A351573 -id:A351573 - cp's OEIS frontend

A351571 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is an exponentially odd number.

0, 1, 4, 0, 5, 16, 12, 8, 0, 21, 16, 4, 9, 44, 44, 0, 21, 1, 24, 5, 80, 60, 44, 92, 0, 41, 68, 12, 31, 156, 80, 51, 112, 81, 112, 0, 21, 92, 92, 123, 41, 272, 48, 16, 5, 156, 112, 4, 0, 1, 156, 9, 81, 244, 156, 244, 176, 123, 92, 44, 33, 272, 12, 0, 124, 384, 72, 21, 272, 384, 156, 8, 39, 101, 4, 24, 272, 332, 176, 5

Offset: 1

Antti Karttunen, Feb 23 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A268335 (exponentially odd numbers), A342925, A350389, A351569, A351570, A351573.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351571(n) = A003415(sigma(A350389(n)));

a(n) = A003415(A351569(n)) = A342925(A350389(n)) = A003415(A000203(A350389(n))).

A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 32, 0, 6, 0, 4, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 32, 14, 10, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 192, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 10, 4, 0, 0, 0, 32, 108, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 14, 6, 140

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

All terms are even: see comments in A235992 and observe that the terms of A350388 are all either odd or multiples of 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A235992, A350388, A351570, A351573.

f1[p_, e_] := If[EvenQ[e], p^e, 1]; f2[p_, e_] := If[EvenQ[e], e/p, 0]; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]))*(Plus @@ f2 @@@ f); Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351572(n) = A003415(A350388(n));

a(n) = A003415(A350388(n)).

cp's OEIS Frontend

A351571 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is an exponentially odd number.

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