A027760 -id:A027760 - cp's OEIS frontend

A027759 Numerator of Sum_{p prime, p-1|n} 1/p.

1, 5, 1, 31, 1, 41, 1, 31, 1, 61, 1, 3421, 1, 5, 1, 557, 1, 821, 1, 371, 1, 121, 1, 3421, 1, 5, 1, 929, 1, 15745, 1, 557, 1, 5, 1, 2557843, 1, 5, 1, 15541, 1, 1805, 1, 743, 1, 241, 1, 60887, 1, 61, 1, 1673, 1, 821, 1, 929, 1, 301, 1, 79085411, 1, 5, 1, 557

Offset: 1

N. J. A. Sloane

			1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A027760 (denominator).

a[n_] := 1/(Select[Divisors[n], PrimeQ[# + 1]&] + 1) // Total // Numerator;
Array[a, 100] (* Jean-François Alcover, Sep 20 2020 *)

a(n) = numerator(sumdiv(n, d, if (isprime(d+1), 1/(d+1)))); \\ Michel Marcus, May 06 2021

a(2n-1) = 1 and a(2n) = A000146(n)* A002445(n) - A000367(n) for n > 0. - Thomas Ordowski, May 06 2021

a(57)-a(64) from John Cerkan, Mar 21 2018

A330541 Triangle read by rows: T(n,k) = gcd {x^n - x^k : x is an integer}, 0 < k < n.

Original entry on oeis.org

2, 6, 2, 2, 12, 2, 30, 2, 24, 2, 2, 60, 2, 24, 2, 42, 2, 120, 2, 24, 2, 2, 252, 2, 240, 2, 24, 2, 30, 2, 504, 2, 240, 2, 24, 2, 2, 60, 2, 504, 2, 240, 2, 24, 2, 66, 2, 120, 2, 504, 2, 240, 2, 24, 2, 2, 132, 2, 240, 2, 504, 2, 240, 2, 24, 2

Offset: 2

Peter Kagey, Dec 17 2019

All diagonals are weakly increasing, T(n,k) divides T(n+1,k+1), and the m-th diagonal converges to A079612(m).
First column is A027760.
First value where T(n,k) < gcd(2^n - 2^k, 3^n - 3^k) is T(12,1) = 2 < 46.
Maximum value in the n-th row is given by A330542(n).

			Table begins:
  n\k|  1    2    3    4    5    6    7    8   9  10 11
  ---+-------------------------------------------------
   2 |  2;
   3 |  6,   2;
   4 |  2,  12,   2;
   5 | 30,   2,  24,   2;
   6 |  2,  60,   2,  24,   2;
   7 | 42,   2, 120,   2,  24,   2;
   8 |  2, 252,   2, 240,   2,  24,   2;
   9 | 30,   2, 504,   2, 240,   2,  24,   2;
  10 |  2,  60,   2, 504,   2, 240,   2,  24,  2;
  11 | 66,   2, 120,   2, 504,   2, 240,   2, 24,  2;
  12 |  2, 132,   2, 240,   2, 504,   2, 240,  2, 24, 2.

Peter Kagey, Table of n, a(n) for n = 2..10012 (first 141 rows, flattened)
Mathematics Stack Exchange, Computing gcd {n^k - n^l : n in Z}.

Cf. A027760, A079612, A330542.

A108939 Triangle read by rows in which row n lists all primes p such that p-1|n.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 5, 2, 2, 3, 7, 2, 2, 3, 5, 2, 2, 3, 11, 2, 2, 3, 5, 7, 13, 2, 2, 3, 2, 2, 3, 5, 17, 2, 2, 3, 7, 19, 2, 2, 3, 5, 11, 2, 2, 3, 23, 2, 2, 3, 5, 7, 13, 2, 2, 3, 2, 2, 3, 5, 29, 2, 2, 3, 7, 11, 31, 2, 2, 3, 5, 17, 2, 2, 3, 2, 2, 3, 5, 7, 13, 19, 37, 2, 2, 3, 2, 2, 3, 5, 11, 41, 2

Offset: 1

Philippe Deléham, Jul 20 2005

Row 2n-1 contains only the term 2.

			Row n = 1 : 2 because 1|1.
Row n = 2 : 2, 3 because 1|2 and 2|2.
Row n = 3 : 2 because 1|3.
Row n = 4 : 2, 3, 5 because 1|4, 2|4 and 4|4.
Row n = 5 : 2 because 1|5.
Row n = 6 : 2, 3, 7 because 1|6, 2|6 and 6|6.
Row n = 7 : 2 because 1|7.
Row n = 8 : 2, 3, 5 because 1|8, 2|8 and 4|8.
Row n = 9 : 2 because 1|9.
Row n = 10 : 2, 3, 11 because 1|10, 2|10 and 10|10.
Row n = 11 : 2 because 1|11.
Row n = 12 : 2, 3, 5, 7, 13 because 1|12, 2|12, 4|12, 6|12 = and 12|12.

