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

Author

Amiram Eldar, Jan 13 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

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

Programs

Mathematica

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

PARI

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