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 A304407 #20 Jun 09 2025 00:30:36
%S A304407 1,1,2,2,4,2,6,3,4,4,10,4,12,6,8,4,16,4,18,8,12,10,22,6,8,12,6,12,28,
%T A304407 8,30,5,20,16,24,8,36,18,24,12,40,12,42,20,16,22,46,8,12,8,32,24,52,6,
%U A304407 40,18,36,28,58,16,60,30,24,6,48,20,66,32,44,24,70,12,72,36,16
%N A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).
%H A304407 Andrew Howroyd, <a href="/A304407/b304407.txt">Table of n, a(n) for n = 1..1000</a>
%H A304407 Ilya Gutkovskiy, <a href="/A304407/a304407.jpg">Scatter plot of a(n) up to n=50000</a>.
%H A304407 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A304407 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A304407 a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
%F A304407 a(p^k) = (p - 1)*k where p is a prime and k > 0.
%F A304407 a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
%F A304407 a(A002110(k)) = A005867(k).
%F A304407 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - _Amiram Eldar_, Nov 30 2022
%F A304407 a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - _Ridouane Oudra_, Jun 06 2025
%e A304407 a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.
%p A304407 seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # _Ridouane Oudra_, Jun 06 2025
%t A304407 a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
%t A304407 Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]
%o A304407 (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ _Andrew Howroyd_, Jul 24 2018
%Y A304407 Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.
%Y A304407 Cf. A001221, A076479.
%K A304407 nonn,mult
%O A304407 1,3
%A A304407 _Ilya Gutkovskiy_, May 12 2018