A308324 -id:A308324 - cp's OEIS frontend

A057895 Nonnegative numbers which can be written as m^k-m [with m and k nonnegative and 0^0 taken as 1].

0, 1, 2, 6, 12, 14, 20, 24, 30, 42, 56, 60, 62, 72, 78, 90, 110, 120, 126, 132, 156, 182, 210, 240, 252, 254, 272, 306, 336, 342, 380, 420, 462, 504, 506, 510, 552, 600, 620, 650, 702, 720, 726, 756, 812, 870, 930, 990, 992, 1020, 1022, 1056, 1122, 1190, 1260

Offset: 0

Henry Bottomley, Sep 26 2000

nonn

			a(8)=30 is in the sequence since 30=2^6-2 (and also =6^2-6).

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000918, A057896, A308324.

N:= 1000: # to get all entries up to N
S:= {0,1}:
for m from 2 to floor((1+sqrt(1+4*N))/2) do
     S:= S union {seq(m^k - m,k=2 .. floor(log(N+m)/log(m)))}
end do:
S; # Robert Israel, Feb 12 2013

A057896 Nonnegative numbers that can be written as m^k - m (with m and k nonnegative) in more than one way.

Original entry on oeis.org

0, 6, 30, 210, 240, 2184, 8190, 78120, 24299970

Offset: 1

Henry Bottomley, Sep 26 2000

The next term, if it exists, is at least 2*10^17. - David Wasserman, May 01 2002
a(10) > 10^24, if it exists. The only numbers below 10^24 that can be written as m^k+m in more than one way are 30 = 5^2+5 = 3^3+3 and 130 = 5^3+5 = 2^7+2. - Giovanni Resta, Jun 21 2018
Conjectured to be finite and complete by Bennett (2001).

			30 is in the sequence since 30 = 2^5 - 2 = 6^2 - 6;
2184 is in the sequence since 2184 = 3^7 - 3 = 13^3 - 13.

Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Brady Haran and Matt Parker, Why 1980 was a great year to be born... but 2184 will be better, Numberphile video (2015).
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33. See Conjecture 3.

Cf. A000918, A057895, A308324.

res:= {0}:
for k from 3 to 60 do
  for m from 2 while m^k-m < 2^60 do
     x:= m^k-m;
     if assigned(R[x]) or issqr(4*x+1) then res:= res union {x}
     else R[x]:= [m,k]
     fi
  od
od:
res; # Robert Israel, Oct 07 2015

More terms from Jud McCranie, Oct 01 2000

Offset corrected by Joerg Arndt, Oct 07 2015

A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

Original entry on oeis.org

0, 2, 6, 14, 20, 24, 30, 42, 62, 78, 110, 120, 126, 156, 240, 254, 272, 336, 342, 506, 510, 620, 726, 812, 930, 1022, 1320, 1332, 1640, 1806, 2046, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4094, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 8190, 9312

Offset: 1

Craig J. Beisel, May 24 2019

The only known terms which have two representations where m is prime are 6 and 2184. It is conjectured by Bennett these are the only terms with this property.

			a(9) = 2^6 - 2 = 62.
For the two terms known to have two representations we have a(3) = 6 = 2^3 - 2 = 3^2 - 3 and a(33)= 2184 = 3^7 - 3 = 13^3 - 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Cf. A057895, A057896, A246068, A308324.

Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).

N:= 10^6; # to get all terms <= N
P:= select(isprime,[2,seq(i,i=3..floor((1+sqrt(1+4*N))/2),2)]):
S:= {0,seq(seq(m^k-m,k=2..floor(log[m](N+m))),m=P)}:
sort(convert(S,list)); # Robert Israel, Aug 11 2019

x=List([]); lim=10000; forprime(m=2, lim, for(k=1, 100, y=(m^k-m); if(y>lim, break, i=setsearch(x, y, 1); if(i>0, listinsert(x, y, i))))); for(i=1, #x, print(x[i]));

isok(n) = {forprime(p=2, oo, my(keepk = 0); for (k=1, oo, if ((x=p^k - p) == n, return(1)); if (x > n, keepk = k; break);); if (keepk == 2, break););} \\ Michel Marcus, Aug 06 2019

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A057895 Nonnegative numbers which can be written as m^k-m [with m and k nonnegative and 0^0 taken as 1].

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A057896 Nonnegative numbers that can be written as m^k - m (with m and k nonnegative) in more than one way.

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A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

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