A088966 Numbers k such that A007947(k) = A007947(m+1) and A007947(m) = A007947(k+1), where k > m.

3, 8, 24, 80, 288, 1088, 4224, 4374, 16640, 66048, 263168, 1050624, 4198400, 16785408, 67125248, 268468224, 1073807360, 4295098368, 17180131328, 68720001024, 274878955520, 1099513724928

Offset: 1

Author

Naohiro Nomoto, Oct 29 2003

Keywords

• nonn
• more

Comments

For every k >= 0, the sequence includes 4^k + 2^(k+1), with m = 2^k + 1. - David Wasserman, Jan 29 2004

So a(13) <= 4198400. - Michel Marcus, Aug 10 2014

Are there other terms like 4374 that are not of this form? - Michel Marcus, Aug 10 2014

Examples

			With n=3 and m=2, rad(3) = rad(3) and rad(2) = rad(4), so 3 is in the sequence.
		

Links

• Christian Hercher, On one of Erdős' Problems - An Efficient Search for Benelux Pairs, arXiv:2506.01099 [math.NT], 2025. See pp. 2, 13.

Crossrefs

Cf. A007947 (rad(n)), A087914 (similar sequence), A091697 (the values of m).

Programs

Maple

rad:= n -> convert(numtheory:-factorset(n),`*`):
count:= 0: lastr:= rad(1):
for n from 2 to 10^7 do
  newr:= rad(n);
  P[lastr,newr]:= n-1;
  if assigned(P[newr,lastr]) then
    count:= count+1; A[count]:= n-1; M[count]:= P[newr,lastr];
  fi;
  lastr:= newr;
od:
seq(A[n],n=1..count); # Robert Israel, Aug 10 2014

Mathematica

(* Recomputation up to a(13), assuming m of the form 2^k+1 *)
rad[n_] := rad[n] = Select[Divisors[n], SquareFreeQ][[-1]];
okQ[n_] := Module[{r = rad[n], r1 = rad[n+1], k, m}, For[k = 0, k < Log[2, n-1], k++, m = 2^k+1; If[r == rad[m+1] && rad[m] == r1, Return[True]]]; False];
Reap[For[n = 1, n <= 5*10^6, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 11 2019 *)

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = {v = vector(nn, i, rad(i)); for (n=1, nn-1, ok = 0; if (n % 2, ma = 2, ma = 1); forstep (m = ma, n-1, 2, if ((v[n] == v[m+1]) && (v[m] == v[n+1]), ok = 1; break);); if (ok, print1(n, ", ")););} \\ Michel Marcus, Aug 10 2014

Formula

G.f.: Conjecture: Q(0)/x - 1/x where Q(k)= 1 + 2^k*x/(1 - 2*x/(2*x + 2^k*x/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013

Extensions

More terms from David Wasserman, Jan 29 2004

a(13) confirmed by Robert Israel, Aug 10 2014

a(14)-a(22) from Bill McEachen, Jul 02 2025