A057643 -id:A057643 - cp's OEIS frontend

A377950 Numbers k that divide A057643(k) = lcm{d+1 : d|k}.

1, 2, 6, 12, 42, 60, 84, 120, 140, 156, 168, 210, 220, 240, 280, 312, 360, 420, 440, 462, 468, 504, 600, 630, 660, 720, 770, 780, 840, 924, 936, 1008, 1064, 1092, 1170, 1200, 1260, 1320, 1404, 1428, 1540, 1560, 1680, 1683, 1800, 1806, 1848, 1860, 1980, 2016, 2160

Offset: 1

Amiram Eldar, Nov 12 2024

After the first term a(1) = 1, the next odd term is a(44) = 1683, the next term that is coprime to 6 is a(159) = 10465, and the next term that is coprime to 30 is a(1359) = 151487.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A057643.
A377951 is a subsequence.
Similar sequences: A056954, A355331, A377952.

Select[Range[2500], Divisible[LCM @@ (Divisors[#] + 1), #] &]

is(k) = !(lcm(apply(x->x+1, divisors(k))) % k);

A377951 Numbers k such that k | A057643(k) and (k+1) | A057643(k+1).

Original entry on oeis.org

1, 799799, 1204280, 2460975, 3382379, 6116175, 7050120, 8070699, 13339424, 20966049, 28460600, 41265680, 41463135, 52404624, 66108399, 68919080, 72946224, 81102944, 84479680, 102971924, 106663304, 110791736, 112375899, 115225439, 118333215, 131115984, 132073424

Offset: 1

Amiram Eldar, Nov 12 2024

nonn

Numbers k such that k and k+1 are both terms in A377950.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A057643.
Subsequence of A377950.
Similar sequences: A355332, A377949, A377953.

q[n_] := q[n] = Divisible[LCM @@ (Divisors[n] + 1), n]; Select[Range[4*10^6], q[#] && q[# + 1] &]

is1(k) = !(lcm(apply(x->x+1, divisors(k))) % k);
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A378056 a(n) = gcd(A057643(n), A084190(n)) = gcd(lcm{d+1 : d|n}, lcm{d-1 : d > 1 and d|n}).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 3, 4, 6, 2, 30, 2, 6, 4, 15, 2, 20, 2, 6, 4, 6, 2, 210, 6, 6, 4, 6, 2, 84, 2, 15, 4, 6, 12, 420, 2, 6, 4, 126, 2, 60, 2, 30, 8, 6, 2, 210, 8, 6, 4, 30, 2, 20, 12, 90, 4, 6, 2, 4620, 2, 6, 40, 45, 6, 84, 2, 6, 4, 36, 2, 420, 2, 6, 24, 30, 12

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A057643, A084190, A378053, A378057.

a[n_] := Module[{d = Divisors[n]}, GCD[LCM @@ (d + 1), LCM @@ (Rest @ d - 1)]]; a[1] = 1; Array[a, 100]

a(n) = {my(d = divisors(n)); gcd(lcm(apply(x->x+1, d)), lcm(apply(x -> if(x > 1, x-1, x), d)));}

a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).

a(p) = 2 for an odd prime p. Composite numbers k such that a(k) = 2 are listed in A378057.

A083344 a(n) = A082457(n) - A066715(n) = gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 30, 0, 0, 0, 0, 0, 0, 4, 0, 0

Offset: 1

Labos Elemer, Apr 25 2003

sign

			n=22: 2n+1 = 45, A057643(45) = 5520, a(22) = gcd(45,5520) = 15 while A066715(45) = 3; a(22) = 15-3 = 12; sites where nonzero terms appear see in A082452.

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A057643, A066715, A000203, A082457, A082452.

di[x_] := Apply[LCM, Divisors[x]+1] (*A066715=*)t1=Table[GCD[2*n+1, DivisorSigma[1, 2*n+1]], {n, 1, 2048}]; (*A082457=*)t2=Table[GCD[2*w+1, di[1+2*w]], {w, 1, 2048}]; (*A083344=*)t2-t1;

a(n)=gcd(lcm(apply(d->d+1,divisors(2*n+1))),2*n+1)-gcd(sigma(2*n+1),2*n+1) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = gcd(2n+1, lcm(1+D(2n+1))) - gcd(2n+1, sigma(2n+1)), gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)), where D(x) is the set of divisors of x.

