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a(n) is the least prime p that starts a run of 2n+1 consecutive primes whose product is a sum of the same number of (others or same) consecutive primes.
1

%I #72 Jan 14 2023 12:16:17

%S 2,29,293,229,3119,67,18121,59629,10247,15391,5903,24007,11783,39359,

%T 21013,104917,38273,61129,23663,2423

%N a(n) is the least prime p that starts a run of 2n+1 consecutive primes whose product is a sum of the same number of (others or same) consecutive primes.

%H Jean-Marc Rebert, <a href="/A352065/a352065_2.txt">doubleDecomposition</a>

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1077.htm">Puzzle 1077. These numbers that are...</a>, The Prime Puzzles and Problems Connection.

%e a(0) = 2, because 2 = 2, and there is no smaller prime.

%e a(1) = 29, because 29 * 31 * 37 = 33263 = 11083 + 11087 + 11093, and there is no smaller prime that starts a run of 3 consecutive primes whose product is a sum of 3 consecutive primes.

%e a(2) = 293, because 293 * 307 * 311 * 313 * 317 = 2775683761181 = 555136752211 + 555136752221 + 555136752227 + 555136752251 + 555136752271, and there is no smaller prime that starts a run of 5 consecutive primes whose product is a sum of 5 consecutive primes.

%e Let y be the product of the 2n+1 consecutive primes starting with a(n) and let q be the first prime in the sum of 2n+1 consecutive primes. For n = 0..3 we have:

%e .

%e n 2n+1 a(n) y #dgts(y) q #dgts(q)

%e - ---- ---- ----------------- -------- ---------------- --------

%e 0 1 2 2 1 2 1

%e 1 3 29 33263 5 11083 5

%e 2 5 293 2775683761181 13 555136752211 12

%e 3 7 229 52139749485151463 17 7448535640735789 16

%e .

%e For more examples, see the "doubleDecomposition" link.

%o (Python)

%o from math import prod

%o from sympy import prime, nextprime, prevprime

%o def A352065(n):

%o plist = [prime(k) for k in range(1,2*n+2)]

%o pd = prod(plist)

%o while True:

%o mlist = [nextprime(pd//(2*n+1)-1)]

%o for _ in range(n):

%o mlist = [prevprime(mlist[0])]+mlist+[nextprime(mlist[-1])]

%o if sum(mlist) <= pd:

%o while (s := sum(mlist)) <= pd:

%o if s == pd:

%o return plist[0]

%o mlist = mlist[1:]+[nextprime(mlist[-1])]

%o else:

%o while (s := sum(mlist)) >= pd:

%o if s == pd:

%o return plist[0]

%o mlist = [prevprime(mlist[0])]+mlist[:-1]

%o pd //= plist[0]

%o plist = plist[1:] + [nextprime(plist[-1])]

%o pd *= plist[-1] # _Chai Wah Wu_, Apr 21 2022

%Y Cf. A203619, A323052.

%K nonn,hard,more

%O 0,1

%A _Jean-Marc Rebert_, Mar 05 2022

%E a(15)-a(19) from _Chai Wah Wu_, Apr 21 2022