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.

A326578 a(n) = n^2*denominator(n*Bernoulli(n-1))/denominator(Bernoulli(n-1)) = n*A326478(n).

Original entry on oeis.org

1, 2, 3, 16, 5, 36, 7, 64, 27, 100, 11, 144, 13, 196, 75, 256, 17, 324, 19, 400, 147, 484, 23, 576, 125, 676, 243, 784, 29, 900, 31, 1024, 363, 1156, 1225, 1296, 37, 1444, 507, 1600, 41, 1764, 43, 1936, 135, 2116, 47, 2304, 343, 2500, 867, 2704, 53, 2916, 3025
Offset: 1

Views

Author

Peter Luschny, Jul 16 2019

Keywords

Comments

Conjecture: If n is Carmichael then a(n) = n.
Are the fixed points of this sequence the numbers satisfying Korselt's criterion?

Links

Crossrefs

Cf. A326478, A326579, A326577, A027641/A027642 (Bernoulli), A002997 (Carmichael), A324050 (Korselt).

Programs

  • Maple
    A326578 := n -> n*A326478(n): seq(A326578(n), n=1..55);
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> n^2/igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..55);
  • Mathematica
    a[n_] := Module[{b =  BernoulliB[n - 1]}, n^2 * Denominator[n * b] / Denominator[b]]; Array[a, 60] (* Amiram Eldar, Apr 26 2024 *)
  • PARI
    a(n) = n^2*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ Michel Marcus, Jul 17 2019

Formula

a(prime(n)) = prime(n).
a(n) = n^2/gcd(n*N(n-1), D(n-1)), with N(k)/D(k) = B(k) the k-th Bernoulli number.

A326577 a(n) = (2*n - 1) / A326478(2*n - 1).

Original entry on oeis.org

1, 3, 5, 7, 3, 11, 13, 3, 17, 19, 3, 23, 5, 3, 29, 31, 3, 1, 37, 3, 41, 43, 15, 47, 7, 3, 53, 1, 3, 59, 61, 3, 5, 67, 3, 71, 73, 3, 1, 79, 3, 83, 5, 3, 89, 7, 3, 1, 97, 3, 101, 103, 15, 107, 109, 3, 113, 1, 3, 1, 11, 3, 5, 127, 3, 131, 7, 3, 137, 139, 3, 1, 5
Offset: 1

Views

Author

Peter Luschny, Jul 16 2019

Keywords

Links

Crossrefs

Cf. A326478, A326578, A027641/A027642 (Bernoulli).

Programs

  • Maple
    A326577 := n -> (2*n - 1)/A326478(2*n - 1): seq(A326577(n), n=1..72);
db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(db(2*n-2), (2*n-1)*nb(2*n-2)): seq(a(n), n=1..72);
  • Mathematica
    a[n_] := Module[{b =  BernoulliB[2*n -2]}, (2* n - 1) * Denominator[b] / ((2 * n - 1) * Denominator[(2 * n - 1) * b])]; Array[a, 100] (* Amiram Eldar, Apr 26 2024 *)
  • PARI
    f(n) = n*denominator(n*bernfrac(n-1))/denominator(bernfrac(n-1)); \\ A326478
a(n) = (2*n-1)/f(2*n-1); \\ Michel Marcus, Jul 17 2019

Formula

If 2*n - 1 is prime then a(n) is prime.
a(n) = gcd((2*n-1)*N(2*n-2), D(2*n-2)), with N(k)/D(k) = B(k) the k-th Bernoulli number.

A121707 Numbers n > 1 such that n^3 divides Sum_{k=1..n-1} k^n = A121706(n).

