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A301939 Integers such that their arithmetic derivative is equal to their Dedekind function. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If n=Prod_{k=1..j}{p_k^i_k} with each p_k prime, then psi(n) = n*Prod{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);

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.

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

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Last modified January 19 18:13 EST 2020. Contains 331051 sequences. (Running on oeis4.)