A099226 -id:A099226 - cp's OEIS frontend

A057897 Numbers which can be written as m^k-k, with m, k > 1.

2, 5, 7, 12, 14, 23, 24, 27, 34, 47, 58, 61, 62, 77, 79, 98, 119, 121, 122, 142, 167, 194, 213, 223, 238, 248, 252, 254, 287, 322, 340, 359, 398, 439, 482, 503, 509, 527, 574, 621, 623, 674, 723, 726, 727, 782, 839, 898, 959, 997, 1014, 1019, 1022, 1087, 1154

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].

All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - T. D. Noe, Oct 06 2004

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* T. D. Noe, Oct 06 2004 *)

ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020

upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ Andrew Howroyd, Oct 20 2020

A099225 Numbers of the form m^k+k, with m and k > 1.

Original entry on oeis.org

6, 11, 18, 20, 27, 30, 37, 38, 51, 66, 67, 70, 83, 85, 102, 123, 128, 135, 146, 171, 198, 219, 227, 248, 258, 260, 264, 291, 326, 346, 363, 402, 443, 486, 515, 521, 531, 578, 627, 629, 678, 731, 732, 735, 786, 843, 902, 963, 1003, 1026, 1029, 1034, 1091, 1158

Offset: 1

T. D. Noe, Oct 06 2004

nonn

For n=11, there are two representations: 2^3+3 and 3^2+2. All other numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = y-x for y > x > 1 and b > a > 1, which appears to have only one solution.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A057897 (numbers of the form m^k-k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

N:= 2000: # for terms <= N
S:= {}:
for k from 2 to floor(log[2](N)) do
  S:= S union {seq(m^k+k, m=2..floor((N-k)^(1/k)))}
od:
sort(convert(S,list)); # Robert Israel, Apr 28 2019

nLim=2000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst]

A253914 Numbers that can be represented as both a^x + x and b^y + b, for some a, b, x, y > 1.

Original entry on oeis.org

6, 18, 20, 30, 66, 258, 260, 732, 1026, 3130, 4098, 4100, 16386, 19686, 46662, 65538, 65540, 65552, 262146, 531444, 823550, 1048578, 1048580, 4194306, 9765630, 14348910, 16777218, 16777220, 16777224, 67108866, 268435458, 268435460, 387420492, 387420498, 1073741826

Offset: 1

Alex Ratushnyak, Jan 18 2015

nonn

Intersection of A099225 and A253913.

Includes a^(a*b)+a = (a^b)^a+a for a,b > 1. - Robert Israel, Apr 28 2019

			a(1) = 6 = 2^2 + 2, in this case a = b = x = y = 2.
a(2) = 18 = 2^4 + 2 = 4^2 + 2.
a(8) = 732 = 3^6 + 3 = 9^3 + 3.

Cf. A099225, A253913, A099226.

A247336 Numbers of the form m^k - k - 1 with k > 0 and m > 1.

Original entry on oeis.org

0, 1, 4, 6, 11, 13, 22, 23, 26, 33, 46, 57, 60, 61, 76, 78, 97, 118, 120, 121, 141, 166, 193, 212, 222, 237, 247, 251, 253, 286, 321, 339, 358, 397, 438, 481, 502, 508, 526, 573, 620, 622, 673, 722, 725, 726, 781, 838, 897, 958, 996, 1013, 1018, 1021, 1086, 1153

Offset: 1

Juri-Stepan Gerasimov, Sep 13 2014

Primes in a(n): 11, 13, 23, 61, 97, 193, 251, 397, 673, 1013, 1021, 1153, ...

			0 is in this sequence because 2^1 - 1 - 1 = 0,
1 is in this sequence because 2^2 - 2 - 1 = 1.

Cf. A057897, A099226, A099228.

a(n + 1) = A057897(n) - 1.

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A057897 Numbers which can be written as m^k-k, with m, k > 1.

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A099225 Numbers of the form m^k+k, with m and k > 1.

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A253914 Numbers that can be represented as both a^x + x and b^y + b, for some a, b, x, y > 1.

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A247336 Numbers of the form m^k - k - 1 with k > 0 and m > 1.

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