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 1-7 of 7 results.

A157438 Primes p such that p^2 divides A085606((p-1)/2) = ((p-1)/2-1)^((p-1)/2) - 1.

Original entry on oeis.org

5, 127, 607
Offset: 1

Views

Author

Alexander Adamchuk, Mar 01 2009

Keywords

Comments

Primes p such that (-3/2)^((p-1)/2) == 1 - p/6 (mod p^2).[Max Alekseyev, Dec 05 2010]
Primes p that divide A085606((p-1)/2) are listed in A157437.
No other terms below 10^11. [From Max Alekseyev, Dec 05 2010.]

Crossrefs

Cf. A085606, A157437.

A155838 Primes p such that p^2 divides A085606(2p-2) = (2p-3)^(2p-2) - 1.

Original entry on oeis.org

2, 5, 13, 61681, 1235573, 27544452149
Offset: 1

Views

Author

Alexander Adamchuk, Jan 28 2009

Keywords

Comments

For p>3, (2p-3)^(2p-2) - 1 is divisible by p.
Primes p such that 3^p == 3-2p (mod p^2).
No other terms below 10^11.

Crossrefs

Cf. A085606.

Extensions

27544452149 from Max Alekseyev, Oct 14 2009

A157437 Primes congruent to 1, 5, 7, or 11 modulo 24.

Original entry on oeis.org

5, 7, 11, 29, 31, 53, 59, 73, 79, 83, 97, 101, 103, 107, 127, 131, 149, 151, 173, 179, 193, 197, 199, 223, 227, 241, 251, 269, 271, 293, 313, 317, 337, 347, 367, 389, 409, 419, 433, 439, 443, 457, 461, 463, 467, 487, 491, 509, 557, 563, 577, 587, 601, 607, 631
Offset: 1

Views

Author

Alexander Adamchuk, Mar 01 2009

Keywords

Comments

Also, primes p that divide A085606((p-1)/2) = ((p-1)/2-1)^((p-1)/2) - 1. Primes p such that p^2 divide A085606((p-1)/2) are listed in A157438.
Also, primes p>3 such -6 is a square modulo p. - Max Alekseyev, May 30 2009
Rational primes that decompose in the field Q(sqrt(-6)). - N. J. A. Sloane, Dec 25 2017

Links

Crossrefs

Cf. A085606, A157438.

Programs

  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#, 6] == 1 &] (* T. D. Noe, Aug 17 2011 *)
Select[Prime[Range[200]],MemberQ[{1,5,7,11},Mod[#,24]]&] (* Harvey P. Dale, May 25 2018 *)

Extensions

Extended by Max Alekseyev, May 30 2009
New name from Max Alekseyev, Jan 04 2018

A254029 Positive solutions of Monkey and Coconuts Problem for the classic case (5 sailors, 1 coconut to the monkey): a(n) = 15625*n - 4 for n >= 1.

Original entry on oeis.org

15621, 31246, 46871, 62496, 78121, 93746, 109371, 124996, 140621, 156246, 171871, 187496, 203121, 218746, 234371, 249996, 265621, 281246, 296871, 312496, 328121, 343746, 359371, 374996, 390621, 406246, 421871, 437496, 453121, 468746
Offset: 1

Views

Author

Luciano Ancora, Mar 14 2015

Keywords

Comments

The sequence lists the numbers of coconuts originally collected on a pile. This is the case s=5, c=1 in the general formula b(n) = n*s^(s+1) - c*(s-1).
{a(n) = 5^6*n - 4}{n>=1} gives the positive solutions to the following problem: co(k) = (4/5)*(co(k-1) - 1), for k >= 0, with co(0) = a, and the requirement c0(5) - 1 == 0 (mod 5). This gives co(5) - 1 = (2^10*a - 7*3^3*61)/5^5, with a = a(n), n >= 1. See a formula below. - Richard S. Fischer and _Wolfdieter Lang, Jun 01 2023

References

  • Charles S. Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pages 52-54.
  • Miodrag S. Petković, "The sailors, the coconuts, and the monkey", Famous Puzzles of Great Mathematicians, Amer. Math. Soc.(AMS), 2009, pages 52-56.

Links

Crossrefs

Cf. A002021, A002022, A006091, A014293, A085283, A085606, A158421.

Programs

  • Mathematica
    s = 5; c = 1; Table[n s^(s + 1) - c (s - 1), {n, 1, 30}] (* or *)
CoefficientList[Series[(15621 + 4 x)/(-1 + x)^2, {x, 0, 29}], x]

Formula

G.f.: x*(15621 + 4*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) = a(n-1) + 15625, with a(0) = -4 and a(-1) = -(4 + 5^6). a(n) = 5^6*n - 4.
a(n) = (15*c(n) + 11) + 265*(c(n) + 1)/2^10, with c(n) = A158421(n) = 2^10*n - 1, for n >= 1. - Richard S. Fischer and Wolfdieter Lang, Jun 01 2023

A123591 a(n) = ((2^n - 1)^(2^n) - 1)/(2^n)^2.

