A361175 - OEIS

%I #15 Mar 11 2023 06:22:20

%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 The sum of the exponential infinitary divisors of n.

%C First differs from A322857 at n = 256.

%C The exponential infinitary divisors of n = Product_i p(i)^e(i) are all the numbers of the form Product_i p(i)^d(i) where d(i) is an infinitary divisor of e(i).

%C The number of exponential infinitary divisors of n is A307848(n).

%H Amiram Eldar, <a href="/A361175/b361175.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrew V. Lelechenko, <a href="https://doi.org/10.1007/s11253-017-1289-7">Exponential and infinitary divisors</a>, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; <a href="https://arxiv.org/abs/1405.7597">arXiv preprint</a>, arXiv:1405.7597 [math.NT] (2014).

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.

%F Multiplicative with a(p^e) = Sum_{d infinitary divisor of e} p^d.

%t idivs[1] = {1}; idivs[n_] := Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];

%t f[p_, e_] := Total[p^idivs[e]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ _Michel Marcus_ at A077609

%o ff(p, e) = sumdiv(e, d, if(isidiv(d, factor(e)), p^d, 0));

%o a(n) = {my(f=factor(n)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])); }

%Y Cf. A077609, A307848.

%Y Similar sequences: A051377, A322857, A323309, A361174.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Mar 03 2023