A082457 a(n) = gcd(1+2n, A057643(1+2n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 5, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 3, 1, 5, 1, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1

Offset: 1

Labos Elemer, Apr 25 2003

nonn

This and A066715 are mostly equal; for differences see A083344.

			n=22: 2n+1=45, A057643(45)=5520, a(22)=gcd(45,5520)=15 while A066715(45)=3.

Cf. A057643, A066715, A000203, A083344.

di[x_] := Apply[LCM, Divisors[x]+1] Table[GCD[2*w+1, di[1+2*w]], {w, 1, 2048}];

A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)(b+1)...*(k+1).

Original entry on oeis.org

2, 6, 8, 30, 12, 168, 16, 270, 80, 396, 24, 10920, 28, 720, 768, 4590, 36, 31920, 40, 41580, 1408, 1656, 48, 2457000, 312, 2268, 2240, 104400, 60, 5499648, 64, 151470, 3264, 3780, 3456, 76767600, 76, 4680, 4480, 15343020, 84, 19071360, 88, 372600, 353280, 6768

Offset: 1

Amarnath Murthy, Jun 01 2003

Named "Vandiver's arithmetical function" by Sándor (2021), after the American mathematician Harry Schultz Vandiver (1882-1973). - Amiram Eldar, Jun 29 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38.
Harry S. Vandiver, Problem 116, American Mathematical Monthly, Vol. 11, No. 2 (1904), pp. 38-39.

Cf. A027750, A000005, A003959, A007955.
Cf. A057643 (LCM instead of product).
Cf. A299436 (exp).

a020696 = product . map (+ 1) . a027750_row'
-- Reinhard Zumkeller, Mar 28 2015

a:= n-> mul(d+1, d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jun 30 2022~

Table[Times @@ (Divisors[n] + 1), {n, 43}] (* Ivan Neretin, May 27 2015 *)

a(n) = {d = divisors(n); return (prod(i=1, #d, d[i]+1));} \\ Michel Marcus, Jun 12 2013

from math import prod
from sympy import divisors
def A020696(n): return prod(d+1 for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 30 2022

a(p) = 2(p+1), a(p^2) = 2(p+1)(p^2+1) for primes p.
a(n) = Product_{k = 1..A000005(n)} (A027750(n,k) + 1). - Reinhard Zumkeller, Mar 28 2015
a(n) = Product_{d|n} (d+1). - Amiram Eldar, Jun 29 2022

Edited by Don Reble, Jun 05 2003

A164020 Denominators of Bernoulli numbers interleaved with even numbers.

Original entry on oeis.org

1, 2, 6, 4, 30, 6, 42, 8, 30, 10, 66, 12, 2730, 14, 6, 16, 510, 18, 798, 20, 330, 22, 138, 24, 2730, 26, 6, 28, 870, 30, 14322, 32, 510, 34, 6, 36, 1919190, 38, 6, 40, 13530, 42, 1806, 44, 690, 46, 282, 48, 46410, 50, 66, 52, 1590, 54, 798, 56, 870, 58, 354, 60, 56786730

Offset: 0

Paul Curtz, Aug 08 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A027642, A141056, A160014.

[IsEven(n) select Denominator(Bernoulli(n)) else n+1: n in [0..100]]; // Vincenzo Librandi, Sep 08 2017

a[n_]:=If[OddQ[n], n+1, BernoulliB[n] // Denominator]; Table[a[n], {n, 0, 60}](* Jean-François Alcover, Dec 29 2012 *)
With[{nn=60},Riffle[Denominator[BernoulliB[Range[0,nn,2]]],Range[2,nn,2]]] (* Harvey P. Dale, Jul 18 2015 *)

a(2*n) = A002445(n).
a(2*n+1) = 2*(n+1).
a(n) divides A057643(n). Franklin T. Adams-Watters, Aug 03 2012

Extended by R. J. Mathar, Sep 23 2009

A082452 a(n)=2n+1 where n is such that A083344(n) is not zero.

