This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385197 #7 Jun 22 2025 16:33:43
%S A385197 1,2,3,3,5,5,7,7,8,9,11,9,13,13,14,15,17,16,19,15,20,21,23,21,24,25,
%T A385197 26,21,29,22,31,31,32,33,34,24,37,37,38,35,41,32,43,33,40,45,47,45,48,
%U A385197 48,50,39,53,52,54,49,56,57,59,42,61,61,56,63,64,52,67,51
%N A385197 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a noncomposite number.
%H A385197 Amiram Eldar, <a href="/A385197/b385197.txt">Table of n, a(n) for n = 1..10000</a>
%F A385197 The unitary convolution of A047994 (the unitary totient phi) with A080339 (the characteristic function of noncomposite numbers): a(n) = Sum_{d | n, gcd(d, n/d) == 1} A047994(d) * A080339(n/d).
%F A385197 a(n) = uphi(n) * (1 + Sum_{p || n} (1/(p-1))), where uphi = A047994, and p || n denotes that p unitarily divides n (i.e., the p-adic valuation of n is 1).
%F A385197 a(n) = A385196(n) + A047994(n).
%F A385197 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = c1 * c2 = 0.92334965064835578762..., c1 = Product_{p prime}(1 - 1/(p*(p+1))) = A065463, and c2 = 1 + Sum_{p prime}((p^2-1)/(p^2*(p^2+p-1))) = 1.31075288978811405615... .
%e A385197 For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 5 of the values are noncomposite numbers, and therefore a(6) = 5.
%t A385197 f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, (Times @@ f @@@ fct)*(1 + Total[Boole[# == 1] & /@ fct[[;; , 2]]/(fct[[;; , 1]] - 1)])]; Array[a, 100]
%o A385197 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2]-1) * (1 + sum(i = 1, #f~, (f[i,2] == 1)/(f[i,1] - 1)));}
%Y A385197 The unitary analog of A349338.
%Y A385197 Cf. A047994, A065463, A077610, A080339, A384047.
%Y A385197 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough), A385195 (1 or 2), A385196 (prime), this sequence (noncomposite), A385198 (prime power), A385199 (1 or prime power).
%K A385197 nonn,easy
%O A385197 1,2
%A A385197 _Amiram Eldar_, Jun 21 2025