A050873 -id:A050873 - cp's OEIS frontend

A169582 Complement of A169581.

3, 6, 8, 10, 15, 17, 18, 19, 21, 28, 30, 32, 34, 36, 39, 42, 45, 47, 49, 50, 51, 53, 55, 66, 68, 69, 70, 72, 74, 75, 76, 78, 91, 93, 95, 97, 98, 99, 101, 103, 105, 108, 110, 111, 114, 115, 117, 120, 122, 124, 126, 128, 130, 132, 134, 136, 153, 155, 156, 157, 159, 161

Offset: 1

Reinhard Zumkeller, Dec 02 2009

nonn

A054521(a(n)) = 0;

GCD(A002260(a(n)),A002024(a(n))) = A050873(a(n)) > 1.

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

A181569 Greatest common divisor of n! and n+1.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 8, 9, 10, 1, 12, 1, 14, 15, 16, 1, 18, 1, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 1, 32, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 1

Reinhard Zumkeller, Oct 31 2010

From Wilson's theorem, it follows that a(n) = 1 when n + 1 is prime, a(n) > 1 otherwise. - Alonso del Arte, Feb 25 2014

			a(6) = 1 because 6! and 7 are coprime.
a(7) = 8 because 7! = 5040 and gcd(5040, 8) = 8.
a(8) = 9 because 8! = 40320 and gcd(40320, 9) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000142, A006093, A010051, A050873.

Cf. A088140, A135683.

[GCD(Factorial(n), n+1): n in [1..80]]; // Vincenzo Librandi, Mar 03 2014

A181569:=n->gcd(n!,n+1): seq(A181569(n), n=1..100); # Wesley Ivan Hurt, Aug 13 2015

Table[GCD[n!, n + 1], {n, 80}] (* Alonso del Arte, Feb 25 2014 *)

a(n)= n!/denominator(polcoeff((x+1)*exp(x+x*O(x^n)), n));  \\ Gerry Martens, Aug 12 2015

A181569(n)=gcd(n!,n+1) \\ M. F. Hasler, Aug 16 2015

a(n) = A050873(A000142(n), n + 1);
a(A006093(n)) = 1;
for n > 3: a(n) = (n + 1) / (n*A010051(n+1) + 1).
a(n) = (n+1)/A014973(n+1). - Michel Marcus, Aug 14 2015

A245717 Triangle read by rows: T(n,k) = gcd(n,k^2), 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 4, 1, 8, 1, 4, 1, 8, 1, 1, 9, 1, 1, 9, 1, 1, 9, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Reinhard Zumkeller, Jul 30 2014

			First rows and their sums (A078430):
.    1:   1                                           1
.    2:   1, 2                                        3
.    3:   1, 1, 3                                     5
.    4:   1, 4, 1, 4                                 10
.    5:   1, 1, 1, 1, 5                               9
.    6:   1, 2, 3, 2, 1,  6                          15
.    7:   1, 1, 1, 1, 1,  1, 7                       13
.    8:   1, 4, 1, 8, 1,  4, 1, 8                    28
.    9:   1, 1, 9, 1, 1,  9, 1, 1, 9                 33
.   10:   1, 2, 1, 2, 5,  2, 1, 2, 1, 10             27
.   11:   1, 1, 1, 1, 1,  1, 1, 1, 1,  1, 11         21
.   12:   1, 4, 3, 4, 1, 12, 1, 4, 3,  4,  1, 12     50

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A050873, A002024, A133819, A078430 (row sums).

a245717 n k = a245717_tabl !! (n-1) !! (k-1)
a245717_row n = a245717_tabl !! (n-1)
a245717_tabl = zipWith (zipWith gcd) a002024_tabl a133819_tabl

Table[GCD[n,k^2],{n,15},{k,n}]//Flatten (* Harvey P. Dale, Nov 05 2022 *)

row(n) = vector(n, k, gcd(n, k^2)); \\ Michel Marcus, Jan 24 2022

A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 17, 23, 33, 47, 71, 92, 129, 169, 235, 299, 408, 525, 691, 885, 1147, 1427, 1832, 2312, 2878, 3635, 4519, 5631, 7002, 8637, 10514, 13055, 15864, 19396, 23530, 28702, 34746, 42210, 50671, 61224, 73506, 88394, 105447, 126398, 150588, 179075

