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.

Previous Showing 31-38 of 38 results.

A213623 Numbers n such that the denominator of the Bernoulli polynomial B(n,x) equals the Clausen number C(n), {n | A144845(n) = A141056(n)}.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 16, 24, 28, 30, 36, 48, 60, 120
Offset: 0

Views

Author

Peter Luschny, Jun 16 2012

Keywords

Comments

Is this a finite sequence?

Crossrefs

Cf. A141056, A144845, A213621.

Programs

  • Maple
    # Clausen(n, k) defined in A160014.
seq(`if`(denom(bernoulli(i,x))=Clausen(i,1),i,NULL), i=0..120);
  • Mathematica
    Clausen[n_, k_] := If[n == 0, 1, Times @@ (Select[Divisors[n], PrimeQ[# + k]&] + k)];
Select[Range[0, 120], Denominator[BernoulliB[#, x] // Together] == Clausen[#, 1]&] (* Jean-François Alcover, Aug 13 2019 *)

A225480 a(n) = B2(n) * C(n) where B2(n) are generalized Bernoulli numbers and C(n) the Clausen numbers.

Original entry on oeis.org

1, 0, -2, 0, 14, 0, -62, 0, 254, 0, -5110, 0, 2828954, 0, -114674, 0, 237036478, 0, -11499383114, 0, 183092554714, 0, -3584085584926, 0, 3965530936622474, 0, -573989008898786, 0, 6375197353574922166, 0, -9251189109760413581110, 0, 33111281730973040956798, 0
Offset: 0

Views

Author

Peter Luschny, May 30 2013

Keywords

Comments

The Clausen numbers C(n) are T(n, 1) in A160014.

Examples

			The numerators of 1/1, 0/2, -2/6, 0/2, 14/30, 0/2, -62/42, 0/2, 254/30, 0/2, -5110/66, 0/2, 2828954/2730, ... (the denominators are the Clausen numbers).

Links

Crossrefs

Cf. A141056, A160014.

Programs

  • Maple
    B := (n, m) -> add(add(add(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j, v)*(m*v)^k, v = 0..j), j = 0..k), k = 0..n);
C := proc(n) numtheory[divisors](n);map(i->i+1,%);select(isprime,%);mul(i,i=%) end:
A225480 := n -> B(n, 2)*C(n); seq(A225480(n), n = 0..33);
  • Mathematica
    B[n_, m_] := Sum[((-1)^(n - v)/(j + 1))*Binomial[n, k]*Binomial[j, v]*If[k == 0, 1, (m*v)^k], {k, 0, n}, {j, 0, k}, {v, 0, j}];
c[n_] := Denominator[Sum[Boole[PrimeQ[d + 1]]/(d + 1), {d, Divisors[n]}]];
a[n_] := B[n, 2]*c[n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Aug 02 2019, from Maple *)
  • Sage
    @CachedFunction
def EulerianNumber(n, k, m) :   # The Eulerian numbers
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1,k-1,m)+(m*k+1)*EulerianNumber(n-1,k,m)
@CachedFunction
def B(n, m):   # The generalized Bernoulli numbers
    return add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))*(-1)^k/(k+1) for k in (0..n))
def A225480(n):
    if n == 0: return 1
    C = mul(filter(lambda s: is_prime(s) , map(lambda i: i+1, divisors(n))))
    return C*B(n, 2)
print([A225480(n) for n in (0..33)])

Formula

Let B(n,m) = sum_{k = 0..n} sum_{j = 0..k} sum_{v = 0..j} ((-1)^(n-v)/(j+1)) *binomial(n,k)*binomial(j,v)*(m*v)^k then a(n) = B(n,2)*A141056(n).
Let B2(n) = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) where S_{2}(n, k) the Stirling-Frobenius subset numbers A039755(n, k) then a(n) = B2(n)*A141056(n).

A249306 Denominators A027642(n) of Bernoulli numbers except for a(4*k+5)=2 instead of 1.

