A388970 Lexicographically earliest sequence of strictly increasing numbers such that the number of exponential divisors of its partial products is strictly increasing.

1, 4, 9, 12, 15, 16, 20, 25, 28, 35, 40, 46, 49, 52, 64, 69, 75, 91, 112, 116, 121, 145, 155, 169, 225, 279, 289, 333, 361, 407, 451, 529, 533, 559, 731, 799, 841, 893, 931, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3392, 3481, 3551, 3721, 3953, 4096, 4189, 4331

Offset

1,2

Comments

All the terms are nonprimes.

Links

Amiram Eldar, Table of n, a(n) for n = 1..5000

Example

  n | a(n) | partial products           | number of exponential divisors
  --+------+----------------------------+-------------------------------
  1 |    1 | 1                          | 1
  2 |    4 | 1 * 4 = 4                  | 2
  3 |    9 | 1 * 4 * 9 = 36             | 4
  4 |   12 | 1 * 4 * 9 * 12 = 432       | 6
  5 |   15 | 1 * 4 * 9 * 12 * 15 = 6480 | 9

Mathematica

f[p_, e_] := DivisorSigma[0, e]; d[n_] := Times @@ f @@@ FactorInteger[n];
seq[len_] := Module[{s = {1}, p = 1, k = 2, d0 = 1, c = 1}, While[c < len, While[d[k*p] <= d0, k++]; AppendTo[s, k]; c++; p *= k; d0 = d[p]; k++]; s]; seq[100]

Prog

(PARI) d(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(len) = {my(s = vector(len), p = 1, k = 2, d0 = 1, c = 1); s[1] = 1; while(c < len, while(d(k*p) <= d0, k++); c++; s[c] = k; p *= k; d0 = d(p); k++); s; }

Crossrefs

Cf. A049419, A388845.

Sequence in context: A175251 A386316 A368998 * A386318 A287367 A312850

Adjacent sequences: A388967 A388968 A388969 * A388971 A388972 A388973

Keyword

nonn

Author

Amiram Eldar, Sep 22 2025

Status

approved