A348973 - OEIS

%I #12 May 02 2022 17:28:36

%S 1,1,1,8,1,11,1,20,15,17,1,7,1,23,23,16,1,13,1,22,31,35,1,17,35,41,27,

%T 5,1,61,1,112,47,53,47,2,1,59,55,2,1,83,1,23,7,71,1,40,63,95,71,6,1,

%U 45,71,37,79,89,1,19,1,95,57,256,83,127,1,70,95,43,1,19,1,113,65,13,95,149,1,128,189,125,1,13,107

%N Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

%C It is known that A129283(n) <= A003959(n) for all n (see A348970 for a proof), which implies that each ratio a(n)/A348974(n) is at most 1:  1/1, 1/1, 1/1, 8/9, 1/1, 11/12, 1/1, 20/27, 15/16, 17/18, 1/1, 7/9, 1/1, 23/24, 23/24, 16/27, 1/1, 13/16, 1/1, 22/27, 31/32, 35/36, 1/1, 17/27, 35/36, 41/42, 27/32, 5/6, 1/1, 61/72, 1/1, 112/243, etc.

%H Antti Karttunen, <a href="/A348973/b348973.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = A129283(n) / A348972(n) = A129283(n) / gcd(A003959(n), A129283(n)).

%t f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Numerator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* _Amiram Eldar_, Nov 06 2021 *)

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A348973(n) = { my(u=n+A003415(n)); (u/gcd(A003959(n),u)); };

%Y Cf. A003415, A003959, A129283, A348970, A348972, A348974 (denominators).

%Y Cf. also A345059.

%K nonn,frac

%O 1,4

%A _Antti Karttunen_, Nov 06 2021