 is 2+2+2 = 6, sum of prime factors of 9 is 3+3 = 6, sum of prime factors of 10 is 2+5= 7; 6+6+7 = 19 is prime, hence prime(4) = 7 is a term.
Prime(19) = 67, prime(20) = 71. Sum of prime factors of 68, 69, 70 is resp. 2+2+17 = 21, 3+23 = 26, 2+5+7 = 14; 21+26+14 = 61 is prime, hence prime(19) = 67 is a term.
Prime(26) = 101, prime(27) = 103. Sum of prime factors of 102 = 2*3*17 is 22, which is composite. Hence prime(26) = 101 is not in the sequence.

Maple

p:= 2: count:= 0: R:= NULL:
while count < 100 do
  q:= nextprime(p);
  v:= add(add(t[1]*t[2], t=ifactors(m)[2]), m=p+1..q-1);
  if isprime(v) then count:= count+1; R:= R, p; fi;
  p:= q;
od:
R; # Robert Israel, Oct 08 2020

Mathematica

spfQ[n_]:=Module[{strt=First[n]+1, end=Last[n]-1}, PrimeQ[Total[Times@@@ Flatten[ FactorInteger[ Range[ strt, end]], 1]]]]; Transpose[Select[ Partition[ Prime[ Range[300]], 2, 1], spfQ]][[1]] (* Harvey P. Dale, May 06 2011 *)

Prog

(Magma) [ p: p in [ NthPrime(k): k in [2..258] ] | IsPrime(&+[ &+[ k[1]*k[2]: k in Factorization(c) ]: c in [p+1..NextPrime(p)-1] ] ) ]; /* Klaus Brockhaus, Mar 29 2007 */

Crossrefs

Contains A338083.

Sequence in context: A045968 A066367 A098036 * A071781 A091509 A027728

Adjacent sequences: A127266 A127267 A127268 * A127270 A127271 A127272

Keyword

nonn

Author

J. M. Bergot, Mar 27 2007

Extensions

Edited, corrected and extended by Klaus Brockhaus, Mar 29 2007

Status

approved