Offset: 0

Alois P. Heinz, Dec 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003989, A050873, A096541, A339006, A339011, A339394.

b:= proc(n, i, p, d) option remember; `if`(n=0, igcd(p, d),
      add(b(n-i*j, i-1, p+j, d+signum(j)), j=`if`(i>1, 0..n/i, n)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, GCD[p, d],
     Sum[b[n - i*j, i - 1, p + j, d + Sign[j]],
     {j, If[i > 1, Range[0, n/i], {n}]}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Mar 09 2021, after Alois P. Heinz *)

A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

Original entry on oeis.org

1, 5, 29, 262, 3129, 46705, 823549, 16777544, 387421251, 10000003469, 285311670621, 8916100581446, 302875106592265, 11112006826387025, 437893890391180013, 18446744073743123788, 827240261886336764193, 39346408075299116257065

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A001923, A018804, A050873, A231712, A332517, A341036, A342370, A342389, A342423.

a[n_] := Sum[GCD[k, n]^k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^k);
```

If p is prime, a(p) = p-1 + p^p = A231712(p).

A345416 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = um+vn where u, v are minimal; T(m,n) = v.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, -1, 1, 0, 1, 1, 1, 0, 0, 1, -2, -1, 1, 1, 0, 1, 1, 2, 1, -1, 0, 0, 1, -3, 1, -1, 1, 0, 1, 0, 1, 1, -2, -1, 1, 1, 1, 0, 0, 1, -4, 3, 2, -1, 1, -1, -1, 1, 0, 1, 1, 1, 1, 3, 1, -2, 0, 0, 0, 0, 1, -5, -3, -2, -3, -1, 1, 2, 1, 1, 1, 0, 1, 1, 4, -2, 2, -1, 1, 1, -1, 1, -1, 0, 0

Offset: 1

N. J. A. Sloane, Jun 19 2021

The gcd is given in A003989, and u is given in A345415. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

			The gcd table (A003989) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
...
The u table (A345415) begins:
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
[0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
[0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
[0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
[0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
[0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
[0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
[0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
[0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
[0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
[0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
...
The v table (this entry) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
[1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
[1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
[1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
[1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
[1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
[1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
[1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
[1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
[1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
...

Cf. A003989, A050873, A345415, A345417, A345418.

mygcd:=proc(a,b) local d,s,t; d := igcdex(a,b,`s`,`t`); [a,b,d,s,t]; end;
gcd_rowv:=(m,M)->[seq(mygcd(m,n)[5],n=1..M)];
for m from 1 to 12 do lprint(gcd_rowv(m,16)); od;

T[m_, n_] := Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= 26 && u*m + v*n == GCD[m, n], {u, v}, Integers], #.#&][[1, 2]]];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 27 2023 *)

A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

Original entry on oeis.org

