A377484 -id:A377484 - cp's OEIS frontend

A377949 Numbers k such that k | A377484(k) and (k+1) | A377484(k+1).

156519, 245024, 310155, 524799, 638000, 893024, 1079000, 2055780, 2095975, 2203200, 2566025, 2592512, 2853135, 2934063, 3213375, 3294719, 4056975, 4322240, 4471935, 5746455, 6515145, 7289919, 7316000, 7329608, 7866495, 8459360, 8555624, 8934464, 9035415, 11291091

Offset: 1

Amiram Eldar, Nov 12 2024

nonn

Numbers k such that k and k+1 are both terms in A056954.

			156519 is a term since A377484(156519) is divisible by 156519 and A377484(156520) is divisible by 156520.

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

Cf. A377484.
Subsequence of A056954.
Similar sequences: A355332, A377951, A377953.

q[n_] := q[n] = Divisible[Times @@ (Rest@ Divisors[n] - 1), n]; Select[Range[10^6], q[#] && q[#+1] &]

is1(k) = if(k == 1, 1, my(d = divisors(k)); !(prod(i = 2, #d, d[i]-1) % k));
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 2, 3, 16, 36, 2, 30, 4, 6, 16, 45, 4, 80, 2, 108, 16, 6, 2, 210, 24, 12, 32, 18, 4, 1008, 2, 45, 64, 12, 48, 8400, 4, 18, 16, 2268, 4, 240, 2, 90, 512, 18, 2, 3150, 32, 216, 64, 540, 4, 160, 144, 2430, 32, 12, 2, 166320, 4, 6, 1280, 405, 48, 1344

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

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

Cf. A000079, A002144, A002145, A020696, A377484, A378056.

a[n_] := GCD[Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@ d - 1)]; Array[a, 70]

a(n) = if(n == 1, 1, my(d = divisors(n)); gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)));

a(n) = 2 if and only if n = 6 or n is a prime of the form 4*k+3 (A002145).
a(n) = 4 if and only if n is a prime of the form 4*k+1 (A002144).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079).

A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

Original entry on oeis.org

1, 1, 2, 3, 8, 10, 36, 42, 128, 216, 600, 660, 3456, 3744, 9408, 18900, 61440, 65280, 279936, 295488, 1152000, 2116800, 4878720, 5100480, 31850496, 41472000, 93450240, 163762560, 568995840, 589317120, 3265920000, 3374784000, 11324620800, 19269550080, 42188636160

Offset: 0

N. J. A. Sloane, Simon Plouffe

Product floor(n/1)*floor(n/2)*floor(n/3)*...*floor(n/n).

a(n) is the number of functions f:[n]->[n] where f(x) is a multiple of x for all x in [n]. We note that there are floor[n/x] possible choices for each image of x under f. [Dennis P. Walsh, Nov 06 2014]

			For n=4 the a(4)=8 functions are given by the image sequences <1,2,3,4>, <1,4,3,4>, <2,2,3,4>, <2,4,3,4>, <3,2,3,4>, <3,4,3,4>, <4,2,3,4>, and <4,4,3,4>. [_Dennis P. Walsh_, Nov 06 2014]

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant
Index entries for sequences related to factorial numbers

Cf. A006218, A075885, A092143, A131385, A131387, A377484, A007955, A075993.

a010786 n = product $ map (div n) [1..n]
-- Reinhard Zumkeller, Feb 26 2012

[&*[n div i: i in [1..n]]: n in [1..35]]; // Vincenzo Librandi, Oct 03 2018

Maple
```
a := n -> mul( floor(n/k), k=1..n);
```

Table[Product[Floor[n/k],{k,n}],{n,40}] (* Harvey P. Dale, May 09 2017 *)

