A367696 -id:A367696 - cp's OEIS frontend

A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 46, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 87, 88, 93, 94, 95, 96, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 118, 119, 122, 127, 128

Offset: 1

Amiram Eldar, Nov 27 2023

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 48, 478, 4734, 47195, 471707, 4716892, 47168363, 471681183, 4716806520, ... . Apparently, the asymptotic density of this sequence exists and equals Product_{p prime} (1 - 2/(p*(p+1))) = 0.47168... (A307868).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A268335.
Cf. A307868.
Subsequences: A007674, A325058.
Similar sequences: A071318, A121495, A340152, A367696.

expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; Select[Range[128], And @@ expOddQ /@ {#, # + 1} &]

isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if (!(f[i, 2] % 2), return (0))); 1;}
is(n) = isexpodd(n) && isexpodd(n+1)

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

Original entry on oeis.org

9, 73, 137, 169, 201, 393, 521, 553, 633, 649, 713, 761, 809, 841, 889, 1001, 1033, 1065, 1129, 1145, 1193, 1225, 1273, 1289, 1353, 1385, 1513, 1545, 1577, 1609, 1657, 1769, 1785, 1865, 1897, 1929, 2025, 2089, 2169, 2217, 2297, 2345, 2377, 2409, 2441, 2505, 2569

Offset: 1

Amiram Eldar, Nov 27 2023

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, ... .

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... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A270428 and A367696.

Similar sequences: A007675, A194002, A325058, A328016.

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

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

cp's OEIS Frontend

A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

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