A390933 Primes that are (product minus sum) of a sequence of consecutive primes.

7, 23, 59, 191, 193, 839, 1129, 1439, 1931, 4969, 5039, 8447, 11447, 23399, 26891, 29989, 36479, 41579, 46619, 57119, 59999, 77279, 110879, 163199, 215353, 232307, 323759, 370871, 392761, 414731, 470579, 521267, 566999, 606791, 664199, 678971, 776159, 824459, 835379, 879839, 919631, 1183739

Offset

1,1

Comments

Primes of the form Product_{k=i..j} prime(k) - Sum_{k=i..j} prime(k).

j - i is always odd.

Links

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

Example

a(7) = 1129 is a term because 3, 5, 7, 11 are consecutive primes and 1129 = 3*5*7*11 - (3+5+7+11) is prime.

Maple

N:= 2000; P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
M:= P[-2]^2-1:
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}; Q[v]:= [i, j] 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-ss:
            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(), 42))) # Michael S. Branicky, Nov 30 2025

Crossrefs

Primes in A387946. Contains A096345, A390916 and A390935. Cf. A391078.

Sequence in context: A037165 A126284 A140096 * A096345 A211644 A077037

Adjacent sequences: A390930 A390931 A390932 * A390934 A390935 A390936

Keyword

nonn

Author

Will Gosnell and Robert Israel, Nov 24 2025

Status

approved