Original entry on oeis.org

1, 2, 6, 1, 30, 2, 42, 1, 30, 2, 66, 1, 2730, 2, 6, 1, 510, 2, 798, 1, 330, 2, 138, 1, 2730, 2, 6, 1, 870, 2, 14322, 1, 510, 2, 6, 1, 1919190, 2, 6, 1, 13530, 2, 1806, 1, 690, 2, 282, 1, 46410, 2, 66, 1, 1590, 2, 798, 1, 870, 2, 354, 1
Offset: 0

Views

Author

Paul Curtz, Oct 28 2014

Keywords

Comments

There exist an infinity of 1's, 2's, 6's, 30's, 42's, 66's, ... .
Respective ranks:
0, 3, 7, 11, 15, 19, ...
1, 5, 9, 13, 17, 21, ... (= A016813)
2, 14, 26, 34, 38, 62, ... (= A051222)
4, 8, 68, 76, 124, 152, ... (= A051226)
6, 114, 186, 258, 354, 402, ... (= A051228)
10, 50, 170, 370, 470, 590, ... (= A051230)
12, 24, 1308, 1884, 2004, 2364, ... (= A249134)
etc.
Hence by antidiagonals a permutation of A001477(n).
First column: A248614(n).
a(n) is an alternative sequence for the denominators of the Bernoulli numbers.
First 36 terms of the corresponding clockwise spiral:
.
330------2----138------1---2730------2
| |
| |
1 42------1-----30------2 6
| | | |
| | | |
798 2 1------2 66 1
| | | | |
| | | | |
2 30------1------6 1 870
| | |
| | |
510------1------6------2---2730 2
|
|
1------6------2----510------1--14322

Links

Crossrefs

A variant of the Clausen numbers A141056, A160014. And of A176591.
Cf. A000034, A002445, A016813, A027642, A051222, A051226, A051228, A051230, A090126, A164020, A248614, A249134.

Programs

  • Maple
    Clausen := proc(n) local S, i;
S := numtheory[divisors](n); S := map(i->i+1, S);
S := select(isprime, S); mul(i, i=S) end:
A249306 := n -> `if`(n mod 4 = 3, 1, Clausen(n)):
seq(A249306(n), n=0..59); # Peter Luschny, Nov 10 2014
  • Mathematica
    a[n_] := Denominator[BernoulliB[n]]; a[n_ /; Mod[n, 4] == 1] = 2; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2014 *)

Formula

a(2n) = A002445(n), a(2n+1) = A000034(n+1).

A362989 a(n) = lcm({i + 1, i = 0..n}) / Product_{d | n, d + 1 prime} d.

Original entry on oeis.org

1, 1, 1, 6, 2, 30, 10, 420, 84, 1260, 420, 13860, 132, 180180, 60060, 360360, 24024, 6126120, 291720, 116396280, 705432, 116396280, 38798760, 2677114440, 9806280, 13385572200, 13385572200, 40156716600, 2677114440, 1164544781400, 5041319400, 72201776446800
Offset: 0

Views

Author

Peter Luschny, May 14 2023

Keywords

Crossrefs

Cf. A003418 (lcm), A160014 (Clausen).

Programs

  • Maple
    LCM := n -> ilcm(seq(i + 1, i = 0..n)):
Clausen := n -> if n = 0 then 1 else
mul(i, i = select(isprime, map(i -> i+1, NumberTheory:-Divisors(n)))) fi:
A362989 := n -> LCM(n) / Clausen(n): seq(A362989(n), n = 0..31);

A368117 a(n) = Product_{(s - 2)|n, s prime} s if n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 3, 15, 3, 21, 15, 3, 3, 165, 21, 39, 15, 3, 3, 1785, 3, 57, 165, 3, 21, 345, 39, 3, 15, 21, 3, 4785, 3, 93, 1785, 3, 3, 195, 57, 777, 165, 3, 3, 615, 21, 129, 345, 3, 39, 922845, 3, 3, 15, 3, 21, 15105, 3, 3, 4785, 273, 3, 885, 93, 183, 1785, 3, 3
Offset: 0

Views

Author

Peter Luschny, Dec 13 2023

Keywords

Comments

All terms are squarefree.

Examples

			The divisors of 15 are {1, 3, 5, 15}. Adding 2 to the divisors gives {3, 5, 7, 17}, which are all primes. Therefore a(15) = 3*5*7*17 = 1785.

Crossrefs

Cf. A160014, A007947 (radical of n, m=0), A141056 and A027760 (Clausen numbers, m=1).

Programs

  • Mathematica
    {1}~Join~Array[Times @@ Select[Divisors[#] + 2, PrimeQ] &, 62] (* Michael De Vlieger, Dec 14 2023 *)
  • PARI
    a(n) = if (n>0, my(d=divisors(n)); prod(k=1, #d, if (isprime(p=d[k]+2), p, 1)), 1); \\ Michel Marcus, Dec 15 2023
  • SageMath
    def a(n): return (mul(s for s in map(lambda i: i + 2, divisors(n))
                  if is_prime(s)) if n > 0 else 1)
print([a(n) for n in range(63)])

Formula

a(n) = A160014(2, n).

A212197 Numbers k that divide the 3k-th Clausen number.

