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

%I A383761 #12 May 30 2025 08:02:23
%S A383761 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,
%T A383761 22,23,6,24,5,25,26,3,27,14,28,29,30,31,2,32,33,34,35,6,12,18,36,37,
%U A383761 38,39,10,40,41,42,43,22,44,15,45,46,47,6,12,7,49,10,50
%N A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.
%C A383761 Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .
%C A383761 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.
%H A383761 Amiram Eldar, <a href="/A383761/b383761.txt">Table of n, a(n) for n = 1..15331</a> (first 10000 rows, flattened)
%H A383761 László Tóth, <a href="https://ac.inf.elte.hu/Vol_027_2007/doi/155_27.pdf">On certain arithmetic functions involving exponential divisors, II</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; <a href="https://arxiv.org/abs/0708.3557">arXiv preprint</a>, arXiv:0708.3557 [math.NT], 2007-2009.
%e A383761 The first 10 rows are:
%e A383761   1
%e A383761   2
%e A383761   3
%e A383761   2, 4
%e A383761   5
%e A383761   6
%e A383761   7
%e A383761   2, 8
%e A383761   3, 9
%e A383761   10
%t A383761 sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
%t A383761 expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
%t A383761 expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
%t A383761 Table[expSqfDivs[n], {n, 1, 70}] // Flatten
%Y A383761 Cf. A278908 (row lengths), A361174 (row sums).
%Y A383761 Cf. A322791, A361255, A383760.
%K A383761 nonn,tabf,easy
%O A383761 1,2
%A A383761 _Amiram Eldar_, May 09 2025