A087914 -id:A087914 - cp's OEIS frontend

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

Naohiro Nomoto, Oct 29 2003

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

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

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

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

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

(* 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 *)

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

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

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

A343101 Pairs of integers (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.

Original entry on oeis.org

2, 8, 6, 48, 14, 224, 30, 960, 75, 1215, 62, 3968, 126, 16128, 254, 65024, 510, 261120, 1022, 1046528, 2046, 4190208, 4094, 16769024, 8190, 67092480, 16382, 268402688, 32766, 1073676288, 65534, 4294836224

Offset: 1

Bernard Schott, Apr 05 2021

This sequence was the subject of the 1st problem of the 3rd Benelux Mathematical Olympiad in 2011, where a pair (k, m) is called a 'Benelux pair' (see links).
Every pair (2^q-2, 2^q*(2^q-2)) for q >= 2 is a solution, the next such pairs are (4094, 16769024), (8190, 67092480), (16382, 268402688), (32766, 1073676288), ... hence there exist infinitely many Benelux pairs.
Only one pair is known to be not of this form (75, 1215) (see examples).

			First pairs are (2, 8), (6, 48), (14, 224), (30, 960), (75, 1215), (62, 3968), (126, 16128), ...
Examples corresponding to solutions (2^q-2, 2^q*(2^q-2)):
-> For q = 2, a(1) = 2 = 2^1 and a(2) = 8 = 2^3 while 3 = 3^1 and 9 = 3^2.
-> For q = 3, a(3) = 6 = 2 * 3 and a(4) = 48 = 2^4 * 3 while 7 = 7^1 and 49 = 7^2.
The only known solution not of that form: a(9) = 75 = 3 * 5^2 and a(10) = 1215 = 5 * 3^5 while 76 = 2^2 * 19 and 1216 = 2^6 * 19.

Thomas Bloom, Problem 850, Erdős Problems.
BxMO 2011, Third Benelux Mathematical Olympiad, Problème 1.
Diophante, A1853. Deux miniatures bénéluxiennes (in French).
Christian Hercher, On one of Erdős' Problems - An Efficient Search for Benelux Pairs, arXiv:2506.01099 [math.NT], 2025. See p. 13.
Index to sequences related to Olympiads.

Cf. A000918 (2^n-2), A087914 (2nd column of the array, the m's).

Confirmed a(23)-a(30) and extended with a(31)-a(32) by Martin Ehrenstein, Apr 18 2021

A343889 Integer k of the pairs (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.

Original entry on oeis.org

2, 6, 14, 30, 75, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534

Offset: 1

Bernard Schott, May 03 2021

First column of the array A343101.

Even integer 2^q - 2, q >= 2 is a term (A000918), and only 75 is known to be not of this form.

			The corresponding pair to a(4) = 30 is (30, 960) because 30 = 2 * 3 * 5 and 960 = 2^5 * 3 * 5 while 961 = 31^2.
The corresponding pair to a(5) = 75 is (75, 1215) because 75 = 3 * 5^2 and 1215 = 5 * 3^5, while 76 = 2^2 * 19 and 1216 = 2^6 * 19.

Cf. A000918, A087914 (2nd column), A343101 (array).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A343101 Pairs of integers (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A343889 Integer k of the pairs (k, m) ordered by m with 1 < k < m such that k has the same prime divisors as m, and, k+1 has the same prime divisors as m+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs