cp's OEIS Frontend

A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

%I A383866 #12 May 17 2025 08:15:48
%S A383866 1,3,4,7,6,12,8,11,13,18,12,28,14,24,24,19,18,39,20,42,32,36,24,44,31,
%T A383866 42,31,56,30,72,32,35,48,54,48,91,38,60,56,66,42,96,44,84,78,72,48,76,
%U A383866 57,93,72,98,54,93,72,88,80,90,60,168,62,96,104,79,84,144
%N A383866 The sum of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.
%C A383866 First differs from A383864 at n = 256.
%C A383866 The sum of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
%C A383866 Analogous to the sum of (1+e)-divisors (A051378) as exponential infinitary divisors (A383760, A361175) are analogous to exponential divisors (A322791, A051377).
%C A383866 The number of these divisors is A383865(n).
%H A383866 Amiram Eldar, <a href="/A383866/b383866.txt">Table of n, a(n) for n = 1..10000</a>
%F A383866 Multiplicative with a(p^e) = 1 + Sum_{d infinitary divisor of e} p^d.
%F A383866 a(n) <= A051378(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.
%F A383866 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52187097260174705015..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d infinitary divisor of k} x^(2*k-d))).
%t A383866 infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* _Michael De Vlieger_ at A077609 *)
%t A383866 f[p_, e_] := 1 + Total[p^infdivs[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A383866 (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); }
%o A383866 infdivs(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ _Michel Marcus_ at A077609
%o A383866 a(n) = {my(f = factor(n), d); prod(i = 1, #f~, d = infdivs(f[i, 2]); 1 + sum(j = 1, #d, f[i, 1]^d[j]));}
%Y A383866 Cf. A036537, A051377, A051378, A077609, A322791, A361175, A383760, A383864, A383865.
%K A383866 nonn,easy,mult
%O A383866 1,2
%A A383866 _Amiram Eldar_, May 13 2025