A226040 -id:A226040 - cp's OEIS frontend

A225481 a(n) = product{ p primes <= n+1 such that p divides n+1 or p-1 divides n }.

1, 2, 6, 2, 30, 6, 42, 2, 30, 10, 66, 6, 2730, 14, 30, 2, 510, 6, 798, 10, 2310, 22, 138, 6, 2730, 26, 6, 14, 870, 30, 14322, 2, 5610, 34, 210, 6, 1919190, 38, 78, 10, 13530, 42, 1806, 22, 690, 46, 282, 6, 46410, 10, 1122, 26, 1590, 6, 43890, 14, 16530, 58

Offset: 0

Peter Luschny, May 29 2013

nonn

a(n) is the product over the primes <= n+1 which satisfy the weak Clausen condition. The weak Clausen condition relaxes the Clausen condition (p-1)|n by logical disjunction with p|(n+1).

			a(20) = 2310 = 2*3*5*7*11, because {3, 7} are divisors of 21 and {2, 5, 11} meet the Clausen condition 'p-1 divides n'.

Peter Luschny, Table of n, a(n) for n = 0..100
Peter Luschny, Generalized Bernoulli numbers.

A027760, A160014, A226040.

Cf. A000040.

a225481 n = product [p | p <- takeWhile (<= n + 1) a000040_list,
                         mod n (p - 1) == 0 || mod (n + 1) p == 0]
-- Reinhard Zumkeller, Jun 10 2013

divides := (a, b) -> b mod a = 0; primes := n -> select(isprime, [$2..n]);
A225481 := n -> mul(k,k in select(p -> divides(p,n+1) or divides(p-1,n), primes(n+1))); seq(A225481(n), n = 0..57);

a[n_] := Product[ If[ Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}]; Table[a[n], {n, 0, 57}] (* Jean-François Alcover, Jun 07 2013 *)

def divides(a, b): return b % a == 0
def A225481(n):
    return mul(filter(lambda p: divides(p,n+1) or divides(p-1,n), primes(n+2)))
[A225481(n) for n in (0..57)]

a(n) / A027760(n) = A226040(n) for n > 0.

A381639 Denominators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 15, 1, 1, 11, 17, 7, 3, 1, 19, 13, 5, 1, 21, 1, 11, 5, 23, 1, 3, 1, 5, 17, 13, 1, 3, 11, 7, 19, 29, 1, 15, 1, 31, 7, 1, 13, 33, 1, 17, 23, 35, 1, 3, 1, 37, 5, 19, 11, 39, 1

Offset: 1

Amiram Eldar, Mar 03 2025

First differs from A119288 at n = 30.
First differs from {A226040(n-1)} at n = 35.
Also denominators of the fractions whose numerators are A381641.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Jean-Louis Nicolas, Grandes valeurs de fonctions liées aux diviseurs premiers consécutifs d'un entier, in: Jean-Marie de Koninck and Claude Levesque (eds.), Théorie des nombres / Number Theory, Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, De Gruyter, 1989; alternative link.

Cf. A001221, A119288, A226040, A381638 (numerators), A381640, A381641.

a[n_] := Module[{p = FactorInteger[n][[;; , 1]]}, Denominator[Total[Most[p]/Rest[p]]]]; Array[a, 100]

a(n) = {my(p = factor(n)[,1]); denominator(sum(i = 1, #p-1, p[i]/p[i+1]));}

A226038 Numbers k such that there are no primes p which divide k+1 and p-1 does not divide k.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 15, 16, 18, 22, 24, 26, 28, 30, 31, 36, 40, 42, 44, 46, 48, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 88, 96, 100, 102, 106, 108, 112, 120, 124, 126, 127, 130, 136, 138, 148, 150, 156, 162, 166, 168, 172, 178, 180, 190, 192, 196, 198

Offset: 1

Peter Luschny, May 27 2013

These are the numbers which satisfy the weak Clausen condition but not the Clausen condition.

			A counterexample is n = 14. 5 divides 15 but 4 does not divide 14.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Peter Luschny, Generalized Bernoulli numbers.

Cf. A226039, A226040, A225481.

s := (p,n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
F := n -> select(p -> s(p,n), select('isprime', [$2..n]));
A226038_list := n -> select(k -> [] = F(k), [$0..n]);
A226038_list(200);

s[p_, n_] := Mod[n+1, p] == 0 && Mod[n, p-1] != 0; f[n_] := Select[ Select[ Range[n], PrimeQ], s[#, n] &]; A226038 = Select[ Range[0, 200], f[#] == {} &] (* Jean-François Alcover, Jul 29 2013, after Maple *)
Join[{0}, Select[Range[200], And @@ Divisible[#, FactorInteger[# + 1][[All, 1]] - 1] &]] (* Ivan Neretin, Aug 04 2016 *)

def F(n): return filter(lambda p: ((n+1) % p == 0) and (n % (p-1) != 0), primes(n))
def A226038_list(n): return list(filter(lambda n: not list(F(n)), (0..n)))
A226038_list(200)

A226039 Numbers k such that there exist primes p which divide k+1 and p-1 does not divide k.

Original entry on oeis.org

5, 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103

Offset: 1

Peter Luschny, May 27 2013

nonn

			20 is in this list because 7 divides 21 but 6 does not divide 20.

Peter Luschny, Generalized Bernoulli numbers.

Cf. A226038, A226040.

s := (p,n) -> ((n+1) mod p = 0) and (n mod (p-1) <> 0);
F := n -> select(p -> s(p,n), select('isprime', [$2..n]));
A226039_list := n -> select(k -> [] <> F(k), [$0..n]);
A226039_list(103);

selQ[n_] := AnyTrue[Prime[Range[PrimePi[n+1]]], Divisible[n+1, #] && !Divisible[n, #-1]&];
Select[Range[103], selQ] (* Jean-François Alcover, Jul 08 2019 *)

def F(n): return any(p for p in primes(n) if (n+1) % p == 0 and n % (p-1) != 0)
def A226039_list(n): return list(filter(F, (0..n)))
A226039_list(103)

cp's OEIS Frontend

A225481 a(n) = product{ p primes <= n+1 such that p divides n+1 or p-1 divides n }.

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A381639 Denominators of Sum_{i=1..omega(n)-1} p_{i}/p_{i+1}, where omega(n) = A001221(n) and p_1 < p_2 < ... p_omega(n) are the distinct prime factors of n; a(1) = 1.

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A226038 Numbers k such that there are no primes p which divide k+1 and p-1 does not divide k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

A226039 Numbers k such that there exist primes p which divide k+1 and p-1 does not divide k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage