A359749 Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.

1, 3, 4, 7, 8, 9, 15, 16, 24, 25, 26, 27, 31, 32, 35, 36, 48, 63, 64, 71, 72, 81, 100, 107, 108, 120, 121, 124, 125, 127, 128, 143, 144, 168, 169, 195, 196, 199, 200, 215, 216, 224, 225, 242, 243, 255, 256, 287, 289, 323, 342, 361, 391, 392, 399, 400, 431, 432, 440

Offset

1,2

Comments

Either k or k+1 is a powerful number (A001694). Except for k=8, are there terms k such that both k and k+1 are powerful (i.e., terms that are also in A060355)? None of the terms A060355(n) for n = 2..39 is in this sequence.

A002496(k)-1, A078324(k)-1, A078325(k)-1, and A049533(k)^2 are terms for all k >= 1.

Links

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

Example

3 is a term since 3 has the exponent 1 in its prime factorization, and 3 + 1 = 4 = 2^2 has a different exponent in its prime factorization, 2.

Mathematica

q[n_] := UnsameQ @@ Join @@ (Union[FactorInteger[#][[;; , 2]]]& /@ (n + {0, 1})); Join[{1}, Select[Range[400], q]]

Prog

(PARI) lista(nmax) = {my(e1 = [], e2); for(n = 2, nmax, e2 = Set(factor(n)[, 2]); if(setintersect(e1, e2) == [], print1(n-1, ", ")); e1 = e2); }

Crossrefs

Subsequences: A000079, A000225 \ {0}, A075408, A359747.

Cf. A001694, A002496, A049533, A060355, A078324, A078325.

Sequence in context: A350690 A363751 A129142 * A378659 A075752 A046541

Adjacent sequences: A359746 A359747 A359748 * A359750 A359751 A359752

Keyword

nonn

Author

Amiram Eldar, Jan 13 2023

Status

approved