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

Author

Paolo P. Lava, Mar 29 2018

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

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

Crossrefs

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

Subsequence of A345003.

Programs

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 *)

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

Formula

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