A322791 Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.

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

Table of n, a(n) for n=1..71.

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

Adjacent sequences: A322788 A322789 A322790 * A322792 A322793 A322794

Keyword

nonn,tabf

Author

Amiram Eldar, Dec 26 2018

Status

approved