A194002 -id:A194002 - cp's OEIS frontend

A328016 Numbers k such that k, k+1, ... k+6 are all cubefree (A004709).

1, 9, 17, 33, 41, 57, 65, 73, 89, 97, 113, 137, 145, 153, 169, 177, 193, 201, 209, 217, 225, 233, 257, 273, 281, 289, 305, 313, 329, 353, 361, 385, 393, 409, 417, 425, 433, 441, 449, 465, 473, 489, 505, 521, 529, 545, 553, 569, 577, 585, 601, 609, 633, 641, 649, 657

Offset: 1

Amiram Eldar, Oct 01 2019

There cannot be 8 consecutive cubefree numbers since one of them must be divisible by 8 = 2^3.
All the terms are congruent to 1 mod 8.
The asymptotic density of this sequence is A328017.

			9 is in the sequence since the numbers 9, 10, ... 15 are all cubefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Cf. A004709, A007675, A194002, A325058, A328017.

cubeFreeQ[n_] := FreeQ[FactorInteger[n], {, k /; k > 2}]; aQ[n_] := AllTrue[n + Range[0, 6], cubeFreeQ]; Select[Range[650], aQ]

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

A328016 Numbers k such that k, k+1, ... k+6 are all cubefree (A004709).

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