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%I #11 Sep 05 2019 15:19:34
%S 29,37,53,101,133,181,213,373,453,509,541,613,677,757,893,901,917,997,
%T 1109,1117,1157,1189,1237,1253,1333,1405,1429,1477,1509,1541,1589,
%U 1621,1701,1749,1757,1765,1829,1885,1941,2077,2117,2133,2181,2213,2261,2333,2341
%N Starts of runs of 7 consecutive exponentially odd numbers (A268335).
%C The maximal run of consecutive exponentially odd numbers is of length 7 since numbers of the form 8k + 4 are not exponentially odd. Thus all the terms of this sequence are of the form 8k + 5 with k = 3, 4, 6, 12, 16, 22, 26, 46, 56, 63, 67, 76, 84, 94, ...
%C The number of terms below 10^k for k = 2, 3, ... is 3, 18, 201, 1878, 18902, 189515, 1895392, 18954089, ... Apparently this sequence has an asymptotic density of 0.01895...
%H Amiram Eldar, <a href="/A325058/b325058.txt">Table of n, a(n) for n = 1..10000</a>
%e 29 is in the sequence since 29, 30 = 2 * 3 * 5, 31, 32 = 2^5, 33 = 3 * 11, 34 = 2 * 17 and 35 = 5 * 7 are 7 consecutive exponentially odd numbers, all having prime factorization with only odd exponents.
%t expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; aQ[n_] := AllTrue[8n + Range[5, 11], expOddQ]; 8 * Select[Range[300], aQ] + 5
%Y Cf. A268335.
%K nonn
%O 1,1
%A _Amiram Eldar_, Sep 04 2019