Robert Israel, Table of n, a(n) for n = 1..10000 (rows 1 to 3261, flattened)

Row products are A027760. Row sums are A085020. Cf. A067513, A108077.

with(numtheory): for n from 1 to 20 do div:=divisors(n): A:=[seq(div[j]+1,j=1..tau(n))]: B:={}: for i from 1 to tau(n) do if isprime(A[i])=true then B:=B union {A[i]} else B:=B: fi: od: C:=convert(B,list): b[n]:=C: od: for n from 1 to 20 do b[n]:=b[n] od; # yields sequence in triangular form - Emeric Deutsch, Aug 03 2005

Corrected by Robert Israel, Sep 21 2023

A141321 a(n) = -A141055(n)/(n+1)!.

Original entry on oeis.org

1, 1, 10, 5, 42, 14, 60, 15, 110, 22, 5460, 910, 420, 60, 2040, 255, 11970, 1330, 23100, 2310, 15180, 1380, 163800, 13650, 3276, 252, 8120, 580, 286440, 19096, 314160, 19635, 3570, 210, 11515140, 639730, 103740, 5460

Offset: 0

Paul Curtz, Aug 02 2008

a(n+1)/a(n)= 2/2, 30/3, 2/4, 42/5, 2/6, 30/7, 2/8, 66/9, 2/10, 2730/11, 2/12 = A027760(n+2)/(n+1), see A141410. Numerators are also A141056(n+3).

Cf. A027760, A141055.

A141055 := proc(n) if n = 0 then -1; else procname(n-1)*A027760(n+2) ; end if; end proc:
A141321 := proc(n) -A141055(n)/(n+1)! ; end proc: # R. J. Mathar, Jul 08 2011

(* b = A141055 *) b[n_] := b[n] = b[n-1]*If[OddQ[n], 2, Denominator[BernoulliB[n+2]]]; b[0]=-1; a[n_] := -b[n]/(n+1)!; Table[a[n], {n, 0, 37}] (* Jean-François Alcover, Dec 18 2014 *)

a(n)=if(n, my(pr=a(n-1)); fordiv(n+2, d, if(isprime(d+1), pr*=d+1)); pr, 1)/(n+1) \\ Charles R Greathouse IV, Jul 08 2011

a(2n) / a(2n+1) = n + 1.

A322702 a(n) is the product of primes p such that p+1 divides n.

Original entry on oeis.org

1, 1, 2, 3, 1, 10, 1, 21, 2, 1, 1, 330, 1, 13, 2, 21, 1, 170, 1, 57, 2, 1, 1, 53130, 1, 1, 2, 39, 1, 290, 1, 651, 2, 1, 1, 5610, 1, 37, 2, 399, 1, 5330, 1, 129, 2, 1, 1, 2497110, 1, 1, 2, 3, 1, 9010, 1, 273, 2, 1, 1, 10727970, 1, 61, 2, 651, 1, 10, 1, 201, 2

Offset: 1

Daniel Suteu, Dec 23 2018

nonn

In general, a(n) is the product of A072627(n) distinct prime factors, with a(n) = 1 iff A072627(n) = 0.

			For n=12, the divisors of 12 are {1, 2, 3, 4, 6, 12}. The prime numbers p, such that p+1 is a divisor of 12, are {2, 3, 5, 11}, therefore a(12) = 2 * 3 * 5 * 11 = 330.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A072627, A027760, A322356, A323156.

a:= n-> mul(`if`(isprime(d-1), d-1, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 29 2018

Array[Apply[Times, Select[Divisors@ #, PrimeQ[# - 1] &] - 1 /. {} -> {1}] &, 69] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]-1), d[k]-1, 1));

a(n) = Product_{p prime, p+1 divides n} p.
a(n) = denominator of Sum_{p prime, p+1 divides n} 1/p.
a(n) = Product_{d|n, d-1 is prime} (d-1), where d runs over the divisors of n.
a(2*n + 1) = 2, iff n == 1 (mod 3), else a(2*n + 1) = 1.
A001221(a(n)) = A072627(n). - Antti Karttunen, Jan 12 2019

A141055 The n-th differences of the row A141045(n,.).

