A367695 -id:A367695 - cp's OEIS frontend

A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Amiram Eldar, Nov 27 2023

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 8, 78, 762, 7615, 76113, 761127, 7611222, 76111895, 761119135, 7611190807, ... . Apparently, the asymptotic density of this sequence exists and equals 0.761119... .

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

Subsequence of A270428.
Subsequences: A007674, A367697.
Similar sequences: A071318, A121495, A340152, A367695.

expOdQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ[DigitCount[#, 2, 1]] &]; Select[Range[100], And @@ expOdQ /@ {#, # + 1} &]

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

A374461 Numbers k such that k and k+1 are both nonsquarefree exponentially odd numbers (A374459).

Original entry on oeis.org

135, 296, 343, 351, 375, 512, 999, 1160, 1375, 1431, 1592, 1624, 2079, 2295, 2375, 2456, 2727, 2943, 3104, 3159, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4616, 4832, 4887, 4913, 5048, 5144, 5319, 5480, 5535, 6183, 6344, 6375, 6655, 6858, 7047, 7263, 7479, 7624

Offset: 1

Amiram Eldar, Jul 09 2024

nonn

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 7, 59, 556, 5539, 55329, 553188, 5531116, 55311354, ... . Apparently, the asymptotic density of this sequence exists and equals 0.005531... .

			135 is a term since both 135 = 3^3 * 5 and 136 = 2^3 * 17 are nonsquarefree exponentially odd numbers.

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

Intersection of A068781 and A367695.

Subsequence of A268335 and A374459.

q[n_] := q[n] = Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[10000], q[#] && q[# + 1] &]

is1(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

cp's OEIS Frontend

A367696 Numbers k such that k and k+1 are both exponentially odious numbers (A270428).

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