OFFSET
1,1
COMMENTS
Two positive integers x and y have the "Distinct Prime Exponents" (or DPE) property if their prime factorizations x = Product p_i^e_i and y = Product q_i^f_i are such that the exponent sets {e_i} and {f_i} are disjoint. The present sequence contains the numbers k such that k-1 and k have the DPE property and k and k+1 have the DPE proprty. - N. J. A. Sloane, Oct 03 2025
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10000
EXAMPLE
63 has the prime-factorization 3^2 * 7^1. 64 has the prime-factorization 2^6. And 65 has the prime-factorization 5^1 * 13^1. The exponent, 6, in the prime-factorization of 64 differs from the exponents, 2 and 1, in the prime-factorization of 63 and differs from the exponents, 1 and 1, in the prime-factorization of 65. So 64 is in the sequence.
On the other hand, the prime-factorization of 39 is 3^1 * 13^1. The prime-factorization of 40 is 2^3 * 5^1. 1 occurs as both an exponent in the prime-factorization of 39 and in the prime-factorization of 40. So neither 39 nor 40 is in the sequence.
MATHEMATICA
pfe[n_]:=Last/@FactorInteger[n]; {1}~Join~Select[Range[1600], ContainsNone[pfe[#-1], pfe[#]]&&ContainsNone[pfe[#+1], pfe[#]]&] (* James C. McMahon, Jul 06 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Jun 03 2008
EXTENSIONS
Extended by Ray Chandler, Jun 26 2009
The initial 1 (which did not really belong here) was removed by N. J. A. Sloane, Oct 03 2025
STATUS
approved
