cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A126094 Numbers k such that prime(k) = A123206(n).

Original entry on oeis.org

4, 7, 22, 83, 5878, 12229, 14871, 43360, 120833116, 284116756, 384239518, 586968768, 2697787565123, 11550434172799, 65612899915231, 252252704148332, 2105877470620834, 36504944044141057, 4194084944223361535
Offset: 1

Views

Author

Alexander Adamchuk, Mar 03 2007

Keywords

Links

Crossrefs

Cf. A000720, A123206.

Formula

a(n) = A000720(A123206(n)).

Extensions

a(13)-a(14) from Max Alekseyev, Feb 23 2012
a(15)-a(19) using Kim Walisch's primecount, from Amiram Eldar, Mar 24 2019

A007925 a(n) = n^(n+1) - (n+1)^n.

Original entry on oeis.org

-1, -1, -1, 17, 399, 7849, 162287, 3667649, 91171007, 2486784401, 74062575399, 2395420006033, 83695120256591, 3143661612445145, 126375169532421599, 5415486851106043649, 246486713303685957375, 11877172892329028459041, 604107995057426434824791
Offset: 0

Views

Author

Dennis S. Kluk (mathemagician(AT)ameritech.net)

Keywords

Comments

From Mathew Englander, Jul 07 2020: (Start)
All a(n) are odd and for n even, a(n) == 3 (mod 4); for n odd and n != 1, a(n) == 1 (mod 4).
The correspondence between n and a(n) when considered mod 6 is as follows: for n == 0, 1, 2, or 3, a(n) == 5; for n == 4, a(n) == 3; for n == 5, a(n) == 1.
For all n, a(n)+1 is a multiple of n^2.
For n odd and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs of the above, see the Englander link. (End)

Examples

			a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.

References

  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Links

Crossrefs

Cf. A051442.
Cf. A166326, A099498, A141074, A174379, A123206, A045575, A082754.

Programs

Formula

Asymptotic expression for a(n) is a(n) ~ n^n * (n - e). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
From Mathew Englander, Jul 07 2020: (Start)
a(n) = A111454(n+4) - 1.
a(n) = A055651(n, n+1).
a(n) = A220417(n+1, n) for n >= 1.
a(n) = A007778(n) - A000169(n+1).
(End)
E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022

A024082 7^n-n^7.

Original entry on oeis.org

1, 6, -79, -1844, -13983, -61318, -162287, 0, 3667649, 35570638, 272475249, 1957839572, 13805455393, 96826261890, 678117659345, 4747390650568, 33232662134145, 232630103648534, 1628412985690417, 11398894291501404, 79792265017612001, 558545862282195466
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(20)=79792265017612001 and a(24)=191581231375979942977 are primes, thus terms of A123206. - M. F. Hasler, Aug 20 2014

Links

Crossrefs

Cf. A024012, A024026, A058794, A024040, A024054, A024068.

Programs

Formula

G.f.: (1-9*x-85*x^2-407*x^3+5991*x^4+15665*x^5+8245*x^6+831*x^7+8*x^8)/((1-7*x)*(1-x)^8). - Bruno Berselli, May 16 2011

Extensions

More terms from Stefan Steinerberger, Apr 08 2006

A243100 Primes of the form x^(y+1)-y^x, for x,y > 0.

Original entry on oeis.org

3, 7, 19, 179, 543607, 129136067, 94143168179, 11920928949924493, 36472996377170722403, 61159026180004467059, 1341068619659378429383, 10301051460877537453973547005699, 710542735760100185871124061615149, 17763568394002504646778106434649157
Offset: 1

Views

Author

M. F. Hasler, Aug 19 2014

Keywords

Comments

See A123206 for primes of the form x^y-y^x with x,y>1. If y=1 is allowed, any prime p is obtained for x=p+1; this motivates the "y+1" in the exponent of the present sequence.
See also A086877 (and A098463) for primes of the form (x+1)^x-x^x.
y=0 would give "Primes of the form x", so y>0 is required. y=1 gives x^2-1 = (x-1)*(x+1) which is only prime for x=2. - Jens Kruse Andersen, Aug 23 2014

Links

Crossrefs

See also A072164.

