A060859 -id:A060859 - cp's OEIS frontend

A060355 Numbers k such that k and k+1 are powerful numbers.

8, 288, 675, 9800, 12167, 235224, 332928, 465124, 1825200, 11309768, 384199200, 592192224, 4931691075, 5425069447, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 11968683934831, 13325427460800, 15061377048200, 28821995554247

Offset: 1

Jason Earls, Apr 01 2001

nonn

"Erdős conjectured in 1975 that there do not exist three consecutive powerful integers." - Guy
See Guy for Erdős's conjecture and statement that this sequence is infinite. - Jud McCranie, Oct 13 2002
It is easy to see that this sequence is infinite: if k is in the sequence, so is 4*k*(k+1). - Franklin T. Adams-Watters, Sep 16 2009
The first of a run of three consecutive powerful numbers (conjectured to be empty) are just those in this sequence and A076445. - Charles R Greathouse IV, Nov 16 2012
Jaroslaw Wroblewski (see Prime Puzzles link) shows that there are infinitely many terms k in this sequence such that neither k nor k+1 is a square. - Charles R Greathouse IV, Nov 19 2012
Paul Erdős wrote of meeting Kurt Mahler in 1936: "I almost immediately posed him the following problem: ... are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes! x^2 - 8y^2 = 1 has infinitely many solutions. I was a bit crestfallen since I felt that I should have thought of this myself." - Jonathan Sondow, Feb 08 2015
Of the first 39 terms k, only 7 are such that neither k nor k+1 is a square. - Jon E. Schoenfield, Jun 12 2024

			1825200 belongs to this sequence because both 1825200 = 2^4 * 3^3 * 5^2 * 13^2 and 1825201 = 7^2 * 193^2 = 1351^2 are powerful numbers. - _Labos Elemer_, May 03 2001

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 288, pp. 74, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B16.
P. Shiu, On the number of square-full integers between successive squares, Volume 27, Issue 2 (December 1980), pp. 171-178.

Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
C. K. Caldwell, Powerful Numbers.
P. Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral. Math. Soc. Gaz., 16 (1) (1989), 1-2.
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2018 (in French).
J. J. O'Connor and E. F. Robertson, Biography of Kurt Mahler.
Carlos Rivera, Problem 53. Powerful numbers revisited, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Powerful numbers.

Primitive elements are in A199801.
Cf. A001694, A060859.
Cf. A076446 (first differences of A001694).

a060355 n = a060355_list !! (n-1)
a060355_list = map a001694 $ filter ((== 1) . a076446) [1..]
-- Reinhard Zumkeller, Jun 03 2015, Nov 30 2012

f[n_]:=First[Union[Last/@FactorInteger[n]]];Select[Range[2000000],f[#]>1&&f[#+1]>1&] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
SequencePosition[Table[If[Min[FactorInteger[n][[;;,2]]]>1,1,0],{n,11310000}],{1,1}][[;;,1]] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Mar 27 2024 *)

is(n)=ispowerful(n)&&ispowerful(n+1) \\ Charles R Greathouse IV, Nov 16 2012

def A060355(n):
    a = sloane.A001694
    return a.is_powerful(n) and a.is_powerful(n+1)
[n for n in (1..333333) if A060355(n)] # Peter Luschny, Feb 08 2015

Corrected and extended by Jud McCranie, Jul 08 2001
More terms from Jud McCranie, Oct 13 2002
a(22)-a(23) from Donovan Johnson, Jul 29 2011

A062739 Odd powerful numbers.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 675, 729, 841, 961, 1089, 1125, 1225, 1323, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3087, 3125, 3249, 3267, 3375, 3481, 3721, 3969, 4225

Offset: 1

Labos Elemer, Jul 12 2001

nonn

Smallest term of this sequence not also in A075109 is 675, followed by 1125. - Alonso del Arte, Nov 22 2011

			Consecutive-odd examples from Sentance: {25,27},{70225,70227},{189750625,189750627}

R. K. Guy, Unsolved Problems in Number Theory, B16

Zak Seidov, Table of n, a(n) for n = 1..10000
W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274.

Cf. A001694, A060355, A060859, A060860, A082695.

Cf. A076445 (consecutive odd powerful numbers).

