A248135 Nonprime numbers k that divide the sum of remainders of k' mod k, for k from 1 to k', where k' is the arithmetic derivative of k.

1, 12, 52, 840, 988, 1461, 4926, 21376, 130484, 210840, 297158

Offset

1,2

Links

Table of n, a(n) for n=1..11.

Example

Arithmetic derivative of 12 is 16.
16 == 0 mod 1; 16 == 0 mod 2; 16 == 1 mod 3; 16 == 0 mod 4;
16 == 1 mod 5; 16 == 4 mod 6; 16 == 2 mod 7; 16 == 0 mod 8;
16 == 7 mod 9; 16 == 6 mod 10; 16 == 5 mod 11; 16 == 4 mod 12;
16 == 3 mod 13; 16 == 2 mod 14; 16 == 1 mod 15; 16 == 0 mod 16
and 0 + 0 + 1 + 0 + 1 + 4 + 2 + 0 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 36.
Finally 36 == 0 mod 12.

Maple

isA248135 := proc(n)
    local adir ;
    if isprime(n) then
        false;
    else
        adir := A003415(n) ;
        if modp(add(adir mod k, k=1..adir), n) = 0 then
            true;
        else
            false;
        end if;
    end if;
end proc:
for n from 1 do
    if isA248135(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Oct 10 2014

Mathematica

ad[n_] := Which[n == 0, 0, n == 1, 0, PrimeQ[n], 1, True, Sum[Module[{p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]];
okQ[n_] := !PrimeQ[n] && With[{d = ad[n]}, Divisible[Total[Mod[d, Range[d]]], n]];
For[k = 1, k <= 300000, k++, If[okQ[k], Print[k]]] (* Jean-François Alcover, May 29 2023 *)

Prog

(Python)
from sympy import isprime, factorint
A248135_list = []
for n in range(1, 10**6):
....if not isprime(n):
........a = sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0
........if not sum(a % i for i in range(1, a)) % n:
............A248135_list.append(n)
# Chai Wah Wu, Oct 19 2014

Crossrefs

Cf. A003415, A018252 (nonprime numbers).

Sequence in context: A297757 A223249 A195544 * A307916 A280660 A268186

Adjacent sequences: A248132 A248133 A248134 * A248136 A248137 A248138

Keyword

nonn,more

Author

Paolo P. Lava, Oct 10 2014

Extensions

a(9)-a(11) from Chai Wah Wu, Oct 19 2014

Name corrected by Jean-François Alcover, May 30 2023

Status

approved