A074981 -id:A074981 - cp's OEIS frontend

A099226 Numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1.

27, 248, 2194, 32763

Offset: 1

T. D. Noe, Oct 06 2004

No other terms < 10^15. The intersection of A057897 and A099225. The representation question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x+y for y > x > 1 and b > a > 1.

			27 = 25^2+2 = 32^5-5, 248 = 7^3+3 = 2^8-8, 2194 = 3^7+7 = 13^3-3 and 32763 = 181^2+2 = 8^5-5.

Cf. A074981 (n such that there is no solution to Pillai's equation).

nLim=40000; lst1={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst1, n]; k++ ], {m, 2, Sqrt[nLim]}]; lst2={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst2, n]; k++ ], {m, 2, Sqrt[nLim]}]; Intersection[lst1, lst2]

A103953 Smallest perfect power b^e such that b^e+n is also a perfect power, or 0 if no such perfect power exists.

Original entry on oeis.org

8, 25, 1, 4, 4, 0, 1, 1, 16, 2187, 16, 4, 36, 0, 1, 9, 8, 9, 8, 16, 4, 27, 4, 1, 100, 1, 9, 4, 196, 6859, 1, 4, 16, 0, 1, 64, 27, 1331, 25, 9, 8, 0, 441, 81, 4, 243, 81, 1, 32, 0, 49, 144, 676, 27, 9, 8, 64, 0, 841, 4, 64, 0, 1, 36, 16, 0, 1089

Offset: 1

Max Alekseyev, Feb 22 2005

nonn

a(A074981(n)) = 0.

Cf. A074954, A001597, A076444, A023056.

a(n) = A074954(n)-n, if A074954(n)>0; a(n)=0, if A074954(n)=0.

Offset corrected by Mohammed Yaseen, Aug 09 2023

A110223 Numbers not the absolute difference between a cube and a square.

Original entry on oeis.org

6, 14, 21, 29, 32, 34, 42, 46, 51, 58, 59, 62, 66, 69, 70, 75, 77, 78, 84, 85, 86, 88, 90, 93, 96, 102, 103, 110, 111, 114, 115, 123, 130, 133, 137, 140, 149, 157, 158, 160, 162, 165, 166, 173, 176, 178, 179, 181, 182, 183, 187, 194, 201, 202, 203, 205, 209, 210, 211

Offset: 1

Robert G. Wilson v, Jul 16 2005

See A074981 for references.

T. D. Noe, Table of n, a(n) for n=1..5098

Cf. A074981. Intersection of A081121 and A054504.

Complement[ Range[212], Union[ Flatten[ Table[ Select[ Table[ Abs[n^3 - m^2], {m, 0, 10000}], # < 10^3 &], {n, -5000, 5000}]]]]

A173671 Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.

Original entry on oeis.org

3, 4, 6, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Max Alekseyev, Nov 24 2010

nonn

The complement of this set, i.e., integers of the form 3^m-2^n, is A192111. - M. F. Hasler, Nov 24 2010

H. Gauchman and I. Rosenholtz (Proposers), R. Martin (Solver), Difference of prime powers, Problem 1404, Math. Mag., 65 (No. 4, 1992), 265; Solution, Math. Mag., 66 (No. 4, 1993), 269.
Math Overflow, 3^n - 2^m = +-41 is not possible. How to prove it?, Several contributors, Jun 29 2010.

Cf. A075824, A074981, A014121, A192110, A192111, A227048, A321671.

Deleted unwarranted programs and b-file. - N. J. A. Sloane, Oct 21 2019

A219551 Number of positive integer solutions to the equation |2^x - 3^y| = n.

Original entry on oeis.org

0, 3, 0, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Jonathan Sondow, Dec 09 2012

Pillai (1931) proved that a(n) is finite for all n.
Hershfeld (1936) computed a(n) for n <= 10 and proved that a(n) <= 2 for all large n.
Stroeker and Tijdeman (1982) proved that a(n) <= 2 for all n > 13.
For additional comments, references, and links, see the crossrefs.
a(n) <= 1 except for n=1, 5, 7, 13, 23:  see e,g, Bennett (2003). - Robert Israel, Mar 06 2017

			1 = 2^2 - 3 = 3 - 2 = 3^2 - 2^3.
5 = 2^3 - 3 = 2^5 - 3^3 = 3^2 - 2^2.
7 = 2^4 - 3^2 = 3^2 - 2.
23 = 2^5 - 3^2 = 3^3 - 2^2 and a(n) <= 2 for n > 13, so a(23) = 2.

