A361255 Triangle read by rows: row n lists the exponential unitary 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, 51, 26, 52, 53, 6, 54, 55, 14, 56, 57, 58, 59

Offset: 1

Author

R. J. Mathar, Mar 06 2023

Keywords

• nonn
• tabf

Comments

Starts to differ from A322791 in row n=16, where 4 is an exponential divisor but not an exponential unitary divisor.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
• Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Crossrefs

Cf. A322857 (row sums), A278908 (row lengths), A322791 (includes non-unitary exp divs).

Programs

Maple

A361255 := 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 or igcd(bi,ai/bi) <> 1 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(A361255(n)),n=1..60) ;

Mathematica

udivQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m] && CoprimeQ[m, n/m]);
expuDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[udivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expuDivs[1] = {1};
expuDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expuDivQ[n, #] &]];
Table[expuDivs[n], {n, 1, 70}] // Flatten (* Amiram Eldar, Mar 11 2023 *)