A365887 -id:A365887 - cp's OEIS frontend

A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416

Offset: 1

Amiram Eldar, Sep 22 2023

First differs from A185359 at n = 22.
Numbers k such that A020639(k) < A051904(k).
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/prime(n)^(prime(n)+1)) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/8, 1/162, 1/46875, 4/86472015 and 8/109844993185235.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13119421909731920416... .

			8 = 2^3 is a term since its least prime factor, 2, is smaller than its exponent, 3.

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

Cf. A020639, A051904.
Subsequences: A008590 \ {0}, A365887, A365888.
Subsequence of A185359.

q[n_] := Less @@ FactorInteger[n][[1]]; Select[Range[2, 420], q]

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}

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

Original entry on oeis.org

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

Amiram Eldar, Sep 22 2023

nonn

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).

			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.

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

Subsequence of A004771, A365886 and A365887.

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

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, ", ")));

cp's OEIS Frontend

A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

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