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%I #26 Jan 15 2026 22:49:46
%S 3,5,6,9,11,12,17,19,23,38,39,49,56,82,138,167,199,365,419,607,670,
%T 1421,1449,1957,2196,2489,6276,7331,10813,20554,33242
%N Numbers k such that the product of the first k prime gaps minus 1 is prime.
%e Prime gaps start 1, 2, 2, 4, 2, 4, 2, ...
%e The cumulative products are 1, 2, 4, 16, 32, 128, 256, ...
%e Subtracting 1 gives 0, 1, 3, 15, 31, 127, 255, ...
%e Primes occur at indices 3, 5, 6, 9, 11, 12, 17, ...
%t seq[lim_] := Position[FoldList[Times, Differences[Prime[Range[lim]]]], _?(PrimeQ[# - 1] &)] // Flatten; seq[1000] (* _Amiram Eldar_, Jan 09 2026 *)
%Y Cf. A001223, A081411.
%K nonn,hard,more
%O 1,1
%A _Arvizzigno Gianni_, Jan 08 2026
%E a(22)-a(29) from _Amiram Eldar_, Jan 09 2026
%E a(30)-a(31) from _Michael S. Branicky_, Jan 10 2026