0, 2, 3, 2, 5, 7, 7, 4, 6, 13, 11, 10, 13, 19, 22, 8, 17, 18, 19, 18, 32, 31, 23, 20, 20, 37, 18, 26, 29, 38, 31, 16, 52, 49, 58, 24, 37, 55, 62, 36, 41, 56, 43, 42, 54, 67, 47, 40, 42, 60, 82, 50, 53, 54, 94, 52, 92, 85, 59, 60, 61, 91, 78, 32, 112, 92, 67, 66, 112, 106, 71, 48, 73, 109, 100

Offset: 1

Ilya Gutkovskiy, Aug 18 2021

nonn

a(n) is the sum of the prime terms in row n of A050873.

Moebius transform of A328260.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A001221, A008472, A010051, A018804, A050873, A061397, A078439, A117494, A122411, A255655, A328260, A329354.

Table[DivisorSum[n, MoebiusMu[n/#] # PrimeNu[#] &], {n, 1, 75}]
Table[DivisorSum[n, # EulerPhi[n/#] &, PrimeQ[#] &], {n, 1, 75}]
Table[Sum[Boole[PrimeQ[GCD[n, k]]] GCD[n, k], {k, 1, n}], {n, 1, 75}]

a(n) = sumdiv(n, d, moebius(n/d)*d*omega(d)); \\ Michel Marcus, Aug 18 2021

a(n) = Sum_{d|n} mu(n/d) * d * omega(d).
a(n) = Sum_{p|n, p prime} p * phi(n/p).
a(n) = Sum_{k=1..n} A010051(gcd(n,k)) * gcd(n,k).

A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 6, 1, 2, 2, 1, 6, 1, 5, 5, 10, 15, 10, 1, 6, 6, 3, 2, 3, 30, 1, 7, 7, 14, 21, 14, 35, 42, 1, 2, 2, 1, 6, 1, 10, 3, 14, 1, 3, 3, 6, 1, 6, 15, 2, 21, 6, 1, 10, 10, 5, 30, 5, 2, 15, 70, 5, 30, 1, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  6,  1;
  [ 4]  2,  2,  1,  6,  1;
  [ 5]  5,  5, 10, 15, 10,  1;
  [ 6]  6,  6,  3,  2,  3, 30,  1;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  1;
  [ 8]  2,  2,  1,  6,  1, 10,  3, 14,  1;
  [ 9]  3,  3,  6,  1,  6, 15,  2, 21,  6,  1;
  [10] 10, 10,  5, 30,  5,  2, 15, 70,  5, 30,   1;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1;

Family: A374433 (intersection), this sequence (symmetric difference), A374435 (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms), A050873 (gcd).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374434 := (n, k) -> mul(symmdiff(PF(n), PF(k))):
seq(print(seq(A374434(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Times @@ SymmetricDifference[s[k], s[n]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374434 defined in A374433.
for n in range(11): print([A374434(n, k) for k in range(n + 1)])

From Michael De Vlieger, Jul 11 2024: (Start)
T(0,0) = T(n,0) = rad(n)/rad(0) = 1 where rad = A007947;
T(n,k) = rad(k*n)/rad(gcd(k,n))
       = A007947(k*n)/A007947(S(n,k)) where S = A050873
       = A374436(n,k)/A374433(n,k). (End)

A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -3, 2, -1, 1, -5, -2, 1, 1, 1, -6, 4, 3, -2, -1, 1, -8, -4, -2, 1, 1, 1, 1, -9, 6, -5, -3, 2, 2, -1, 1, -11, -6, 7, 2, 1, -1, -2, 1, 1, -14, 8, 4, 5, 6, -5, -2, -1, -1, 1, -15, 10, -9, -8, -3, 1, 2, 3, 2, -1, 1, -18, -10, 6, 10, 7, 4, -3, -4, -3, -1, 1, 1

Offset: 1

N. J. A. Sloane, Jun 19 2021

The gcd is 1 unless m=n when it is m; u is given in A345417. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

			The u table (A345417) begins:
[0, -1, -2, -3, -5, -6, -8, -9, -11, -14, -15, -18, -20, -21, -23, -26]
[1,  0,  2, -2,  4, -4,  6, -6,   8,  10, -10, -12,  14, -14,  16,  18]
[1, -1,  0,  3, -2, -5,  7,  4,  -9,   6,  -6,  15,  -8, -17,  19, -21]
[1,  1, -2,  0, -3,  2,  5, -8,  10,  -4,   9,  16,   6,  -6, -20, -15]
[1, -1,  1,  2,  0,  6, -3,  7,  -2,   8, -14, -10,  15,   4, -17, -24]
[1,  1,  2, -1, -5,  0,  4,  3,  -7,   9,  12, -17,  19,  10, -18,  -4]
[1, -1, -2, -2,  2, -3,  0,  9,  -4,  12,  11, -13, -12,  -5, -11,  25]
[1,  1, -1,  3, -4, -2, -8,  0,  -6,  -3, -13,   2,  13,  -9,   5,  14]
[1, -1,  2, -3,  1,  4,  3,  5,   0,  -5,  -4,  -8, -16,  15,  -2, -23]
[1, -1, -1,  1, -3, -4, -7,  2,   4,   0,  15, -14,  17,   3,  13,  11]
[1,  1,  1, -2,  5, -5, -6,  8,   3, -14,   0,   6,   4, -18,  -3,  12]