vector(50, n, prod(k=1, n, n\k)) \\ Michel Marcus, Nov 10 2014

a(n+1) = a(n)*A208449(n)/A208450(n). - Reinhard Zumkeller, Feb 26 2012
GCD(a(n), a(n+1)) = A208448(n). - Reinhard Zumkeller, Feb 26 2012
From Vaclav Kotesovec, Oct 03 2018: (Start)
log(a(n)) ~ c * (n - log(2*Pi*n)/2), where c = 0.7885...
Conjecture: c = A085361. (End)
From Ridouane Oudra, Jan 18 2025: (Start)
a(n) = Product_{k=1..n} ((k+1)/k)^floor(n/(k+1)).
a(n) = Product_{k=1..n} k^A075993(n, k).
a(n) = A092143(n)/f(n), where f(n) = Product_{k=1..n} ((floor(n/k)-1)!).
a(n) = A092143(n)/g(n), where g(n) = Product_{k=1..n} A377484(k).
a(n)/a(n-1) = A007955(n)/A377484(n). (End)

More terms from Hieronymus Fischer, Jul 08 2007
Edited by N. J. A. Sloane, Jul 05 2008 at the suggestion of Rick L. Shepherd
a(0)=1 prepended by Alois P. Heinz, Oct 30 2023

A056954 Numbers k such that k^2 divides A056819(k).

Original entry on oeis.org

1, 30, 60, 90, 105, 120, 132, 144, 168, 180, 210, 240, 252, 264, 280, 336, 360, 380, 396, 420, 495, 504, 520, 528, 540, 546, 552, 560, 612, 616, 630, 660, 720, 728, 756, 760, 792, 840, 858, 870, 900, 924, 990, 1008, 1040, 1050, 1056, 1080, 1092, 1104

Offset: 1

Leroy Quet, Sep 06 2000

From Amiram Eldar, Nov 12 2024: (Start)
Equivalently, numbers k that divide A377484(k) = Product_{d|k, d>1} (d - 1).
After the first term a(1) = 1, the next odd term is a(5) = 105, the next term that is coprime to 6 is a(228) = 6545, and the next term that is coprime to 30 is a(574) = 19019. (End)

			30 is a term because 30^2 divides A056819(30) = 5320224000.

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

Cf. A056819, A377484.
A377949 is a subsequence.
Similar sequences: A355331, A377950, A377952.

Select[Range[1000], Divisible[Times @@ (Rest@ Divisors[#] - 1), #] &] (* Amiram Eldar, Nov 12 2024 *)

is(k) = if(k == 1, 1, my(d = divisors(k)); !(prod(i = 2, #d, d[i]-1) % k)); \\ Amiram Eldar, Nov 12 2024

A208449 Numerator of A010786(n+1) / A010786(n).

Original entry on oeis.org

2, 3, 8, 5, 18, 7, 64, 27, 25, 11, 288, 13, 98, 225, 1024, 17, 729, 19, 2000, 147, 242, 23, 55296, 125, 169, 729, 10976, 29, 1125, 31, 32768, 1089, 289, 1225, 209952, 37, 722, 507, 640000, 41, 64827, 43, 42592, 91125, 1058, 47, 14155776, 343, 15625, 2601

Offset: 1

Reinhard Zumkeller, Feb 26 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A208450 (denominator).
Cf. A010786, A208448.
Cf. A007955, A377484.

import Data.Ratio ((%), numerator)
a208449 n = a208449_list !! (n-1)
a208449_list = map numerator $
   zipWith (%) (tail a010786_list) a010786_list

A208449[n_] := Times @@ # / GCD[Times @@ #, Times @@ (# - 1)] & [Rest[Divisors[n + 1]]];
Array[A208449, 100] (* Paolo Xausa, Feb 20 2025 *)

f(n) = prod(k=1, n, n\k); \\ A010786
a(n) = numerator(f(n+1)/f(n)); \\ Michel Marcus, Feb 08 2025

a(n) = A010786(n+1) / A208448(n).

a(n) = A007955(n+1)/gcd(A007955(n+1), A377484(n+1)). - Ridouane Oudra, Feb 03 2025

A208450 Denominator of A010786(n+1) / A010786(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 21, 16, 9, 10, 55, 12, 39, 112, 315, 16, 170, 18, 513, 80, 105, 22, 8855, 96, 75, 416, 3159, 28, 203, 30, 9765, 640, 132, 816, 32725, 36, 333, 304, 140049, 40, 13325, 42, 13545, 39424, 495, 46, 2080925, 288, 5292, 1600, 11475, 52, 117130

Offset: 1

Reinhard Zumkeller, Feb 26 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A208449 (numerator).

