OFFSET
1,1
COMMENTS
The values of the least k such that d(n*k) = d(k^2) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 12, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ...
Perfect power terms in this sequence are 27, 125, 169, 289, 343, 361, 529, 729, 841, 961, 1000, 1024, 1331, 1369, 1681, 1849, 2187, 2197, 2209, ...
From Robert Israel, Aug 08 2016: (Start)
No terms are squarefree.
Contains p^2 where p is a prime >= 13 (with k = 144).
Contains p^3 where p is an odd prime (with k = 4 p).
Contains p^4 where p is a prime >= 11 (with k = 3600).
If n is in the sequence, with d(n*k) = d(k^2), and GCD(n*k,m) = 1, then n*m is in the sequence. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..1000
EXAMPLE
27 is a term because d(27*k) = d(k^2) with k = 12.
MAPLE
f:= proc(n) local k, r, S;
S:= select(t -> t[2]::odd, isqrfree(n)[2]);
r:= mul(t[1], t=S);
for k from r to n-1 by r do
if numtheory:-tau(n*k)=numtheory:-tau(k^2) then return true fi
od;
false
end proc:
select(f, [$1..2000]); # Robert Israel, Aug 08 2016
MATHEMATICA
f[n_] := Module[{k = 1}, While[DivisorSigma[0, k n] != DivisorSigma[0, k^2], k++]; k];
Reap[For[n = 1, n <= 1500, n++, If[f[n] < n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 11 2020, after PARI *)
PROG
(PARI) a(n) = {my(k = 1); while (numdiv(k*n) != numdiv(k^2), k++); k; }
lista(nn) = for(n=1, nn, if(a(n) < n, print1(n, ", ")));
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Jul 27 2016
STATUS
approved