login
Sphenic numbers that are also the sum of three consecutive primes.
1

%I #25 Nov 25 2022 08:38:41

%S 1015,1533,1645,2233,2737,2915,3219,3515,3745,3815,4301,4503,4565,

%T 4623,4697,4921,5289,5621,6055,6095,6213,6251,6409,7055,7347,7657,

%U 7847,8099,8455,8569,8687,8729,9499,9581,9955,10105,10153,10295,10735,11155,11297,11315,11803,12665,12805,12845

%N Sphenic numbers that are also the sum of three consecutive primes.

%C Intersection of A007304 and A034961.

%C Includes 15*p where p, 5*p-14, 5*p-2 and 5*p+16 are consecutive primes. Dickson's conjecture implies there are infinitely many such terms. - _Robert Israel_, Nov 24 2022

%H Robert Israel, <a href="/A335969/b335969.txt">Table of n, a(n) for n = 1..10000</a>

%e 1015 = A007304(140) = A034961(67), 1533 = A007304(226) = A034961(96).

%p P:= select(isprime, [seq(i,i=3..10^4,2)]):

%p P3:= P[1..-3] + P[2..-2] + P[3..-1]:

%p filter:= proc(t) local F; F:= ifactors(t)[2]; nops(F) = 3 and F[1,2]=1 and F[2,2] = 1 and F[3,2]=1 end proc:

%p select(filter, P3); # _Robert Israel_, Nov 24 2022

%t Intersection[ Select[Range[105, 40000,2], 3 == PrimeOmega[#] == PrimeNu[#] &], Total /@ Partition[Prime[Range[40000]], 3, 1]]

%Y Cf. A007304, A034961.

%K nonn

%O 1,1

%A _Zak Seidov_, Jul 04 2020