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A359749
Numbers k such that k and k+1 do not share a common exponent in their prime factorizations.
1
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 13 2023
STATUS
approved