A194345 - OEIS

%I #12 Jun 07 2023 04:35:01

%S 1,7,8,9,10,11,12,14,15,16,17,18,19,21,24,33,38,39,82,97,158,166,180

%N Numbers k for which the largest prime factor of p(k) divides p(1)*p(2)*...*p(k-1), where p(k) is the number of partitions of k.

%C It appears that for all k > 180, the largest prime factor of p(k) does not divide p(1)*p(2)*...*p(k-1). This has been checked up to k = 2000. [Checked up to k = 10000, using A071963 b-file. - _Pontus von Brömssen_, Jun 05 2023]

%C See A071963 and A194259 for links and additional comments.

%e 1 is in the sequence because p(1) = 1 and 1 has no prime factor, so the condition is vacuously true.

%e For k = 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 24, 33, 38, 39, 82, 97, every prime factor of p(k) divides p(1)*p(2)*...*p(k-1).

%e For k = 158, 166, 180, not every prime factor of p(k) divides p(1)*p(2)*...*p(k-1), but the largest one does.

%Y Cf. A000041, A071963, A087173, A192885, A194259, A194260, A194261, A194262.

%K nonn,more

%O 1,2

%A _Jonathan Sondow_, Aug 21 2011