A361255 - OEIS

%I #14 Mar 11 2023 15:08:31

%S 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,

%T 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 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

%N Triangle read by rows: row n lists the exponential unitary divisors of n.

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

%H Amiram Eldar, <a href="/A361255/b361255.txt">Table of n, a(n) for n = 1..15331</a> (first 10000 rows, flattened)

%H Nicusor Minculete and László Tóth, <a href="http://ac.inf.elte.hu/Vol_035_2011/elte_annales_35_jav_vagott.pdf#page=205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

%p A361255 := proc(n)

%p     local expundivs ,d,isue,p,ai,bi;

%p     expudvs := {} ;

%p     for d in numtheory[divisors](n) do

%p         isue := true ;

%p         for p in numtheory[factorset](n) do

%p             ai := padic[ordp](n,p) ;

%p             bi := padic[ordp](d,p) ;

%p             if bi > 0 then

%p                 if modp(ai,bi) <>0 or igcd(bi,ai/bi) <> 1 then

%p                     isue := false;

%p                 end if;

%p             else

%p                 isue := false ;

%p             end if;

%p         end do;

%p         if isue then

%p             expudvs := expudvs union {d} ;

%p         end if;

%p     end do:

%p     sort(expudvs) ;

%p end proc:

%p seq(op(A361255(n)),n=1..60) ;

%t udivQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m] && CoprimeQ[m, n/m]);

%t expuDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[udivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expuDivs[1] = {1};

%t expuDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expuDivQ[n, #] &]];

%t Table[expuDivs[n], {n, 1, 70}] // Flatten (* _Amiram Eldar_, Mar 11 2023 *)

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

%K nonn,tabf

%O 1,2

%A _R. J. Mathar_, Mar 06 2023