A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

8, 81, 108, 2500, 2700, 3375, 5292, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772

Offset

1,1

Comments

If n = Product_{k=1..j} p_k ^ i_k with each p_k prime, then psi(n) = n * Product_{k=1..j} (p_k + 1)/p_k and n' = n*Sum_{k=1..j} i_k/p_k.

Thus every number of the form p^(p+1), where p is prime, is in the sequence.

The sequence also contains every number of the form 108*p^2 where p is a prime > 3, or 108*p^3*(p+2) where p > 3 is in A001359. - Robert Israel, Mar 29 2018

Links

Paolo P. Lava, Table of n, a(n) for n = 1..100

Formula

Solutions of the equation n' = psi(n).

Example

5292 = 2^2 * 3^3 * 7^2.
n' = 5292*(2/2 + 3/3 + 2/7) = 12096,
psi(n) = 5292*(1 + 1/2)*(1 + 1/3)*(1 + 1/7) = 12096.

Maple

with(numtheory): P:=proc(n) local a, p; a:=ifactors(n)[2];
if add(op(2, p)/op(1, p), p=a)=mul(1+1/op(1, p), p=a) then n; fi; end:
seq(P(i), i=1..10^6);

Mathematica

selQ[n_] := Module[{f = FactorInteger[n], p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, f}] == Sum[{p, e} = pe; (n/p)e, {pe, f}]];
Select[Range[10^6], selQ] (* Jean-François Alcover, Oct 16 2020 *)

Prog

(PARI) dpsi(f) = prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
ader(n, f) = sum(i=1, #f~, n/f[i, 1]*f[i, 2]);
isok(n) = my(f=factor(n)); dpsi(f) == ader(n, f); \\ Michel Marcus, Mar 29 2018

Crossrefs

Cf. A001359, A001615, A003415, A166374, A342458. A345005 (gives the odd terms).

Subsequence of A345003.

Sequence in context: A078292 A210134 A274855 * A302417 A227227 A303184

Adjacent sequences: A301936 A301937 A301938 * A301940 A301941 A301942

Keyword

nonn,easy

Author

Paolo P. Lava, Mar 29 2018

Status

approved