A383760 Irregular triangle read by rows in which the n-th row lists the exponential infinitary divisors of n.

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 48, 7, 49, 10, 50

Offset

1,2

Comments

First differs from A322791 and A383761 at rows 16, 48, 80, 81, 112, 144, 162, ... and from A361255 at rows 256, 768, 1280, 1792, ... .

An exponential infinitary divisor d of a number n is a divisor d of n such that for every prime divisor p of n, the p-adic valuation of d is an infinitary divisor of the p-adic valuation of n.

Links

Amiram Eldar, Table of n, a(n) for n = 1..15379 (first 10000 rows, flattened)

Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014.

Example

The first 10 rows are:
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Mathematica

infDivQ[n_, 1] = True; infDivQ[n_, d_] := n > 0 && d > 0 && BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e;
expInfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[infDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expInfDivs[1] = {1};
expInfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expInfDivQ[n, #] &]];
Table[expInfDivs[n], {n, 1, 70}] // Flatten

Crossrefs

Cf. A307848 (row lengths), A361175 (row sums).

Cf. A077609, A322791, A361255, A383761.

Sequence in context: A284318 A322791 A383761 * A361255 A304745 A353853

Adjacent sequences: A383757 A383758 A383759 * A383761 A383762 A383763

Keyword

nonn,tabf,easy

Author

Amiram Eldar, May 09 2025

Status

approved