A340465 Primes of the form prime(i)*prime(i+1)+prime(i+2)*prime(i+3)+...+prime(k-1)*prime(k).

41, 313, 2137, 6569, 7853, 10133, 10847, 12401, 13757, 14747, 17569, 17911, 24001, 24049, 27901, 31307, 38729, 43177, 43961, 44819, 51607, 69191, 81517, 88379, 104683, 107099, 130631, 137177, 138239, 145967, 154487, 154723, 158777, 162947, 175463, 184409, 192853, 196169, 232499, 243137, 261983

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 08 2021

Keywords

• nonn

Comments

A prime that has more than one expression of the given form is included only once. The first such prime is a(14353) = 6858604873 = 1979*1987+...+7109*7121 = 19949*19961+...+20231*20233.

Examples

			a(1) = 2*3+5*7 = 41.
a(2) = 3*5+7*11+13*17 = 313.
a(3) = 17*19+23*29+31*37 = 2137.
a(4) = 5*7+11*13+17*19+23*29+31*37+41*43+47*53 = 6569.
a(5) = 41*43+47*53+59*61 = 7853.
		

Links

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

Crossrefs

Includes A340464.

Programs

Maple

S1:= [0,seq(ithprime(2*i)*ithprime(2*i+1),i=1..100)]:
P1:= ListTools:-PartialSums(S1):
S2:= [0,seq(ithprime(2*i-1)*ithprime(2*i),i=1..100)]:
P2:= ListTools:-PartialSums(S2):
M:= 2*max(S1):
S:= select(t -> t < M and isprime(t), {seq(seq(P1[i]-P1[j],j=i mod 2 + 1 .. i-2,2),i=1..101)} union {seq(seq(P2[i]-P2[j],j=i mod 2 + 1..i-2,2),i=1..101)} union {seq(P2[i],i=1..101,2)}):
sort(convert(S,list));

Python

from sympy import isprime, nextprime, prime
def sp2(lst):
  ans = 0
  for i in range(0, len(lst), 2): ans += lst[i]*lst[i+1]
  return ans
def aupto(nn):
  alst, i = [], 1
  while True:
    consec2i = [prime(j+1) for j in range(2*i)]; sp = sp2(consec2i)
    if sp > nn: break
    while sp <= nn:
      if isprime(sp): alst.append(sp)
      consec2i = consec2i[1:] + [nextprime(consec2i[-1])]; sp = sp2(consec2i)
    i += 1
  return sorted(alst)
print(aupto(261983)) # Michael S. Branicky, Jan 08 2021