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A240521 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.

%I A240521 #44 Feb 19 2025 12:34:40
%S A240521 6,12,20,35,63,99,143,208,272,323,437,575,725,899,1147,1517,1763,2021,
%T A240521 2303,2597,3127,3599,4087,4757,5183,5767,6399,6723,7387,8633,9797,
%U A240521 10403,11021,11663,12317,13673,15367,16637,17947,19043,20711,22499,23707,25591
%N A240521 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 A240521 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 A240521 This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.
%C A240521 S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - _Peter Munn_, Aug 10 2021
%C A240521 A minimal set of generators for A000379 as a group under A059897(.,.). - _Peter Munn_, Aug 11 2019
%H A240521 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>.
%H A240521 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>.
%F A240521 a(n) = A052330(3*2^(n-1)) = A329330(A050376(n), 6). - _Peter Munn_, Aug 10 2021
%o A240521 (Python)
%o A240521 from sympy import primepi, integer_nthroot
%o A240521 def A240521(n):
%o A240521     def bisection(f,kmin=0,kmax=1):
%o A240521         while f(kmax) > kmax: kmax <<= 1
%o A240521         kmin = kmax >> 1
%o A240521         while kmax-kmin > 1:
%o A240521             kmid = kmax+kmin>>1
%o A240521             if f(kmid) <= kmid:
%o A240521                 kmax = kmid
%o A240521             else:
%o A240521                 kmin = kmid
%o A240521         return kmax
%o A240521     def f(x): return n+x-sum(primepi(integer_nthroot(x,1<<i)[0]) for i in range(x.bit_length().bit_length()))
%o A240521     return (m:=bisection(f,n,n))*bisection(lambda x:f(x)+1,m,m) # _Chai Wah Wu_, Feb 18-19 2025
%Y A240521 Positions of 3's in A240535.
%Y A240521 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 A240521 Cf. A000379, A030101, A052330, A059897, A329330.
%K A240521 nonn
%O A240521 1,1
%A A240521 _Vladimir Shevelev_, Apr 07 2014
%E A240521 More terms from _Peter J. C. Moses_, Apr 18 2014