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A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).

%I A367697 #13 Nov 28 2023 00:22:38
%S A367697 9,73,137,169,201,393,521,553,633,649,713,761,809,841,889,1001,1033,
%T A367697 1065,1129,1145,1193,1225,1273,1289,1353,1385,1513,1545,1577,1609,
%U A367697 1657,1769,1785,1865,1897,1929,2025,2089,2169,2217,2297,2345,2377,2409,2441,2505,2569
%N A367697 Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).
%C A367697 The maximal length of a run of consecutive exponentially odious numbers is 15 since numbers of the form 16*k + 8 are not exponentially odious. Thus all the terms of this sequence are of the form 16*k + 9 with k = 0, 4, 8, 10, 12, 24, 32, 34, 39, 40, ... .
%C A367697 The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 15, 176, 1821, 18120, 181277, 1812917, 18129256, 181290721, ... . Apparently, the asymptotic density of this sequence exists and equals 0.018129... .
%H A367697 Amiram Eldar, <a href="/A367697/b367697.txt">Table of n, a(n) for n = 1..10000</a>
%t A367697 expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; q[n_] := AllTrue[16*n + Range[9, 23], expOdQ]; 16 * Select[Range[0, 160], q] + 9
%o A367697 (PARI) isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}
%o A367697 is(n) = {my(k = (n-9)/16); if(denominator(k) > 1, return(0)); for(i=9, 23, if(!isexpod(16*k + i), return(0))); 1;}
%Y A367697 Subsequence of A270428 and A367696.
%Y A367697 Similar sequences: A007675, A194002, A325058, A328016.
%K A367697 nonn,base
%O A367697 1,1
%A A367697 _Amiram Eldar_, Nov 27 2023