Cf. A007955, A377484.

import Data.Ratio ((%), denominator)
a208450 n = a208450_list !! (n-1)
a208450_list = map denominator $
   zipWith (%) (tail a010786_list) a010786_list

A208450[n_] := Times @@ (# - 1) / GCD[Times @@ #, Times @@ (# - 1)] & [Rest[Divisors[n + 1]]];
Array[A208450, 100] (* Paolo Xausa, Feb 20 2025 *)

a(n) = A010786(n) / A208448(n).

a(n) = A377484(n+1)/gcd(A007955(n+1), A377484(n+1)). - Ridouane Oudra, Feb 08 2025

A378054 Numbers k that divide A378053(k) = gcd(Product_{d|k} (d + 1), Product_{d|k, d>1} (d - 1)).

Original entry on oeis.org

1, 60, 90, 120, 144, 168, 180, 210, 240, 252, 280, 336, 360, 420, 504, 540, 560, 630, 660, 720, 840, 900, 924, 990, 1008, 1056, 1080, 1092, 1200, 1260, 1320, 1404, 1440, 1512, 1560, 1680, 1800, 1848, 1872, 1890, 1980, 2016, 2100, 2112, 2160, 2184, 2310, 2376, 2400

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

After the first term a(1) = 1, the next odd term is a(71) = 3465, the next term that is coprime to 6 is a(1058) = 95095, and the next term that is coprime to 30 is a(12174) = 2263261.

			60 is a term since A378053(60) = 166320 = 60 * 2772 is divisible by 60.

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

Intersection of A056954 and A355331.
A378055 is a subsequence.
Cf. A020696, A377484, A378053, A378058.

Select[Range[2500], And @@ Divisible[{Times @@ ((d = Divisors[#]) + 1), Times @@ (Rest @ d - 1)}, #] &]

is(k) = if(k == 1, 1, my(d = divisors(k)); !(gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)) % k));

A378055 Numbers k such that k | A378053(k) and (k+1) | A378053(k+1).

Original entry on oeis.org

638000, 13466816, 14753024, 16092999, 19494189, 38137749, 63668079, 80061344, 86119704, 107232255, 112375899, 121550624, 127205000, 154466675, 294147854, 391738599, 553140224, 561712095, 682199595, 728999999, 871651143, 879207615, 911062844, 920985624, 1017609999

Offset: 1

Amiram Eldar, Nov 15 2024

nonn

Intersection of A355332 and A377949.
Subsequence of A378054.
Cf. A020696, A377484, A378053, A378059.

q[n_] := q[n] = And @@ Divisible[{Times @@ ((d = Divisors[n]) + 1), Times @@ (Rest@d - 1)}, n]; Select[Range[2*10^7], q[#] && q[# + 1] &]

is1(k) = if(k == 1, 1, my(d = divisors(k)); !(gcd(prod(k=1, #d, d[k]+1), prod(k=2, #d, d[k]-1)) % k));
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

cp's OEIS Frontend

A377949 Numbers k such that k | A377484(k) and (k+1) | A377484(k+1).

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A378053 a(n) = gcd(Product_{d|n} (d + 1), Product_{d|n, d>1} (d - 1)) = gcd(A020696(n), A377484(n)).

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A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

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A056954 Numbers k such that k^2 divides A056819(k).

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A208449 Numerator of A010786(n+1) / A010786(n).

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A208450 Denominator of A010786(n+1) / A010786(n).

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A378054 Numbers k that divide A378053(k) = gcd(Product_{d|k} (d + 1), Product_{d|k, d>1} (d - 1)).

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