A080262 -id:A080262 - cp's OEIS frontend

A227802 Positive numbers of the form a^b +- 1 where a is in the sequence and b>1.

1, 2, 3, 5, 7, 8, 9, 10, 15, 17, 24, 26, 28, 31, 33, 48, 50, 63, 65, 80, 82, 99, 101, 124, 126, 127, 129, 224, 226, 242, 244, 255, 257, 288, 290, 342, 344, 511, 513, 575, 577, 624, 626, 675, 677, 728, 730, 783, 785, 960, 962, 999, 1001, 1023, 1025, 1088, 1090

Offset: 1

Alexander E. Koren, Jul 31 2013

nonn

			1 is the seed value of the sequence. 2 is in the sequence because 2 = 1^2 + 1 and 1 is in the sequence. 3 is in the sequence because 3 = 2^2 - 1 and 2 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A080262.

is(n)=if(n<4,1,my(t);(ispower(n-1,,&t)&&is(t)) || (ispower(n+1,,&t)&&is(t))) \\ Charles R Greathouse IV, Aug 08 2013

list(lim)=my(v=List([1,2]),i=1);while(i++<=#v, for(e=2,log(lim+2)\log(v[i]), listput(v,v[i]^e-1);listput(v,v[i]^e+1))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Aug 08 2013

def func(i):
 a=[2]
 pows={2:2}
 for rep in range(i):
  for x in a[:]:
   if not x**pows[x]-1 in a:
    a.append(x**pows[x]-1)
    pows[x**pows[x]-1]=2
   if not x**pows[x]+1 in a:
    a.append(x**pows[x]+1)
    pows[x**pows[x]+1]=2
   pows[x]+=1
  a.sort()
 return a

A292538 Lucas-Carmichael numbers of the form k^2 - 1.

Original entry on oeis.org

399, 2915, 7055, 63503, 147455, 1587599, 1710863, 2249999, 2924099, 6656399, 9486399, 14288399, 19289663, 25603599, 26936099, 28451555, 31270463, 32148899, 45158399, 49280399, 71368703, 91011599, 105884099, 111513599, 144288143, 146894399, 150405695, 152028899, 175827599

Offset: 1

Amiram Eldar, Sep 18 2017

nonn

Intersection of A005563 and A006972.
The numbers k such that k^2 - 1 is a Lucas-Carmichael number are 20, 54, 84, 252, 384, 1260, 1308, 1500, 1710, 2580, 3080, 3780, 4392, ...
From David A. Corneth, Aug 26 2023: (Start)
As k^2 - 1 = (k - 1)*(k + 1) and k is even we have k-1 and k+1 are coprime. So we can factor k-1 and k+1 separately when checking if k^2 - 1 is a term.
Possible other ideas are factoring an odd number only once, keeping it for the factorization of k^2 - 1 and (k + 2)^2 - 1. Alternatively dodging k = 18m +- 8, 18m +- 10 or 50m +- 24, 50m +- 26 to not get numbers that are multiples of odd primes squared. (End)
Wagstaff (2024) found that among the first 10^4 Lucas-Carmichael numbers there are 164 that are also Cunningham numbers (A080262) and that all of them are in this sequence. Below 10^15 there are 682 Lucas-Carmichael numbers that are also Cunningham numbers, and all of them are in this sequence (checked using the list of Lucas-Carmichael numbers by Daniel Suteu at A006972). - Amiram Eldar, Dec 29 2024

David A. Corneth, Table of n, a(n) for n = 1..2391 (terms <= 4*10^17; first 164 terms from Amiram Eldar)
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.

Cf. A005563, A006972, A080262.

filter:= t ->
  andmap(f -> f[2]=1 and (t+1) mod (f[1]+1) = 0, ifactors(t)[2]):
select(filter, [seq(k^2-1, k=3..10^5)]); # Robert Israel, Sep 24 2017

lcQ[n_] := !PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]] + 1]] == {0}; Select[Range[2, 10^4]^2 - 1, lcQ]

More terms from David A. Corneth, Aug 26 2023

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A227802 Positive numbers of the form a^b +- 1 where a is in the sequence and b>1.

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A292538 Lucas-Carmichael numbers of the form k^2 - 1.

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