A256706 Numbers whose arithmetic derivative A003415 is a square different from 1.

0, 1, 4, 12, 14, 39, 46, 54, 55, 94, 138, 155, 158, 183, 203, 256, 291, 295, 297, 299, 320, 323, 334, 426, 432, 446, 512, 579, 583, 695, 718, 799, 878, 943, 955, 959, 979, 1002, 1029, 1080, 1146, 1191, 1254, 1255, 1269, 1404, 1411, 1454, 1678, 1703, 1920

Offset

1,3

Comments

If n' = 1 (<=> n is prime) were not excluded, the sequence would mainly consist of the primes.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

Disjoint from sequence A157037 of numbers whose arithmetic derivative is a prime.

Maple

b:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, -1, a(n-1)) while
      (p-> p=1 or not issqr(p))(b(k)) do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 08 2015

Mathematica

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; {0}~Join~Flatten@ Position[Array[f, 2000], n_ /; IntegerQ[Sqrt@ n] && n != 1] (* Michael De Vlieger, Apr 08 2015, after Michael Somos at A003415 *)

Prog

(PARI) is(n)=!isprime(n)&&issquare(A003415(n))

Crossrefs

Cf. A003415, A157037.

Sequence in context: A123076 A310563 A260257 * A340869 A103020 A215011

Adjacent sequences: A256703 A256704 A256705 * A256707 A256708 A256709

Keyword

nonn

Author

M. F. Hasler, Apr 08 2015

Status

approved