S. Pillai, On the inequality 0 < a^x - b^y <= n, Journal Indian Math. Soc., 19 (1931), 1-11.
R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Part 2, Math. Cent. Tracts, 155 (1982), 321-369.

M. A. Bennett, On Some Exponential Equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
M. A. Bennett, Pillai’s conjecture revisited, J. Number Theory, 98 (2003) 228-235.
A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
M. Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 2009.

Cf. A053289, A074981, A076438, A207079.

Clear[seq]; seq[m_] := seq[m] = (Clear[a]; a[A219551%20=%20seq%5Bm%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; Do[n = Abs[2^x - 3^y]; a[n] = a[n] + 1, {x, 1, m}, {y, 1, m}]; Table[a[n], {n, 0, 10}]); seq[m = 1]; While[seq[m] != seq[m - 1], m = 2*m]; A219551 = seq[m] (* _Jean-François Alcover, Dec 13 2012 *)

a(2n) = a(3n) = 0.

a(n) <= 2 for n > 13.

a(11) - a(30) from Robert Israel, Mar 06 2017

A074954 Least perfect power b^e such that b^e-n is also a perfect power, or 0 if no such perfect power exists.

Original entry on oeis.org

9, 27, 4, 8, 9, 0, 8, 9, 25, 2197, 27, 16, 49, 0, 16, 25, 25, 27, 27, 36, 25, 49, 27, 25, 125, 27, 36, 32, 225, 6889, 32, 36, 49, 0, 36, 100, 64, 1369, 64, 49, 49, 0, 484, 125, 49, 289, 128, 49, 81, 0, 100, 196, 729, 81, 64, 64, 121, 0, 900, 64, 125, 0, 64, 100, 81, 0, 1156

Offset: 1

Reinhard Zumkeller, Oct 10 2002

nonn

a(A074981) = 0.

			a(30) = 6889: 30 = 83^2 - 19^3.

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597.

A074980 Numbers which are not of the form m^p - n^q where p = 2 or 3, q = 2 or 3.

Original entry on oeis.org

6, 10, 14, 34, 42, 46, 50, 58, 62, 66, 70, 78, 82, 86, 90, 102, 110, 114, 122, 130, 134, 158, 162, 166, 178, 182, 194, 202, 206, 210, 214, 226, 230, 234, 238, 246, 250, 254, 258, 266, 274, 278, 290, 302, 306, 310, 314, 322, 326, 330, 338, 354, 358, 374, 378

Offset: 1

Zak Seidov, Oct 07 2002

nonn

This is a famous hard problem and the terms shown are only conjectured values.

Checked for squares and cubes through 1.134*10^25.

			146 is not in the sequence because 146 = 195^3-2723^2.

Alf van der Poorten, Remarks on the sequence of 'perfect' powers

Cf. A074981.

Corrected and extended by Jud McCranie, Oct 10 2002

A066510 Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with i>1, j>1.

Original entry on oeis.org

6, 14, 34, 42, 58, 62, 66, 70, 78, 86, 90, 102, 110, 114, 130, 158, 178, 182, 202, 210, 230, 238, 254, 258, 266, 274, 278, 302, 306, 310, 314, 322, 326, 330, 358, 374, 378, 390, 394, 398, 402, 410, 418, 422, 426, 430, 434, 438, 446, 450, 454

Offset: 1

Don Reble, Oct 12 2002

This is a famous hard problem and the terms shown are only conjectured values.
The terms shown are not the difference of two powers below 10^19. - Don Reble
One can immediately represent the odd numbers and the multiples of four as differences of two squares. - Don Reble
The terms shown are not the difference of two powers below 10^27. - Mauro Fiorentini, Jan 08 2020

