A057897 -id:A057897 - cp's OEIS frontend

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

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]

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

Original entry on oeis.org

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]

A099228 Primes of the form m^k-k, with m and k > 1.

Original entry on oeis.org

2, 5, 7, 23, 47, 61, 79, 167, 223, 359, 439, 503, 509, 727, 839, 997, 1019, 1087, 1223, 1367, 1847, 2207, 2399, 2741, 3023, 3719, 3967, 4093, 4759, 5039, 5623, 5927, 6553, 7919, 8179, 8647, 10607, 11447, 13687, 14159, 14639, 15619, 16127, 17159, 17573

Offset: 1

T. D. Noe, Oct 06 2004

nonn

It appears that primes of this form are much more common than primes of the form m^k+k (A099227).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A057897 (numbers of the form m^k-k, with m and k > 1), A084746 (least k such that n^k-k is prime).

Cf. A099227.

nLim=32000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Select[Union[lst], PrimeQ]

A057898 Largest number such that n = m^a(n) - a(n) with m a positive integer; i.e., where (n + a(n))^(1/a(n)) is a positive integer.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

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

			a(5) = 3 since 5 = 2^3 - 3.

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

Cf. A057897, A057899.

N:= 200: # for a(1)..a(N)
V:= Vector(N,1):
for k from 2 while 2^k-k <= N do
  for m from 2 do
    v:= m^k-k;
    if v > N then break fi;
    V[v]:= k;
  od;
od:
convert(V,list); # Robert Israel, Sep 04 2020

A057899 Smallest positive integer such that n=a(n)^k-k with k a positive integer.

Original entry on oeis.org

2, 2, 4, 5, 2, 7, 3, 9, 10, 11, 12, 2, 14, 4, 16, 17, 18, 19, 20, 21, 22, 23, 5, 3, 26, 27, 2, 29, 30, 31, 32, 33, 34, 6, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 7, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 2, 60, 61, 4, 8, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Henry Bottomley, Sep 26 2000

nonn

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

			a(5)=2 since 5=2^3-3

Cf. A057897, A057898.

a(n) = (n+A057898(n))^(1/A057898(n)).

A269769 Numbers of the form p^k - k where p is a prime number and k > 1.

Original entry on oeis.org

2, 5, 7, 12, 23, 24, 27, 47, 58, 77, 119, 121, 122, 167, 238, 248, 287, 340, 359, 503, 527, 621, 723, 839, 959, 1014, 1328, 1367, 1679, 1847, 2037, 2180, 2194, 2207, 2397, 2807, 3120, 3479, 3719, 4084, 4487, 4910, 5039, 5327, 6239, 6553, 6856, 6887, 7919, 8179

Offset: 1

Altug Alkan, Mar 04 2016

nonn

Primes of the form p^k - k where p is prime are 2, 5, 7, 23, 47, 167, 359, 503, ...
Subsequence of A057897.
A182474 is a subsequence.
Up to 10^14 all the terms have a unique representation as p^k - k. - Giovanni Resta, Mar 21 2017

			    2 is a term because   2 = 2^2 - 2.
    5 is a term because   5 = 2^3 - 3.
    7 is a term because   7 = 3^2 - 2.
   12 is a term because  12 = 2^4 - 4.
  121 is a term because 121 = 2^7 - 7.

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

Cf. A000961, A057897, A182474.

N:= 10000: # to get all terms <= N
P:= select(isprime, [$1..floor((N+2)^(1/2))]):
S:= {}:
for k from 2 do
  pmax:= floor((N+k)^(1/k));
  if pmax < 2 then break fi;
  S:= S union {seq(p^k-k, p = select(`<=`,P,pmax))};
od:
sort(convert(S,list)); # Robert Israel, Mar 21 2017

A318606 Numbers of the form p^k-k for some prime, p, and integer k >= 0.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 10, 12, 16, 18, 22, 23, 24, 27, 28, 30, 36, 40, 42, 46, 47, 52, 58, 60, 66, 70, 72, 77, 78, 82, 88, 96, 100, 102, 106, 108, 112, 119, 121, 122, 126, 130, 136, 138, 148, 150, 156, 162, 166, 167, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 248, 250, 256, 262

Offset: 1

Jud McCranie, Aug 29 2018

nonn

			2^7-7=121, so 121 is in the sequence.

Jud McCranie, Table of n, a(n) for n = 1..5000
S. P. Hurd and J. S. McCranie, Integers that are Sums of Uniform Powers of all their Prime Factors: the sequence A068916, J. of Int. Seq., vol 22 (2019), article 19.3.4.

Cf. A000040, A057897, A068916, A269769.

With[{nn = 265}, Union@ Flatten@ Table[p^k - k, {p, Prime@ Range@ PrimePi@ nn}, {k, Log[p, nn]}]] (* Michael De Vlieger, Jul 16 2019 *)

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.

cp's OEIS Frontend

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

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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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A099228 Primes of the form m^k-k, with m and k > 1.

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A057898 Largest number such that n = m^a(n) - a(n) with m a positive integer; i.e., where (n + a(n))^(1/a(n)) is a positive integer.

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A057899 Smallest positive integer such that n=a(n)^k-k with k a positive integer.

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A269769 Numbers of the form p^k - k where p is a prime number and k > 1.

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A318606 Numbers of the form p^k-k for some prime, p, and integer k >= 0.

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

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