Original entry on oeis.org

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581, 583, 589, 611
Offset: 1

Views

Author

Alexander Adamchuk, Aug 16 2006

Keywords

Comments

All terms belong to A038509 (Composite numbers with smallest prime factor >= 5). Many but not all terms belong to A060976 (Odd nonprimes, c, which divide Bernoulli(2*c)).
Many terms are semiprimes:
- the non-semiprimes are {275, 455, 475, 539, 575, 715, 775, 875, 935, ...}: see A321487;
- semiprime terms that are multiples of 5 have indices {7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, ...} = A002145 (Primes of form 4*k + 3, except 3, or k > 0; or Primes which are also Gaussian primes);
- semiprime terms that are multiples of 7 have indices {5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, ...} = A003627 (Primes of form 3*k - 1, except 2, or k > 1);
- semiprime terms that are multiples of 11 have indices {5, 7, 13, 17, 19, 23, 37, 43, 47, 53, 59, 67, 73, 79, 83, ...} = Primes of the form 4*k + 1 and 4*k - 1. [Edited by Michel Marcus, Jul 21 2018, M. F. Hasler, Nov 09 2018]
Conjecture: odd numbers n > 1 such that n divides Sum_{k=1..n-1} k^(n-1). - Thomas Ordowski and Robert Israel, Oct 09 2015. Professor Andrzej Schinzel (in a letter to me, dated Dec 29 2015) proved this conjecture. - Thomas Ordowski, Jul 20 2018
Note that n^2 divides Sum_{k=1..n-1} k^n for every odd number n > 1. - Thomas Ordowski, Oct 30 2015
Conjecture: these are "anti-Carmichael numbers" defined; n > 1 such that p - 1 does not divide n - 1 for every prime p dividing n. Equivalently, odd numbers n > 1 such that n is coprime to A027642(n-1). A number n > 1 is an "anti-Carmichael" if and only if gcd(n, b^n - b) = 1 for some integer b. - Thomas Ordowski, Jul 20 2018
It seems that these numbers are all composite terms of A317358. - Thomas Ordowski, Jul 30 2018
a(62) = 697 is the first term not in A267999: see A306097 for all these terms. - M. F. Hasler, Nov 09 2018
If the conjecture from Thomas Ordowski is true, then no term is a multiple of 2 or 3. - Jianing Song, Jan 27 2019
Conjecture: an odd number n > 1 is a term iff gcd(n, A027642(n-1)) = 1. - Thomas Ordowski, Jul 19 2019
Conjecture: Sequence consists of numbers n > 1 such that r = b^n + n - b will produce a prime for one or more integers b > 1. Only when n is in this sequence do one or more prime factors of n fail to divide r for all b. Also, n and b must be coprime for r to be prime. The above also applies to r = b^n - n - b, ignoring n=3, b=2. - Richard R. Forberg, Jul 18 2020
Odd numbers n > 1 such that Sum_{k(even)=2..n-1}2*k^(n-1) == 0 (mod n). - Davide Rotondo, Oct 28 2020
What is the asymptotic density of these numbers? The numbers A267999 have a slightly lower density. The difference between the densities is equal to the density of the numbers A306097. - Thomas Ordowski, Feb 15 2021
The asymptotic density of this sequence is in the interval (0.253, 0.265) (Ordowski, 2021). - Amiram Eldar, Feb 26 2021

Links

Crossrefs

Cf. A000312, A002145, A002997, A027642, A031971, A038509, A060976, A121706, A267999 (probably a subsequence).
Cf. A306097 for terms of this sequence A121707 not in sequence A267999, A321487 for terms which are not semiprimes.
Cf. A191677 (n divides Sum_{k
Cf. A326478 for a conjectured connection with the Bernoulli numbers.

Programs

  • Maple
    filter:= n -> add(k &^ n mod n^3, k=1..n-1) mod n^3 = 0:
select(filter, [$2..1000]); # Robert Israel, Oct 08 2015
  • Mathematica
    fQ[n_] := Mod[Sum[PowerMod[k, n, n^3], {k, n - 1}], n^3] == 0; Select[
Range[2, 611], fQ] (* Robert G. Wilson v, Apr 04 2011 and slightly modified Aug 02 2018 *)
  • PARI
    is(n)=my(n3=n^3);sum(k=1,n-1,Mod(k,n3)^n)==0 \\ Charles R Greathouse IV, May 09 2013
  • PARI
    for(n=2, 1000, if(sum(k=1, n-1, k^n) % n^3 == 0, print1(n", "))) \\ Altug Alkan, Oct 15 2015
  • Sage
    # after Andrzej Schinzel
def isA121707(n):
    if n == 1 or is_even(n): return False
    return n.divides(sum(k^(n-1) for k in (1..n-1)))
[n for n in (1..611) if isA121707(n)] # Peter Luschny, Jul 18 2019

Extensions

Sequence corrected by Robert G. Wilson v, Apr 04 2011

A326579 a(n) = n*denominator(n*Bernoulli(n-1)) for n >= 1 and a(0) = 0.