Powerful[n_Integer] := (n ==1) || Min[Transpose[FactorInteger[n]][[2]]]>=2; Select[Range[5000],OddQ[ # ]&&Powerful[ # ]&] (* T. D. Noe, May 04 2006 *)
Join[{1},Select[Range[3,4301,2],Min[FactorInteger[#][[All,2]]]>1&]] (* Harvey P. Dale, Jan 08 2021 *)

It is not true that a(n) = A001694(2n-1).

Sum_{n>=1} 1/a(n) = (2/3) * Sum_{n>=1} 1/A001694(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) = (2/3) * A082695 = 1.2957309... - Amiram Eldar, Jun 23 2020

Checked by T. D. Noe, May 04 2006

A060860 Numbers k such that k^2-1 and k^2 are consecutive powerful numbers.

Original entry on oeis.org

3, 17, 26, 99, 485, 577, 1351, 3363, 19601, 24335, 70226, 114243, 470449, 665857, 930249, 2862251, 3650401, 3880899, 22619537, 39480499, 130576328, 131836323, 189750626, 456335045, 768398401, 1184384449, 4478554083, 9863382151, 10850138895, 26102926097

Offset: 1

Labos Elemer, May 04 2001

nonn

a(31) > 10^11. - Donovan Johnson, Nov 15 2011

a(n) - 1 is a term of A335851. - Amiram Eldar, Feb 23 2024

			592192224 = 2^5*3^2*13^2*23^3 = 24334*24336, 592192225 = 5^2*31^2*157^2 = 24335^2.

Amiram Eldar, Table of n, a(n) for n = 1..55
Index entries for sequences related to powerful numbers.

Cf. A060355, A060859, A001694, A335851.

seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q, i}, q = Union[p, 2*Select[p, # <= max && OddQ[#] &]]; i = Position[Differences[q], 2] // Flatten; Sqrt[q[[i]]*(q[[i]] + 2) + 1]]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)

a(n) = sqrt(A060859(n) + 1). - Amiram Eldar, Feb 23 2024

Corrected and extended by Jud McCranie, Jul 08 2001
a(21)-a(24) from Donovan Johnson, Apr 27 2008
a(25)-a(26) from Donovan Johnson, Dec 07 2008
a(27)-a(28) from Donovan Johnson, Jun 17 2011
a(29)-a(30) from Donovan Johnson, Nov 15 2011

A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

Original entry on oeis.org

21, 960540, 16418498901144294337512360, 436066841882071117095002459324085167366543342937477344818646196279385305441506861017701946929489111120

Offset: 1

Michel Lagneau, Feb 20 2010

nonn

A. Nitaj proved Erdős's conjecture (1975) and claimed that there exist infinitely many triples of 3-powerful numbers a,b,c with (a,b) = 1, such that a+b=c, because the equation x^3 + y^3 = 6z^3 admits an infinite number of solutions, and given by the recurrence equations (see formula). It is proved that a=x(k)^3, b=y(k)^3, and c=6c^3, and are 3-powerful numbers for each k >= 1.

			37^3 + 17^3 = 6*21^3.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 348.
Mordell, L. J. (1969). Diophantine equations. Academic Press. ISBN 0-12-506250-8

P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]
A. Nitaj, On a conjecture of Erdős on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
Wikipedia, Diophantine equation

Cf. A050240, A050241, A057521, A060859, A113839, A115645, A115651, A115676, A115686, A115687, A115689, A115691, A115693, A115695, A115697, A116064, A140172.

x0:=37:y0:=17:z0:=21: for p from 1 to 5 do: x1:=x0*(x0^3+ 2*y0^3):y1:=-y0*(2*x0^3+ y0^3):z1:=z0*(x0^3- y0^3): print(z1) : x0 :=x1 :y0 :=y1 :z0 :=z1 :od :

We generate the solutions (x(k),y(k),z(k)) from the initial solution x(0) = 37, y(0)=17, z(0)=21 x(k+1) = x(k)*(x(k)^3 + 2*y(k)^3) y(k+1) = -y(k)*(2*x(k)^3 + y(k)^3) z(k+1) = z(k)*(x(k)^3 - y(k)^3).

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A060355 Numbers k such that k and k+1 are powerful numbers.

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A062739 Odd powerful numbers.

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A060860 Numbers k such that k^2-1 and k^2 are consecutive powerful numbers.

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A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

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