Programs

  • PARI
    a=[];for(S=1,199,for(x=1,S-1,ispseudoprime(p=x^(1+y=S-x)-y^x)&&a=concat(a,p)));vecsort(a) \\ The list calculated this way is probably not complete up to the last terms. E.g., a 46 digit prime is found for x=3, y=97 after three larger terms for smaller S=x+y.
  • PARI
    m=300; a=[]; for(x=1, m+5, for(y=1, m+5, p=x^(y+1)-y^x; if(p<2^m && ispseudoprime(p), a=concat(a, p)))); a=vecsort(a) \\ Compute all terms below 2^m. Jens Kruse Andersen, Aug 23 2014

A243114 Primes of the form 6^x-x^6.

Original entry on oeis.org

5, 162287, 13055867207, 1719070799748422589190392551, 174588755932389037098918153698589675008087, 307180606913594390117978657628360735703373091543821695941623353827100004182413811352186951
Offset: 1

Views

Author

M. F. Hasler, Aug 20 2014

Keywords

Comments

The next term is too large to include.
See A117706 for the corresponding numbers x.
The next term has 113 digits. - Harvey P. Dale, Jan 17 2018

Crossrefs

Cf. A024068, A123206, A163319.
Cf. A072180, A109387, A117705, A117706, A128447, A128448, A128449, A128450, A128451, A122003, A128453, A128454.

Programs

  • Mathematica
    Select[Table[6^x-x^6,{x,200}],PrimeQ] (* Harvey P. Dale, Jan 17 2018 *)
  • PARI
    for(x=1,1e5,ispseudoprime(p=6^x-x^6)&&print1(p","))

A242113 a(n) = number of primes of the form k^n - m^k where k > m > 0.

Original entry on oeis.org

0, 1, 2, 6, 7, 2, 14, 7, 11, 10, 33, 10, 42, 35, 47, 39, 122, 22, 248, 113, 247, 236, 751, 75, 1268, 812, 1422, 1531, 4543, 87, 8669, 5750, 8884, 10983, 29084, 2274, 58841, 41242, 58030, 74646, 216647, 11656, 419147, 313237, 364925, 617742, 1576642, 75542, 3071839, 2299620
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 15 2014

Keywords

Comments

It would be good to have a proof that a(n) is always finite. - N. J. A. Sloane, Sep 06 2014

Examples

			a(2) = 1 because  2^2 - 1^2 = 3 is prime;
a(3) = 2 because  2^3 - 1^2 = 7 is prime and 3^3 - 2^3 = 19 is prime, but 2^3 - 2^3 < 0, 5^3 - 2^5 = 93 is not prime, 5^3 - 2^7 = 215 is not prime, 9^3 - 2^9 = 217 is not prime, 11^3 - 2^11 < 0.
More generally, primes of the form k^r - m^k where  k > m > 0:
r = 2: 3;
r = 3: 7, 19;
r = 4: 7, 17, 73, 593, 2273, 20369;
r = 5: 7, 23, 31, 179, 58537, 1951811, 1986949;
r = 6: 4818617, 24006497;
r = 7: 7, 47, 79, 103, 127, 1137, 2179, 77101, 162287, 543607, 1706527, 9940951, 6069961193, 25365130463;
r = 8: 31, 6553, 141793, 49046209, 815722529, 16983038753, 499709542049;
r = 9: 71, 151, 223, 431, 463, 487, 503, 4521799, 133227103, 10604491181, 1175888158183;
r = 10: 4177, 37097, 58049, 58537, 1803001, 2486784401, 3486783889, 41426502825041, 819626139497153, 52458394747474721.

Crossrefs

Cf. A045575, A123206, A245459, A245643.

Programs

  • Mathematica
    f[r_] := Length@ Rest@ Union@ Flatten@ Table[ If[ PrimeQ[k^r - m^k], k^r - m^k, 0], {k, 2, 10000000}, {m, Floor[k^(r/k)]}]; Do[ Print[ f[r]], {r, 2, 50}] (* Robert G. Wilson v, Aug 25 2014 *)

Formula

a(n) >= A245459(n).

Extensions

a(10)-a(50) from Robert G. Wilson v, Aug 25 2014
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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