			Examples showing that certain numbers are not in the sequence: 10 = 13^3-3^7, 22 = 7^2 - 3^3, 29 = 15^2 - 14^2, 31 = 2^5 - 1, 52 = 14^2 - 12^2, 54 = 3^4 - 3^3, 60 = 2^6 - 2^2, 68 = 10^2 - 2^5, 72 = 3^4 - 3^2, 76 = 5^3 - 7^2, 84 = 10^2 - 2^4, ...
50 = 7^2 - -1^3, 82 = 9^2 - -1^3, 226 = 15^2 - -1^3, 246 = 11^2 - -5^3, 290 = 17^2 - -1^3, ... [Typos corrected by _Gerry Myerson_, May 14 2008]

R. K. Guy, Unsolved Problems in Number Theory, Sections D9 and B19.

Mauro Fiorentini, Table of n, a(n) for n = 1..119
Alf van der Poorten, Remarks on the sequence of 'perfect' powers.

Cf. A074980, A023057.

For sequence with similar definition, but restricted to positive values of r and s, see A074981.

A075824 Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.

Original entry on oeis.org

9, 11, 17, 19, 21, 25, 27, 33, 35, 39, 41, 43, 45, 49, 51, 53, 57, 59, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 103, 105, 107, 109, 111, 113, 115, 117, 121, 123, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157

Offset: 1

Felice Russo, Oct 14 2002

nonn

All listed terms can be certified by considering 2^k - 3^m modulo 2552550. [Max Alekseyev, Feb 08 2010]

			5 doesn't belong to the sequence because it can be expressed as 2^3 - 3^1.

R. K. Guy, Unsolved Problems in Number Theory, D9.
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

T. Metsankyla, Catalan's Conjecture : Another old Diophantine problem solved, Bull. Amer. Math. Soc. 41 (2004), 43-57.
Wikipedia, Catalan's conjecture

Cf. A074981, A192110, A328077.

Inserted "odd" in definition. - N. J. A. Sloane, Jan 30 2009
Jon E. Schoenfield observed that 49 was missing, Jan 30 2009
More terms from Max Alekseyev, Feb 08 2010

A075789 Value of i, when n is written as r^i - s^j with the smallest possible r^i (with minimal i) and r, s > 0, i, j > 1; or 0 if n is not of this form.

Original entry on oeis.org

2, 3, 2, 3, 2, 0, 3, 2, 2, 3, 3, 2, 2, 0, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 3, 2, 5, 2, 2, 5, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 2, 3, 2, 2, 7, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 5, 2, 3, 2, 0, 7, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 7, 0, 7, 2, 2

Offset: 1

Zak Seidov, Oct 13 2002

nonn

The zeros are only conjectures (cf. A074981).

"minimal i" means that, if r^i = a^b with composite b, then i is the smallest prime factor of b; e.g., r^i = 3^4 = 9^2, i.e., r = 9, i = 2.

			1 = 3^2 - 2^3, 2 = 3^3 - 5^2, 3 = 2^2 - 1^2, 4 = 2^3 - 2^2, etc.
a(10) = 3 because 10 = 13^3 - 3^7.

Cf. A074981 (not difference of powers), A075788, A075790, A075791.

a(n,LIM=999*n)=for(k=1,LIM,(ispower(k)||k==1)&&ispower(n+k)&&return(factor(ispower(n+k))[1,1])) \\ M. F. Hasler, May 29 2018

More terms from David Wasserman, Jan 23 2005

Edited and data double-checked with given PARI code by M. F. Hasler, May 29 2018

cp's OEIS Frontend

A099226 Numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1.

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A103953 Smallest perfect power b^e such that b^e+n is also a perfect power, or 0 if no such perfect power exists.

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A110223 Numbers not the absolute difference between a cube and a square.

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Mathematica

A173671 Positive integers that cannot be expressed as 3^m-2^n where m and n are integers.

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Extensions

A219551 Number of positive integer solutions to the equation |2^x - 3^y| = n.

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A074954 Least perfect power b^e such that b^e-n is also a perfect power, or 0 if no such perfect power exists.

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A074980 Numbers which are not of the form m^p - n^q where p = 2 or 3, q = 2 or 3.

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Extensions

A066510 Conjectured list of positive numbers which are not of the form r^i-s^j, where r,s,i,j are integers with i>1, j>1.

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A075824 Odd numbers that cannot be expressed as 2^k - 3^m where k and m are integers.

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