A178671 -id:A178671 - cp's OEIS frontend

A178676 a(n) = 5^n + 5.

6, 10, 30, 130, 630, 3130, 15630, 78130, 390630, 1953130, 9765630, 48828130, 244140630, 1220703130, 6103515630, 30517578130, 152587890630, 762939453130, 3814697265630, 19073486328130, 95367431640630, 476837158203130, 2384185791015630, 11920928955078130

Offset: 0

Vincenzo Librandi, Dec 25 2010

			G.f. = 6 + 10*x + 30*x^2 + 130*x^3 + 630*x^4 + 3130*x^5 + 15630*x^6 + ... - _Michael Somos_, Jan 28 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A034474, A178671, A242328, A242329.

List([0..30], n -> 5^n +5); # G. C. Greubel, Jan 27 2019

Magma
```
[5^n+5: n in [0..35]];
```

Table[5^n + 5, {n, 0, 40}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n)=5^n+5 \\ Charles R Greathouse IV, Oct 07 2015

[5^n+5 for n in range(40)] # G. C. Greubel, Jan 27 2019

a(n) = 5*(a(n-1) - 4) with a(0) = 6.
G.f.: (6-26*x)/((1-5*x)*(1-x)). - R. J. Mathar, Jan 05 2011
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1. - Vincenzo Librandi, Sep 30 2013
E.g.f.: exp(5*x) + 5*exp(x). - G. C. Greubel, Jan 27 2019
a(n) = A000351(n)+5 = A034474(n)+4 = A242328(n)+3 = A242329(n)+1. - Elmo R. Oliveira, Dec 06 2023

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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A178676 a(n) = 5^n + 5.

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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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