A391078 Primes that are (product + sum) of a sequence of consecutive primes.

11, 23, 47, 167, 227, 251, 359, 479, 719, 1181, 1847, 2111, 2591, 3719, 6719, 7559, 8819, 10607, 12539, 14591, 19319, 27551, 29231, 30071, 31319, 51071, 53819, 68111, 97967, 149759, 155219, 172199, 177239, 195359, 199799, 234239, 273527, 305783, 314711, 339863, 356399, 379451, 417311, 423791

Offset

1,1

Comments

Is 23 the only term in common with A390933, i.e. a prime that is both (product - sum) of a sequence of consecutive primes and (product + sum) of a sequence of consecutive primes?

No other term in common for n <= 10^8. - Michael S. Branicky, Dec 01 2025

Links

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

Example

a(3) = 47 is a term because 47 = 5*7 + 5 + 7 is prime, where 5 and 7 are consecutive primes.

Maple

N:= 1000: # for terms <= N^2
P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
M:= N^2:
R:= {}:
for i from 1 to nP do
  p:= P[i]; s:= P[i];
  for j from i+1 to nP do
    p:= p * P[j]; s:= s + P[j];
    v:= p+s;
    if v > M then break fi;
    if isprime(v) then R:= R union {v} fi;
  od
od:
sort(convert(R, list));

Prog

(Python)
import heapq
from itertools import islice
from sympy import isprime, sieve
def agen(): # generator of terms
    pp, ss, nextcount, alst, oldv = 6, 5, 3, [], -1
    h = [(pp+ss, pp, ss, 1, 2)]
    while True:
        (v, pr, sm, s, l) = heapq.heappop(h)
        if v > oldv and isprime(v):
            yield v
        oldv = v
        if v >= pp:
            pp *= sieve[nextcount]
            ss += sieve[nextcount]
            heapq.heappush(h, (pp+ss, pp, ss, 1, nextcount))
            nextcount += 1
        pr //= sieve[s]; sm -= sieve[s]; s += 1; l += 1; pr *= sieve[l]; sm += sieve[l]
        heapq.heappush(h, (pr+sm, pr, sm, s, l))
    return alst
print(list(islice(agen(), 44))) # Michael S. Branicky, Nov 30 2025

Crossrefs

Cf. A387590, A390933.

Sequence in context: A139834 A100558 A126199 * A096342 A066179 A217566

Adjacent sequences: A391075 A391076 A391077 * A391079 A391080 A391081

Keyword

nonn

Author

Will Gosnell and Robert Israel, Nov 27 2025

Status

approved