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 1-5 of 5 results.

A258389 a(n) = (n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)).

Original entry on oeis.org

2, 14, 128, 1504, 21752, 374184, 7464368, 169402496, 4309519952, 121450640200, 3755499322808, 126409853754144, 4600799868451880, 180029930424249416, 7536568838736534752, 336087767194699956736, 15905186914751401828640, 796113699641442496367496
Offset: 1

Views

Author

Daniel Suteu, May 28 2015

Keywords

Examples

			For a(3) = (3^(3+1)-(3-1)^3) + ((3+1)^3-3^(3-1)) = (3^4 - 2^3) + (4^3 - 3^2) = 128.

Links

Crossrefs

Cf. A051442, A084363, A178922.

Programs

  • Magma
    [(n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)): n in [1..20]]; // Vincenzo Librandi, May 29 2015
  • Mathematica
    Array[#^(# + 1) - (# - 1)^# + ((# + 1)^# - #^(# - 1)) &, 20] (* Vincenzo Librandi, May 29 2015 *)
  • Sidef
    func a(n) {
     ((n+1)**n - n**(n-1)) -
     ((n-1)**n - n**(n+1))
};
1.to(Math.inf).each { |n|
    say a(n);
};

Formula

a(n) = (n^(n+1)-(n-1)^n) + ((n+1)^n-n^(n-1)) = A084363(n) + A178922(n).
a(n) = A051442(n) - A051442(n-1). - Mathew Englander, Jul 08 2020

A263323 The greater of maximal exponent and maximal prime index in the prime factorization of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 2, 3, 5, 2, 6, 4, 3, 4, 7, 2, 8, 3, 4, 5, 9, 3, 3, 6, 3, 4, 10, 3, 11, 5, 5, 7, 4, 2, 12, 8, 6, 3, 13, 4, 14, 5, 3, 9, 15, 4, 4, 3, 7, 6, 16, 3, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 3, 21, 12, 3, 8, 5, 6, 22, 4
Offset: 1

Views

Author

Alexei Kourbatov, Oct 14 2015

Keywords

Comments

Also: minimal m such that n divides (prime(m)#)^m. Here prime(m)# denotes the primorial A002110(m), i.e., the product of all primes from 2 to prime(m). - Charles R Greathouse IV, Oct 15 2015
Also: minimal m such that n is the product of at most m distinct primes not exceeding prime(m), with multiplicity at most m.
By convention, a(1)=0, as 1 is the empty product.
Those n with a(n) <= k fill a k-hypercube whose 1-sides span from 0 to k.
A263297 is a similar construction, with a k-simplex instead of a hypercube.
Each nonnegative integer occurs finitely often; in particular:
- Terms a(n) <= k occur A000169(k+1) = (k+1)^k times.
- The term a(n) = 0 occurs exactly once.
- The term a(n) = k > 0 occurs exactly A178922(k) = (k+1)^k - k^(k-1) times.

Examples

			a(36)=2 because 36 is the product of 2 distinct primes (2*2*3*3), each not exceeding prime(2)=3, with multiplicity not exceeding 2.

Links

Crossrefs

Cf. A000169, A001221, A002110, A051903, A061395, A178922, A263297.

Programs

  • Mathematica
    f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Max @@ Last /@ FactorInteger@n]; Array[f, 80]
  • PARI
    a(n) = if (n==1, 0, my(f = factor(n)); max(vecmax(f[,2]), primepi(f[#f~,1]))); \\ Michel Marcus, Oct 15 2015

Formula

a(n) = max(A051903(n), A061395(n)).
a(n) <= pi(n), with equality if n=1 or prime.

A218089 a(n) = n*((n+1)^n - n^(n-1)).

Original entry on oeis.org

1, 14, 165, 2244, 35755, 659238, 13856521, 327596552, 8612579511, 249374246010, 7887780406957, 270660921021516, 10015416945711619, 397588957529910734, 16855928678721845265, 760132325936960344080, 36333256253671504279279
Offset: 1

Views

Author

Brian J. Tyrrell, Oct 20 2012

Keywords

Programs

  • Maple
    A178922 := proc(n)
    (n+1)^n-n^(n-1) ;
end proc:
A218089 := proc(n)
    n*A178922(n) ;
end proc: # R. J. Mathar, Oct 21 2012
  • Mathematica
    a[n_] := n*((n + 1)^n - n^(n - 1))
Table[n*((n + 1)^n - n^(n - 1)), {n, 100}]
  • Maxima
    A218089[n]:=n*((n+1)^n-n^(n-1))$ makelist(A218089[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

Formula

a(n) = n*A178922(n).

A258387 a(n) = (n+1)^n + n^(n-1).

Original entry on oeis.org

3, 11, 73, 689, 8401, 125425, 2214801, 45143873, 1043046721, 26937424601, 768945795289, 24041093493169, 817012858376625, 29986640798644769, 1182114430632237601, 49814113380273715457, 2234572751614363400449, 106313261857649938064809
Offset: 1

Views

Author

Daniel Suteu, May 28 2015

Keywords

Examples

			For n=3 the a(3) = 73.
(3+1)^3 + 3^(3-1) = 4^3 + 3^2.
4^3 + 3^2 = 64 + 9 = 73.

Links

Crossrefs

Cf. A178922, A258384.

Programs

  • Magma
    [(n+1)^n + n^(n-1): n in [1..20]]; // Vincenzo Librandi, May 29 2015
  • Mathematica
    Array[(# + 1)^# + #^(# - 1) &, 20] (* Vincenzo Librandi, May 29 2015 *)
  • PARI
    vector(10,n,(n+1)^n+n^(n-1)) \\ Derek Orr, Jun 01 2015
  • Sidef
    func a(n) {
    (n+1)**n + n**(n-1)
};
1.to(Math.inf).each { |n|
    say a(n);
};

Formula

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

A158400 Primes of the form (k+1)^k - k^(k-1).

Original entry on oeis.org

7, 7151, 109873, 956953279, 3497141354765072424170242943188801
Offset: 1

Views

Author

Paolo P. Lava & Giorgio Balzarotti, Mar 18 2009

Keywords

Comments

Next term too large to be included (1652 digits)

Examples

			3^2-2^1 = 9-2 = 7.

Crossrefs

Primes in A178922.
Cf. A054463.

Programs

  • Maple
    P:=proc(i) local a,n; for n from 2 by 1 to i do a:=n^(n-1)-(n-1)^(n-2); if isprime(a) then print(a); fi; od; end: P(2000);
  • Mathematica
    Select[Table[(n + 1)^n - n^(n - 1), {n, 1000}], PrimeQ] (* Michael De Vlieger, Apr 22 2015 *)
  • PARI
    select(isprime, vector(9,n,(n+1)^n-n^(n-1))) \\ Charles R Greathouse IV, Apr 22 2015
Showing 1-5 of 5 results.

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