OFFSET
1,2
COMMENTS
The concept of exponential divisor was introduced by Mathukumalli V. Subbarao (1921-2006) in 1971. - Amiram Eldar, Sep 21 2025
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11, p. 126.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, p. 73.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, section 30, p. 71.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, section 10, p. 51.
LINKS
Olivier Bordellès, Arithmetic Tales, Advanced Edition, Springer Cham, 2020. See section 4.2, pp. 167 and 170.
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13, No. 2 (2010), Article 10.3.7.
Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 81, 88, and 93.
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 201-206.
Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2 (1988), pp. 343-349.
I. Kátai and M. V. Subbarao, On the distribution of exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 22 (2003), pp. 161-180.
Andrew V. Lelechenko, Exponential divisor functions, Šiauliai Mathematical Seminar, Vol. 10, No. 18 (2015), pp. 181-197; arXiv preprint, arXiv:1307.3683 [math.NT], 2013-2015.
Lutz Lucht, On the sum of exponential divisors and its iterates, Archiv der Mathematik, Vol. 27 (1976), pp. 383-386.
Nicuşor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28 (2011), pp. 125-133.
Nicuşor Minculete, On certain inequalities about arithmetic functions which use the exponential divisors, International Journal of Number Theory, Vol. 8, No. 6 (2012), pp. 1527-1535; ResearchGate link.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
József Sándor, A note on exponential divisors and related arithmetic functions, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publ. Inst. Math. (Beograd), Vol. 61, No. 75 (1997), pp. 21-32.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471, alternative link.
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29-May 1, 1971, Berlin, Heidelberg: Springer Berlin Heidelberg, 1972, pp. 247-271; alternative link.
M. V. Subbarao and D. Suryanarayana, Exponentially perfect and unitary perfect numbers, abstract, Notices Amer. Math. Soc., Vol. 18 (1971), pp. 798-799; entire volume.
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; arXiv preprint, arXiv:math/0610274 [math.NT], 2006-2009.
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.
László Tóth, An order result for the exponential divisor function, Vol. 71, No. 1-2 (2007), pp. 165-171; arXiv preprint, arXiv:0708.3552 [math.NT], 2007.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1 (1995), pp. 133-141.
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
KEYWORD
nonn,easy,tabf
AUTHOR
Amiram Eldar, Dec 26 2018
STATUS
approved