A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 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, 12, 7, 49, 10, 50

Offset: 1

Author

Amiram Eldar, May 09 2025

Keywords

• nonn
• tabf
• easy

Comments

Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .

An exponential squarefree exponential divisor (or e-squarefree e-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 a squarefree divisor of the p-adic valuation of n.

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
• László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Crossrefs

Cf. A278908 (row lengths), A361174 (row sums).

Cf. A322791, A361255, A383760.

Programs

Mathematica

sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
Table[expSqfDivs[n], {n, 1, 70}] // Flatten