A359746 Numbers k such that k, k+1 and k+2 have the same ordered prime signature.

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

Author

Amiram Eldar, Jan 13 2023

Keywords

• nonn

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

Examples

			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).
		

Links

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

Crossrefs

Subsequence of A052214 and A359745.

Subsequences: A039833, A075039.

Cf. A124010, A175590.

Programs

Mathematica

q[n_] := SameQ @@ (FactorInteger[#][[;; , 2]]& /@ (n + {0, 1, 2})); Select[Range[2, 4000], q]

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); }