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A359746
Numbers k such that k, k+1 and k+2 have the same ordered prime signature.
2
33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865, 3901, 3957
OFFSET
1,1
COMMENTS
First differs from its subsequence A039833 at n = 17, and from its subsequence A075039 at n = 53.
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
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
Subsequence of A052214 and A359745.
Subsequences: A039833, A075039.
Sequence in context: A052214 A063838 A075039 * A039833 A355711 A250732
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 13 2023
STATUS
approved