A365888 -id:A365888 - 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];}

A365887 Numbers k such that k and k+1 are both terms of A365886.

Original entry on oeis.org

80, 567, 728, 1215, 1376, 1863, 2024, 2511, 2672, 3159, 3320, 3807, 3968, 4455, 4616, 5103, 5264, 5751, 5912, 6399, 6560, 7047, 7208, 7695, 7856, 8343, 8504, 8991, 9152, 9639, 9800, 10287, 10448, 10935, 11096, 11583, 11744, 12231, 12392, 12879, 13040, 13527, 13688

Offset: 1

Amiram Eldar, Sep 22 2023

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 3, 31, 310, 3097, 30971, 309711, 3097110, 30971095, 309710953, ... . Apparently, the asymptotic density of this sequence exists and equals 0.003097109... .

			80 = 2^4 * 5 is a term since its least prime factor, 2, is smaller than its exponent, 4, and the least prime factor of 81 = 3^4, 3, is also smaller than its exponent, 4.

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

Subsequence of A365886.

A365888 is a subsequence.

q[n_] := Less @@ FactorInteger[n][[1]]; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[14000]

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}
lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

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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