OFFSET
1,1
COMMENTS
The ordered prime signature of a number n is the list of exponents of the distinct prime factors in the prime factorization of n, in the order of the prime factors (A124010).
Can 4 consecutive integers have the same ordered prime signature? There are no such quadruples below 10^9.
The answer to the question above is no. Two out of every four consecutive numbers are even and their powers of 2 are different. - Ivan N. Ianakiev, Jan 13 2023
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
33 is a term since 33 = 3^1 * 11^1, 34 = 2^1 * 17^1, and 35 = 5^1 * 7^1 have the same ordered prime signature, (1, 1).
4923 is a term since 4923 = 3^2 * 547^1, 4924 = 2^2 * 1231^1, and 4925 = 5^2 * 197^1 have the same ordered prime signature, (2, 1).
603 is a term of A052214 but not a term of this sequence, since 603 = 3^2 * 67^1, 604 = 2^2 * 151^1, and 605 = 5^1 * 11^2 have different ordered prime signatures, (2, 1) or (1, 2).
MATHEMATICA
q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]
PROG
(PARI) lista(nmax) = {my(e1 = [], e2 = factor(2)[, 2]); for(n = 3, nmax, e3 = factor(n)[, 2]; if(e1 == e2 && e2 == e3, print1(n-2, ", ")); e1 = e2; e2 = e3); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 13 2023
STATUS
approved