Original entry on oeis.org

-1, -2, -60, -120, -5040, -10080, -302400, -604800, -39916800, -79833600, -217945728000, -435891456000, -2615348736000, -5230697472000, -2667655710720000, -5335311421440000, -4257578514309120000, -8515157028618240000, -2810001819444019200000

Offset: 0

Paul Curtz, Aug 01 2008

sign

Can be thought of as the second sequence of a family: the first is A091137, the third starts 1, 2, 84, 168.

			The first differences of A141045(2,.) are 11-(-19)=30 and -19-11 = -30. The 2nd difference is a(2)= -30-30= -60.

a[n_] := a[n] = a[n-1]*If[OddQ[n], 2, Denominator[BernoulliB[n+2]]]; a[0]=-1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 17 2014 *)

a(n)=if(n,my(pr=a(n-1));fordiv(n+2,d,if(isprime(d+1),pr*=d+1));pr,-1) \\ Charles R Greathouse IV, Jul 08 2011

a(n+1) = a(n)*A027760(n+3).

Edited and extended by R. J. Mathar, Aug 12 2008

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Original entry on oeis.org

1, 2, 6, 24, 30, 1440, 42, 120960, 30, 7257600, 66, 958003200, 2730, 5230697472000, 6, 62768369664000, 510, 64023737057280000

Offset: 0

Paul Curtz, Sep 28 2009

b(n)=a(2n+1)/a(2n) =2,4,48,2880,241920,145152,= 2*(1,2,24,1440,=1,2*A141421). Among other denominators, A027642,A141056,A164020. 2*A141421 is second bisection of A091137 which is linked to Bernoulli via A027760. See A160014,von Staudt-Clausen theorem.

A225821 a(n) = Product_{p | p is prime and p, p-1 both divide n}.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 10, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 10, 1, 42, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 10, 1, 2, 1, 42

Offset: 1

José María Grau Ribas, Jul 30 2013

nonn

a(n) = 2 iff n is even and is a term of A226872. - Daniel Suteu, Jul 28 2019
From Bernard Schott, Jul 30 2019: (Start)
a(n) = n if n = 1, 2, 6, 42, 1806.
a(n) = 6 if n is of the form 2^i*3^j, i and j >= 1, so if n is a term of A033845.
a(n) = 10 if n is of the form 2^i*5^j, i >= 2 and j >= 1.
a(n) = 30 if n is of the form 2^i*3^j*5^k, i >=2, j >= 1 and k >= 1. (End)

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A173557, A027760.

fa=FactorInteger; d[m_]:= Product[If[IntegerQ[m/(fa[m][[i, 1]]-1)],fa[m][[i, 1]], 1], {i, Length@fa@m}]; Table[d[n], {n, 1, 333}]

a(n)=my(f=factor(n)[,1]); prod(i=1,#f,if(n%(f[i]-1)==0,f[i],1)) \\ Charles R Greathouse IV, Nov 13 2013

def A225821(n) : return prod(p for (p,m) in factor(n) if n%(p-1)==0) # Eric M. Schmidt, Jul 31 2013

a(n) = denominator(A031971(n)/n) = gcd(n, A027642(n)). - Daniel Suteu, Jul 28 2019

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

Peter Luschny, Dec 13 2023

nonn

All terms are squarefree.

			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.

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

{1}~Join~Array[Times @@ Select[Divisors[#] + 2, PrimeQ] &, 62] (* Michael De Vlieger, Dec 14 2023 *)

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

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)])

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

A176493 A091137(n)/(n+1).

Original entry on oeis.org

1, 1, 4, 6, 144, 240, 8640, 15120, 403200, 725760, 43545600, 79833600, 201180672000, 373621248000, 2092278988800, 3923023104000, 1883051089920000, 3556874280960000, 2688996956405760000, 5109094217170944000, 1605715325396582400000, 3065456530302566400000

Offset: 0

Paul Curtz, Apr 19 2010

nonn

a(n) = A174727(n)/A027760(n+1).

cp's OEIS Frontend

A027759 Numerator of Sum_{p prime, p-1|n} 1/p.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A330541 Triangle read by rows: T(n,k) = gcd {x^n - x^k : x is an integer}, 0 < k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A108939 Triangle read by rows in which row n lists all primes p such that p-1|n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A141321 a(n) = -A141055(n)/(n+1)!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A322702 a(n) is the product of primes p such that p+1 divides n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A141055 The n-th differences of the row A141045(n,.).

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

PARI

Formula

Extensions

A165823 Large denominators of Bernoulli numbers. Mix A002445, 2*A141421 .

Views

Author

Keywords

Comments

A225821 a(n) = Product_{p | p is prime and p, p-1 both divide n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs