A212662 Numbers k for which k' = x' + y', where x > 0, k = x + y, and k', x', y' are the arithmetic derivatives of k, x, y.

3, 6, 9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 36, 39, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 66, 69, 72, 75, 78, 81, 82, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 116, 117, 120, 121, 123, 125, 126, 129, 132, 135, 138, 141, 144, 145

Offset

1,1

Links

Paolo P. Lava, Table of n, a(n) for n = 1..5000

Example

k=24, x=8, y=16 and 24=8+16; k'=44, x'=12, y'=32 and 44=12+32.
In more than one way:
k=39, x=4, y=35 and 39=4+35; k'=16, x'=4, y'=12 and 16=4+12;
k=39, x=13, y=26 and 39=13+26; k'=16, x’=1, y'=15 and 16=1+15.
k=255, x=54, y=201 and 255=54+201; k'=151, x'=81, y'=70 and 16=4+12;
k=255, x=85, y=170 and 255=85+170; k'=151, x'=22, y'=129 and 16=1+15;
k=255, x=114, y=141 and 39=13+26; k'=151, x'=101, y'=50 and 16=1+15.

Maple

with(numtheory);
A212662:=proc(q)
local a, b, c, i, n, p, pfs;
for n from 1 to q do
  pfs:=ifactors(n)[2]; a:=n*add(op(2, p)/op(1, p), p=pfs);
  for i from 1 to trunc(n/2) do
    pfs:=ifactors(i)[2]; b:=i*add(op(2, p)/op(1, p), p=pfs);
    pfs:=ifactors(n-i)[2]; c:=(n-i)*add(op(2, p)/op(1, p), p=pfs);
    if a=b+c then print(n); break; fi;
  od;
od; end:
A212662(1000);

Prog

(PARI) ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ Michel Marcus, Aug 27 2022

Crossrefs

Cf. A003415, A212663, A212664.

Cf. A211223, A211224, A211225.

Sequence in context: A003252 A160943 A160930 * A342831 A161351 A328168

Adjacent sequences: A212659 A212660 A212661 * A212663 A212664 A212665

Keyword

nonn

Author

Paolo P. Lava, May 23 2012

Status

approved