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Starts of runs of 15 consecutive integers that are exponentially odious numbers (A270428).
2

%I #13 Nov 28 2023 00:22:38

%S 9,73,137,169,201,393,521,553,633,649,713,761,809,841,889,1001,1033,

%T 1065,1129,1145,1193,1225,1273,1289,1353,1385,1513,1545,1577,1609,

%U 1657,1769,1785,1865,1897,1929,2025,2089,2169,2217,2297,2345,2377,2409,2441,2505,2569

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

%C 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 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 Amiram Eldar, <a href="/A367697/b367697.txt">Table of n, a(n) for n = 1..10000</a>

%t 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 (PARI) isexpod(n) = {my(f = factor(n)); for(i=1, #f~, if (!(hammingweight(f[i, 2]) % 2), return (0))); 1;}

%o 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 Subsequence of A270428 and A367696.

%Y Similar sequences: A007675, A194002, A325058, A328016.

%K nonn,base

%O 1,1

%A _Amiram Eldar_, Nov 27 2023