[1,  1, -2, -3,  3,  6,  6, -1,   5,  11,  -5,   0,  10,   7,  14, -10]
...
The v table (this entry) begins:
[  1,   1,  1,  1,   1,   1,   1,   1,  1,   1,   1,  1,   1,  1,   1,  1]
[ -1,   1, -1,  1,  -1,   1,  -1,   1, -1,  -1,   1,  1,  -1,  1,  -1, -1]
[ -2,   2,  1, -2,   1,   2,  -2,  -1,  2,  -1,   1, -2,   1,  2,  -2,  2]
[ -3,  -2,  3,  1,   2,  -1,  -2,   3, -3,   1,  -2, -3,  -1,  1,   3,  2]
[ -5,   4, -2, -3,   1,  -5,   2,  -4,  1,  -3,   5,  3,  -4, -1,   4,  5]
[ -6,  -4, -5,  2,   6,   1,  -3,  -2,  4,  -4,  -5,  6,  -6, -3,   5,  1]
[ -8,   6,  7,  5,  -3,   4,   1,  -8,  3,  -7,  -6,  6,   5,  2,   4, -8]
[ -9,  -6,  4, -8,   7,   3,   9,   1,  5,   2,   8, -1,  -6,  4,  -2, -5]
[-11,   8, -9, 10,  -2,  -7,  -4,  -6,  1,   4,   3,  5,   9, -8,   1, 10]
[-14,  10,  6, -4,   8,   9,  12,  -3, -5,   1, -14, 11, -12, -2,  -8, -6]
[-15, -10, -6,  9, -14,  12,  11, -13, -4,  15,   1, -5,  -3, 13,   2, -7]
[-18, -12, 15, 16, -10, -17, -13,   2, -8, -14,   6,  1,  -9, -6, -11,  7]
...

Cf. A003989, A050873, A345415, A345416, A345417, A345419, A345420, A345421, A345422.

A051194 Triangular array T read by rows: T(n,k) = number of positive integers that divide both n and k.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2

Offset: 1

Clark Kimberling

The function T(n,k) is defined for all integer n, k but only the values for 1<=k<=n as a triangular array are listed here.

			Triangle begins:
{1};
{1,2};
{1,1,2};
{1,2,1,3};
{1,1,1,1,2};
{1,2,2,2,1,4};
...

Math StackExchange, A question on the discrete Fourier Transform of some function

Cf. A050873 (gcd), A000005 (number of divisors), A077478 (as square array).

Sum of numbers in row n matches A000203. Sum of numbers in first n rows matches A024916.

T[ n_, k_] := Length[Intersection[Divisors @ If[n == 0, 1, n], Divisors @ If[k == 0, 1, k]]] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = sum( i=1, min( abs(n), abs(k)),(n%i == 0) && (k%i == 0))} /* Michael Somos, Jul 18 2011 */

T(n,k) = A000005(A050873(n,k)). - Reinhard Zumkeller, Jun 28 2010

T(n,k) = T(k,n) = T(-n,k) = T(n,-k) = T(n,n+k) = T(n+k,k). - Michael Somos, Jul 18 2011

cp's OEIS Frontend

A169582 Complement of A169581.

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A181569 Greatest common divisor of n! and n+1.

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Magma

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PARI

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A245717 Triangle read by rows: T(n,k) = gcd(n,k^2), 1 <= k <= n.

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Haskell

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A339312 Sum over all partitions of n of the GCD of the number of parts and the number of distinct parts.

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Maple

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A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

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A345416 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = u*m+v*n where u, v are minimal; T(m,n) = v.

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Maple

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A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

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A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

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A345416 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = um+vn where u, v are minimal; T(m,n) = v.

A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.