A394866 - OEIS

%I #45 May 28 2026 23:21:55

%S 1,1,1,7,1,27,1,408,19,254,1,17091,1,3074,191,2060123,1,725434,1,

%T 13590448,1380,693870,1,7021506649,194,11557671,341926,15572577478,1,

%U 50062082037,1,69848972741943,121055,3524329794,1538,330291266156667,1,63367446007,1289195

%N a(1) = 1; for n > 1, a(n) is the number of integers m in [2^(n-1)..3^(n-1) - 1] such that A000005(m) = n.

%C a(n) = 1 if n is a noncomposite number (A008578) and a(n) > 1 if n is a composite number (A002808).

%C --------------------------------------------------------------------------

%C \                  |   |   |   |     |   |     |   |      |      |       |

%C  \ with only tau = | 1 | 2 | 3 |  4  | 5 |  6  | 7 |  8   |  9   |  10   |

%C   ---------------  |   |   |   |     |   |     |   |      |      |       |

%C       bounds on  \ |   |   |   |     |   |     |   |      |      |       |

%C      the region = \|   |   |   |     |   |     |   |      |      |       |

%C -------------------------------------------------------------------------|

%C       1..2^(n-1)-1 | 0 | 0 | 0 |  1  | 0 |  4  | 0 |  15  |   4  |   12  | A393179(n)-1

%C 2^(n-1)..3^(n-1)-1 | 1 | 1 | 1 |  7  | 1 | 27  | 1 | 408  |  19  |  254  |       a(n)

%C 3^(n-1)..5^(n-1)-1 | 0 |   |   |     |   |     |   |      |      |       |

%C 5^(n-1)..7^(n-1)-1 |...|...|...| ... |...| ... |...| ...  |  ... |  ...  |

%C --------------------------------------------------------------------------

%C                                A390951   A394226   A395379 A395380 A395381

%e a(1) = 1 because only contains 1 integer 1 such that A000005(m) = 1;

%e a(2) = 1 because [2..2] contains 1 integer 2 such that A000005(m) = 2;

%e a(3) = 1 because [4..8] contains 1 integer 4 such that A000005(m) = 3;

%e a(4) = 7 because [8..26] contains 7 integers 8, 10, 14, 15, 21, 22, 26 such that A000005(m) = 4.

%t a[n_] := If[CompositeQ[n], Sum[Boole[DivisorSigma[0, k] == n], {k, 2^(n-1), 3^(n-1)-1}], 1]; Array[a, 15] (* _Amiram Eldar_, May 07 2026 *)

%o (Magma) [1] cat [#[m: m in [2^(n-1)..3^(n-1) - 1] | #Divisors(m) eq n]: n in [2..10]];

%o (PARI) a(n) = if (n==1, 1, my(nb=0); forfactored(f=2^(n-1), 3^(n-1)-1, if(numdiv(f[2])==n, nb++)); nb); \\ _Michel Marcus_, May 07 2026

%Y Cf. A000079, A024023.

%Y Cf. A000005, A002808, A008578.

%Y Cf. A390951, A394226, A395379, A395380, A395381.

%K nonn

%O 1,4

%A _Juri-Stepan Gerasimov_, May 02 2026

%E a(18)-a(19) from _Michel Marcus_, May 07 2026

%E a(20)-a(23) from _Amiram Eldar_, May 07 2026

%E a(24)-a(39) from _Daniel Suteu_, May 25 2026