Original entry on oeis.org

1, 2, 6, 14, 42, 114, 602, 798, 1806, 5334, 34314, 101346, 229362, 4357878, 9786714, 12198858, 168241542, 185947566, 231778302, 524550894
Offset: 1

Views

Author

Peter Luschny, May 05 2012

Keywords

Comments

The classical Clausen numbers are given in A141056. See A160014 for generalizations. Related sequences are A014117 and A106741.

Links

Crossrefs

Cf. A014117, A027760, A106741, A141056, A160014.

Programs

  • Mathematica
    (* This program is not convenient for more than 15 terms *) c[n_] := Sum[Boole[PrimeQ[d+1]]/(d+1), {d, Divisors[n]}] // Denominator; Reap[For[n = 1, n < 10^7, n++, If[Divisible[c[3*n], n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 21 2013 *)
  • PARI
    A212197_list(searchlimit) =
{
    for (n=1, searchlimit,
        p = 1;
        fordiv(3*n, d,
            r = d + 1;
            if (isprime(r), p = p*r;)
        );
        if (Mod(p, n) == 0, print1(n, ", "));
    );
}

A325147 Reduced Clausen numbers.

Original entry on oeis.org

10, 546, 2, 46, 6630, 76670, 211659630, 6, 261870, 111418, 46, 13589784390, 524588442, 114, 1138240087314330, 2, 276742830, 26805565070, 1909802752494, 3210, 15370, 177430547680928732190, 358, 5760551069383110, 76004922, 1126, 4347631610092420338, 81366
Offset: 1

Views

Author

Peter Luschny, May 21 2019

Keywords

Comments

Let P(m) denote the prime factors of m and C(m) = Clausen(m-1, 1) (cf. A160014) then Product_{p in P(C(m)) setminus P(m)} p is in this sequence provided P(m) is a subset of P(C(m)).

Examples

			Let n = 561 then P(561) = {3, 11, 17} and P(Clausen(560,1)) = {2, 3, 5, 11, 17, 29, 41, 71, 113, 281}. Since P(561) is a subset of P(Clausen(560, 1)), a(18) = 2*5*29*41*71*113*281 = 26805565070.

Crossrefs

Weak Carmichael numbers are A225498. Clausen numbers are in A160014.
A324977 is a subsequence.

Programs

  • Maple
    with(numtheory): a := proc(n) if isweakCarmichael(n) then # cf. A225498 and A160014
mul(m, m in factorset(Clausen(n-1, 1)) minus factorset(n)) else NULL fi end:
seq(a(n), n=2..1350);
  • Mathematica
    pf[n_] := FactorInteger[n][[All, 1]];
Clausen[0, ] = 1; Clausen[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[# + k]&] + k);
weakCarmQ[n_] := If[EvenQ[n] || PrimeQ[n], Return[False], pf[n] == (pf[n] ~Intersection~ pf[Clausen[n - 1, 1]])];
f[n_] := Times @@ Complement[pf[Clausen[n - 1, 1]], pf[n]];
f /@ Select[Range[2, 2000], weakCarmQ] (* Jean-François Alcover, Jul 21 2019 *)

A356655 Clausen numbers based on the strictly proper divisors of n, 1 < d < n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 1, 15, 1, 3, 1, 105, 1, 3, 1, 15, 1, 21, 1, 165, 1, 3, 1, 1365, 1, 3, 1, 15, 1, 231, 1, 255, 1, 3, 1, 25935, 1, 3, 1, 165, 1, 21, 1, 345, 1, 3, 1, 23205, 1, 33, 1, 15, 1, 399, 1, 435, 1, 3, 1, 465465, 1, 3, 1, 255, 1, 483, 1, 15, 1, 33, 1
Offset: 0

Views

Author

Peter Luschny, Aug 20 2022

Keywords

Crossrefs

Cf. A160014, A166120.

Programs

  • Maple
    clausen := proc(n) numtheory[divisors](n) minus {1, n};
map(i -> i+1, %); select(isprime, %); mul(i, i=%)  end:
seq(clausen(n), n = 0..80);
  • Mathematica
    a[n_] := Product[If[1 < d < n && PrimeQ[d + 1], d + 1, 1], {d, Divisors[n]}]; Array[a, 100, 0] (* Amiram Eldar, Aug 20 2022 *)
  • PARI
    a(n) = if (n, vecprod(select(isprime, apply(x->x+1, setminus(divisors(n), [1,n])))), 1); \\ Michel Marcus, Aug 21 2022

Formula

a(n) = Product_{d | n} (d + 1), where d + 1 is prime and 1 < d < n.
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