A391078 - OEIS

%I #17 Dec 07 2025 05:57:33

%S 11,23,47,167,227,251,359,479,719,1181,1847,2111,2591,3719,6719,7559,

%T 8819,10607,12539,14591,19319,27551,29231,30071,31319,51071,53819,

%U 68111,97967,149759,155219,172199,177239,195359,199799,234239,273527,305783,314711,339863,356399,379451,417311,423791

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

%C 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?

%C No other term in common for n <= 10^8. - _Michael S. Branicky_, Dec 01 2025

%H Robert Israel, <a href="/A391078/b391078.txt">Table of n, a(n) for n = 1..10000</a>

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

%p N:= 1000: # for terms <= N^2

%p P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):

%p M:= N^2:

%p R:= {}:

%p for i from 1 to nP do

%p   p:= P[i]; s:= P[i];

%p   for j from i+1 to nP do

%p     p:= p * P[j]; s:= s + P[j];

%p     v:= p+s;

%p     if v > M then break fi;

%p     if isprime(v) then R:= R union {v} fi;

%p   od

%p od:

%p sort(convert(R, list));

%o (Python)

%o import heapq

%o from itertools import islice

%o from sympy import isprime, sieve

%o def agen(): # generator of terms

%o     pp, ss, nextcount, alst, oldv = 6, 5, 3, [], -1

%o     h = [(pp+ss, pp, ss, 1, 2)]

%o     while True:

%o         (v, pr, sm, s, l) = heapq.heappop(h)

%o         if v > oldv and isprime(v):

%o             yield v

%o         oldv = v

%o         if v >= pp:

%o             pp *= sieve[nextcount]

%o             ss += sieve[nextcount]

%o             heapq.heappush(h, (pp+ss, pp, ss, 1, nextcount))

%o             nextcount += 1

%o         pr //= sieve[s]; sm -= sieve[s]; s += 1; l += 1; pr *= sieve[l]; sm += sieve[l]

%o         heapq.heappush(h, (pr+sm, pr, sm, s, l))

%o     return alst

%o print(list(islice(agen(), 44))) # _Michael S. Branicky_, Nov 30 2025

%Y Cf. A387590, A390933.

%K nonn

%O 1,1

%A _Will Gosnell_ and _Robert Israel_, Nov 27 2025