A351572 -id:A351572 - cp's OEIS frontend

A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 22, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 22, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 22, 1, 38

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

Observation: There seems to be no terms in range 2..19 in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A342925, A350388, A351568, A351571, A351572, A351575 (positions of ones).

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); };
A351570(n) = A003415(sigma(A350388(n)));

a(n) = A003415(A351568(n)) = A342925(A350388(n)) = A003415(A000203(A350388(n))).

A351573 Arithmetic derivative of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 1, 12, 0, 7, 1, 1, 1, 9, 8, 0, 1, 1, 1, 1, 10, 13, 1, 44, 0, 15, 27, 1, 1, 31, 1, 80, 14, 19, 12, 0, 1, 21, 16, 68, 1, 41, 1, 1, 1, 25, 1, 1, 0, 1, 20, 1, 1, 81, 16, 92, 22, 31, 1, 8, 1, 33, 1, 0, 18, 61, 1, 1, 26, 59, 1, 12, 1, 39, 1, 1, 18, 71, 1, 1, 0, 43, 1, 10, 22, 45, 32, 140, 1, 7, 20, 1, 34

Offset: 1

Antti Karttunen, Feb 23 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A350388, A268335 (exponentially odd numbers), A351571, A351572.

f1[p_, e_] := If[OddQ[e], p^e, 1]; f2[p_, e_] := If[OddQ[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]));
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); };
A351573(n) = A003415(A350389(n));

a(n) = A003415(A350389(n)).

cp's OEIS Frontend

A351570 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is a square.

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