A020696 -id:A020696 - cp's OEIS frontend

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

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));

A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

Original entry on oeis.org

1, 2, 12, 96, 2880, 34560, 5806080, 92897280, 25082265600, 2006581248000, 794606174208000, 19070548180992000, 208250386136432640000, 5831010811820113920000, 4198327784510482022400000, 3224315738504050193203200000, 14799609239733590386802688000000

Offset: 1

Ilya Gutkovskiy, Jun 26 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 1..178

Cf. A010786, A020696, A092143, A131385.

[(&*[Factorial(Ceiling(n/(n-j+1))): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Mar 08 2023

seq(mul(ceil(n/k)!, k=1..n), n=1..30); # Ridouane Oudra, Apr 10 2023

a[n_] := Product[Ceiling[n/k]!, {k, 1, n}]; Table[a[n], {n, 1, 17}]

a(n) = prod(k=1, n, ceil(n/k)!); \\ Michel Marcus, Jun 27 2019

def A308820(n): return product( factorial(ceil(n/(n-k+1))) for k in range(1,n+1))
[A308820(n) for n in range(1,21)] # G. C. Greubel, Mar 08 2023

a(n) = Product_{k=1..n-1} Product_{d|k} (d + 1).
a(n) = Product_{k=1..n-1} (k + 1)^floor((n-1)/k). - Ridouane Oudra, Apr 10 2023
a(n) = A131385(n)*A092143(n-1). - Ridouane Oudra, Sep 20 2024

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);}

A333121 a(1) = 1; a(n+1) = Product_{d|n} (1 + a(d)).

Original entry on oeis.org

1, 2, 6, 14, 90, 182, 7686, 15374, 1383750, 19372514, 10577393190, 21154786382, 2438935322096070, 4877870644192142, 224977149851430019446, 286620888910721844775478, 396611655030211352708069066250, 793223310060422705416138132502, 8436334593920261958919014477018674175558

Offset: 1

Ilya Gutkovskiy, Mar 08 2020

nonn

Cf. A007018, A020696, A066843, A068334.

a[1] = 1; a[n_] := a[n] = Product[1 + a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 19}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A308820 a(n) = Product_{k=1..n} ceiling(n/k)!.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A333121 a(1) = 1; a(n+1) = Product_{d|n} (1 + a(d)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica