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%I #34 Feb 24 2022 02:04:13
%S 6,12,20,35,63,99,143,208,272,323,437,575,725,899,1147,1517,1763,2021,
%T 2303,2597,3127,3599,4087,4757,5183,5767,6399,6723,7387,8633,9797,
%U 10403,11021,11663,12317,13673,15367,16637,17947,19043,20711,22499,23707,25591
%N a(n) = A050376(n)*A050376(n+1) where A050376(n) is the n-th number of the form p^(2^k) with p is prime and k >= 0.
%C Let m be an odd positive number. Let S_m denote the sequence {Product_{i=1..r} q_(n+t_i)}_{n>=1}, where {q_i} is sequence A050376 and Sum_{i=1..r} 2^(t_1 - t_i) is the binary representation of m, such that t_1 > t_2 > ... > t_r = 0. Note that {S_1, S_3, S_5, ...} is a partition of all integers > 1. Then S_1=A050376, which is obtained when we set r=1, t_1 = 0. [Formula made compatible with A240535 data by _Peter Munn_, Aug 10 2021]
%C This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.
%C S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - _Peter Munn_, Aug 10 2021
%C A minimal set of generators for A000379 as a group under A059897(.,.). - _Peter Munn_, Aug 11 2019
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Group.html">Group</a>.
%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.
%F a(n) = A052330(3*2^(n-1)) = A329330(A050376(n), 6). - _Peter Munn_, Aug 10 2021
%Y Positions of 3's in A240535.
%Y Sequences for other parts of the partition described in the first comment: A050376 (S_1), A240522 (S_5), A240524 (S_7), A240536 (S_9), A241024 (S_11), A241025 (S_13).
%Y Cf. A000379, A030101, A052330, A059897, A329330.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Apr 07 2014
%E More terms from _Peter J. C. Moses_, Apr 18 2014
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