Original entry on oeis.org

-1, 0, 5, 90075, 25657845139503479, 516742576223066713590751888575037849059948015
Offset: 0

Views

Author

Alexander Adamchuk, Nov 13 2006

Keywords

Comments

The next term is too large to include.
Last digit of a(n) is 5 or 9 for n>1. It appears that a(4k) == 4 mod 5 and a(4k+1) == a(4k+2) == a(4k+3) == 0 mod 5.
p divides a(p) for prime p>2. Composite numbers n such that n divides a(n) are listed in A127643 = {15,51,65,85,185,221,255,341,451,533,561,595,645,679,771,...}. - Alexander Adamchuk, Jan 22 2007

Links

Crossrefs

Cf. A085606 (n-1)^n - 1.
Cf. A127643.

Programs

  • Magma
    [((2^n - 1)^(2^n) - 1)/(2^n)^2: n in [0..7]]; // G. C. Greubel, Oct 26 2017
  • Mathematica
    Table[((2^n-1)^(2^n)-1)/(2^n)^2,{n,0,7}]
  • PARI
    for(n=0,7, print1(((2^n - 1)^(2^n) - 1)/(2^n)^2, ", ")) \\ G. C. Greubel, Oct 26 2017

Formula

a(n) = ((2^n - 1)^(2^n) - 1)/(2^n)^2.
a(n) = A085606(2^n)/(2^n)^2.

Extensions

More terms from Alexander Adamchuk, Jan 22 2007

A127643 Composite numbers k that divide A123591(k) = ((2^k - 1)^(2^k) - 1)/(2^k)^2.

Original entry on oeis.org

15, 51, 65, 85, 185, 221, 255, 341, 451, 533, 561, 595, 645, 679, 771, 1059, 1095, 1105, 1271, 1285, 1313, 1387, 1455, 1581, 1729, 1905, 2045, 2047, 2091, 2307, 2465, 2701, 2755, 2821, 2895, 3201, 3205, 3277, 3281, 3341, 3603, 3655, 3723, 3855, 4033, 4039
Offset: 1

Views

Author

Alexander Adamchuk, Jan 22 2007

Keywords

Comments

p divides A123591(p) for prime p > 2.
Odd composite numbers k such that (2^k-1)^(2^k) == 1 (mod k). - Robert Israel, Jul 06 2017

Links

Crossrefs

Cf. A123591, A085606.

Programs

  • Maple
    select(n -> not isprime(n) and (2^n-1) &^ (2^n) mod n = 1, [seq(i,i=9..10000,2)]); # Robert Israel, Jul 06 2017
  • Mathematica
    Do[f=PowerMod[(2^n-1),(2^n),n]-1;If[ !PrimeQ[n]&&IntegerQ[(n+1)/2]&&IntegerQ[f/n],Print[n]],{n,2,10000}]

A085283 a(n) = n*n^n - (n-1)*(n-1)^n.

Original entry on oeis.org

1, 1, 7, 65, 781, 11529, 201811, 4085185, 93864121, 2413042577, 68618940391, 2138428376721, 72470493235141, 2653457921150425, 104382202543721467, 4390455017903519489, 196621779843659466481, 9340717969198079777313
Offset: 0

Views

Author

Paul Barry, Jun 26 2003

Keywords

Comments

The system of equations
x(0) = n*x(1) + 1,
(n-1)*x(1) = n*x(2) + 1,
...
(n-1)*x(n) = n*x(n+1) + 1.
relates to the Monkey-And-Coconuts problem and reduces to the single equation
A007778(n-1)*x(0) = A007778(n)*x(n+1) + a(n),
whose solutions {x(0),x(n+1)} are given by {A014293(n), A085606(n)=A007778(n-1) - 1}. - Lekraj Beedassy, Jul 15 2003
For n >= 1, a(n) is equal to the number of functions f: {1,2,...,n+1}->{1,2,...,n} such that Im(f) contains a fixed element. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Links

Crossrefs

Cf. A045531, A055869.

Programs

  • Mathematica
    Join[{1},Table[n*n^n-(n-1)(n-1)^n,{n,20}]] (* Harvey P. Dale, Sep 08 2016 *)

Formula

E.g.f.: -(x + 2*x*W(-x) + W(-x)^2)/(W(-x)*(1 + W(-x))^3), where W(x) is the Lambert W function. - Fabian Pereyra, Sep 26 2023
Showing 1-7 of 7 results.

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