Original entry on oeis.org

0, 1, 2, 6, 4, 30, 6, 42, 8, 90, 10, 66, 12, 2730, 14, 30, 16, 510, 18, 798, 20, 2310, 22, 138, 24, 13650, 26, 54, 28, 870, 30, 14322, 32, 5610, 34, 210, 36, 1919190, 38, 78, 40, 13530, 42, 1806, 44, 2070, 46, 282, 48, 324870, 50, 1122, 52, 1590, 54, 43890, 56
Offset: 0

Views

Author

Peter Luschny, Jul 17 2019

Keywords

Comments

Conjecture: For n>1: denominator(Bernoulli(n-1)) = n*denominator(n*Bernoulli(n-1)) <=> n is Korselt <=> n is prime or n is Carmichael.

Links

Crossrefs

Cf. A326578, A326478, A326577, A027641/A027642 (Bernoulli), A002997 (Carmichael), A324050 (Korselt).

Programs

  • Maple
    A326579 := n -> `if`(n = 0, 0, n*denom(n*bernoulli(n-1))): seq(A326579(n), n=0..56);
  • Mathematica
    a[n_] := n * Denominator[n * BernoulliB[n - 1]]; a[0] = 0; Array[a, 60, 0] (* Amiram Eldar, Apr 26 2024 *)
  • PARI
    a(n) = if (n, n*denominator(n*bernfrac(n-1)), 0); \\ Michel Marcus, Jul 19 2019

Formula

a(2*n) = 2*n.

A326580 a(n) = (2*n+1)*denominator((2*n+1)*Bernoulli(2*n)).

Original entry on oeis.org

1, 6, 30, 42, 90, 66, 2730, 30, 510, 798, 2310, 138, 13650, 54, 870, 14322, 5610, 210, 1919190, 78, 13530, 1806, 2070, 282, 324870, 1122, 1590, 43890, 16530, 354, 56786730, 126, 6630, 64722, 690, 4686, 140100870, 150, 2310, 3318, 6210270, 498, 57873270, 174
Offset: 0

Views

Author

Peter Luschny, Jul 17 2019

Keywords

Links

Crossrefs

Cf. A326579, A326578, A326478, A027641/A027642 (Bernoulli).

Programs

  • Maple
    A326580 := n -> (2*n+1)*denom((2*n+1)*bernoulli(2*n)):
seq(A326580(n), n=0..43);
  • Mathematica
    Table[(2n+1)Denominator[(2n+1)BernoulliB[2n]],{n,0,50}] (* Harvey P. Dale, Jul 31 2021 *)
  • PARI
    a(n) = (2*n+1)*denominator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Jul 19 2019

Formula

a(n) = A326579(2*n + 1).

A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 3, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 1, 3, 1, 1, 1, 37, 1, 3, 1, 41, 1, 43, 1, 15, 1, 47, 1, 7, 1, 3, 1, 53, 1, 1, 1, 3, 1, 59, 1, 61, 1, 3, 1, 5, 1, 67, 1, 3, 1, 71, 1, 73, 1, 3, 1, 1, 1, 79, 1
Offset: 1

Views

Author

Peter Luschny, Jul 19 2019

Keywords

Comments

Conjectures:
(1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).
(2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.
(3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).

Examples

			a(559) =   1 and 559 is in A121707.
a(561) = 561 and 561 is Carmichael.
a(563) = 563 and 563 is prime.

Links

Crossrefs

Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).
Cf. A326578, A326478, A326579, A326577.

Programs

  • Maple
    db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);
  • Mathematica
    a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)
  • PARI
    a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019

Formula

a(n) divides n, n/a(n) = A326478(n).
Showing 1-6 of 6 results.

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