A365888 Starts of run of 3 consecutive integers that are terms of A365886.

3484375, 6640623, 13609375, 16765623, 23734375, 26890623, 33859375, 37015623, 43984375, 47140623, 54109375, 57265623, 64234375, 67390623, 74359375, 77515623, 84484375, 87640623, 94609375, 97765623, 104734375, 107890623, 114859375, 118015623, 124984375, 128140623

Offset: 1

Author

Amiram Eldar, Sep 22 2023

Keywords

• nonn

Comments

Numbers k such that k, k+1 and k+2 are all terms of A365886.

Numbers of the form 4*k+2 are not terms of A365886. Therefore there are no runs of 4 or more consecutive integers. Since the middle integer in each triple is even it must be and divisible by 8, so all the terms of this sequence are of the form 8*k+7.

The numbers of terms not exceeding 10^k, for k = 7, 8, ..., are 2, 20, 198, 1979, 19796, ... . Apparently, the asymptotic density of this sequence exists and equals 1.979...*10^(-7).

Examples

			3484375 = 5^6 * 223 is a term since its least prime factor, 5, is smaller than its exponent, 6, the least prime factor of 3484376 = 2^3 * 7 * 43 * 1447, 2, is smaller than its exponent, 3, and the least prime factor of 3484377 = 3^5 * 13 * 1103, 3, is also smaller than its exponent, 5.
		

Links

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

Crossrefs

Subsequence of A004771, A365886 and A365887.

Programs

Mathematica

q[n_] := Less @@ FactorInteger[n][[1]]; Select[8 * Range[10^6] + 7, AllTrue[# + {0, 1, 2}, q] &]

PARI

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}
lista(kmax) = forstep(k = 7, kmax, 8, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));