Original entry on oeis.org

45, 75, 117, 147, 189, 195, 225, 231, 245, 315, 325, 345, 363, 385, 405, 441, 483, 495, 507, 525, 561, 567, 585, 595, 645, 663, 675, 693, 715, 735, 765, 775, 777, 795, 819, 845, 847, 867, 875, 891, 931, 945, 957, 975, 1001, 1035, 1071, 1083, 1089, 1095, 1125

Offset: 1

Labos Elemer, Apr 25 2003

nonn

			n=22: 2n+1=45, A057643(45)=5520, a(22)=GCD[45,5520]=15 while A066715[45]=3; a(22)=15-3=12.

Cf. A057643, A066715, A000203, A082457, A083344.

di[x_] := Apply[LCM, Divisors[x]+1] (*A066715=*)t1=Table[GCD[2*n+1, DivisorSigma[1, 2*n+1]], {n, 1, 2048}]; (*A082457=*)t2=Table[GCD[2*w+1, di[1+2*w]], {w, 1, 2048}]; (*A083344=*)t3=t2-t1; (*A082452=*)1+2*Flatten[Position[Abs[Sign[t3]], 1]];

A082453 a(n)=2n+1 where n is such that A083344(n) is zero.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137

Offset: 1

Labos Elemer, Apr 25 2003

nonn

			First two missing odd numbers are 45 and 75; missing terms are in A082452.

Cf. A057643, A066715, A000203, A082457, A083344.

di[x_] := Apply[LCM, Divisors[x]+1] (*A066715=*)t1=Table[GCD[2*n+1, DivisorSigma[1, 2*n+1]], {n, 1, 2048}]; (*A082457=*)t2=Table[GCD[2*w+1, di[1+2*w]], {w, 1, 2048}]; (*A083344=*)t3=t2-t1; (*A082453=*)1+2*Flatten[Position[Abs[Sign[t3]], 0]];

A119250 Smallest number greater than n having a maximal number of divisors d such that d-1 are divisors of n.

Original entry on oeis.org

2, 6, 4, 6, 6, 12, 8, 30, 20, 66, 12, 60, 14, 24, 48, 30, 18, 84, 20, 30, 88, 276, 24, 60, 78, 42, 140, 30, 30, 84, 32, 90, 204, 630, 72, 60, 38, 60, 280, 90, 42, 84, 44, 60, 60, 1128, 48, 60, 200, 66, 468, 210, 54, 84, 168, 120, 580, 1770, 60, 420, 62, 96, 440, 90, 462, 84

Offset: 1

Reinhard Zumkeller, May 10 2006

nonn

Let x(0)=1 and x(k) = lcm(x(k-1), d(k)) with 1 <= k <= A000005(n), where d(k) = (k-th divisor of n): A057643(n)=x(A000005(n)) and a(n) = Min{x(i): x(i)>n}.
a(n) = n+1 iff n is an odd prime;
A057643(n) = x(A000005(n)) and a(n) = Min{x(i): x(i)>n}.

Cf. A027750.

cp's OEIS Frontend

A377950 Numbers k that divide A057643(k) = lcm{d+1 : d|k}.

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A377951 Numbers k such that k | A057643(k) and (k+1) | A057643(k+1).

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A378056 a(n) = gcd(A057643(n), A084190(n)) = gcd(lcm{d+1 : d|n}, lcm{d-1 : d > 1 and d|n}).

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A083344 a(n) = A082457(n) - A066715(n) = gcd(2n+1, A057643(2n+1)) - gcd(2n+1, A000203(2n+1)).

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A082457 a(n) = gcd(1+2n, A057643(1+2n)).

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A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)*(b+1)*...*(k+1).

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Maple

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A164020 Denominators of Bernoulli numbers interleaved with even numbers.

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A082452 a(n)=2n+1 where n is such that A083344(n) is not zero.

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A020696 Let a,b,c,...k be all divisors of n; a(n) = (a+1)(b+1)...*(k+1).