A322857 - OEIS

%I #15 Mar 06 2023 06:49:52

%S 1,2,3,6,5,6,7,10,12,10,11,18,13,14,15,18,17,24,19,30,21,22,23,30,30,

%T 26,30,42,29,30,31,34,33,34,35,72,37,38,39,50,41,42,43,66,60,46,47,54,

%U 56,60,51,78,53,60,55,70,57,58,59,90,61,62,84,78,65,66,67

%N a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

%C The exponential unitary (or e-unitary) divisors of n = Product p(i)^a(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of a(i).

%H Ivan Neretin, <a href="/A322857/b322857.txt">Table of n, a(n) for n = 1..10000</a>

%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.

%F Multiplicative with a(p^e) = Sum_{d|e, gcd(d, e/d)==1} p^d.

%t f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eusigma, 100]

%o (PARI) ff(p, e) = sumdiv(e, d, if (gcd(d, e/d)==1, p^d));

%o a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = ff(f[k,1], f[k,2]); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Dec 29 2018

%Y Cf. A361255, A051377, A077610, A278908 (number of exponential unitary divisors).

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Dec 29 2018