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A322791
Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.
18
1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 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, 12, 48, 7, 49
OFFSET
1,2
LINKS
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13, No. 2 (2010), Article 10.3.7.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471, alternative link.
Eric Weisstein's World of Mathematics, e-Divisor
EXAMPLE
The table starts
1
2
3
2, 4
5
6
7
2, 8
3, 9
10
MAPLE
A322791 := proc(n)
local expundivs , d, isue, p, ai, bi;
expudvs := {} ;
for d in numtheory[divisors](n) do
isue := true ;
for p in numtheory[factorset](n) do
ai := padic[ordp](n, p) ;
bi := padic[ordp](d, p) ;
if bi > 0 then
if modp(ai, bi) <>0 then
isue := false;
end if;
else
isue := false ;
end if;
end do;
if isue then
expudvs := expudvs union {d} ;
end if;
end do:
sort(expudvs) ;
end proc:
seq(op(A322791(n)), n=1..40) ; # R. J. Mathar, Mar 06 2023
MATHEMATICA
divQ[n_, m_] := (n > 0 && m>0 && Divisible[n, m]); expDivQ[n_, d_] := Module[ {f=FactorInteger[n]}, And@@MapThread[divQ, {f[[;; , 2]], IntegerExponent[ d, f[[;; , 1]]]} ]]; expDivs[1]={1}; expDivs[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&] ]; Table[expDivs[n], {n, 1, 50}] // Flatten
PROG
(PARI) isexpdiv(f, d) = { my(e); for (i=1, #f~, e = valuation(d, f[i, 1]); if(!e || (e && f[i, 2] % e), return(0))); 1; }
row(n) = {my(d = divisors(n), f = factor(n), ediv = []); if(n == 1, return([1])); for(i=2, #d, if(isexpdiv(f, d[i]), ediv = concat(ediv, d[i]))); ediv; } \\ Amiram Eldar, Mar 27 2023
CROSSREFS
Cf. A049419 (row lengths), A051377 (row sums).
Cf. A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary).
Sequence in context: A304728 A369609 A284318 * A361255 A304745 A353853
KEYWORD
nonn,tabf
AUTHOR
Amiram Eldar, Dec 26 2018
STATUS
approved