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A301939
Integers whose arithmetic derivative is equal to their Dedekind function.
4
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
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
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava, Mar 29 2018
STATUS
approved