A018804 -id:A018804 - cp's OEIS frontend

A071645 a(n) = A051193(A072109(n))/A018804(A072109(n)).

1, 3, 72, 1915, 25795, 19162, 20236, 75259, 1554957, 1491432, 17313036, 16732368, 24256243, 36578268, 44514226, 75661104, 292402908, 5482272832, 1178829937, 5302997764, 18154748952, 28439854393, 16773808684, 47254132795, 53265315697, 74423911503, 141806992844

Offset: 1

Vladeta Jovovic, Jun 22 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A018804, A051193, A072109.

More terms from Amiram Eldar, Apr 26 2023

A127626 Triangle T(n,k) = A018804(k) if k|n, else T(n,k)=0.

Original entry on oeis.org

1, 1, 3, 1, 0, 5, 1, 3, 0, 8, 1, 0, 0, 0, 9, 1, 3, 5, 0, 0, 15, 1, 0, 0, 0, 0, 0, 13, 1, 3, 0, 8, 0, 0, 0, 20, 1, 0, 5, 0, 0, 0, 0, 0, 21, 1, 3, 0, 0, 9, 0, 0, 0, 0, 27, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 3, 5, 8, 0, 15, 0, 0, 0, 0, 0, 40, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25

Offset: 1

Gary W. Adamson, Jan 20 2007

Inverse Mobius transform of a matrix with A018804 in the main diagonal and the rest zeros.

			First few rows of the triangle are:
1;
1, 3;
1, 0, 5;
1, 3, 0, 8;
1, 0, 0, 0, 9;
1, 3, 5, 0, 0, 15;
...

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

Cf. A051731, A038040 (row sums), A018804 (diagonal).

Cf. A127093.

a127626 n k = a127626_tabl !! (n-1) !! (k-1)
a127626_row n = a127626_tabl !! (n-1)
a127626_tabl = map
   (map (\x -> if x == 0 then 0 else a018804 x)) a127093_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127627 Triangle T(n,k) = A054525(n,k)*A018804(k), read by rows 1<=k<=n.

Original entry on oeis.org

1, -1, 3, -1, 0, 5, 0, -3, 0, 8, -1, 0, 0, 0, 9, 1, -3, -5, 0, 0, 15, -1, 0, 0, 0, 0, 0, 13, 0, 0, 0, -8, 0, 0, 0, 20, 0, 0, -5, 0, 0, 0, 0, 0, 21, 1, -3, 0, 0, -9, 0, 0, 0, 0, 27, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 3, 0, -8, 0, -15, 0, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 20 2007

			First few rows of the triangle are:
1;
-1, 3;
-1, 0, 5;
0, -3, 0, 8;
-1, 0, 0, 0, 9;
1, -3,-5, 0, 0, 15;
...

Cf. A054525, A018804, A029935 (row sums).

A018804 := proc(n)
    sum(igcd(n, j), j=1..n):
end proc:
A127627 := proc(n,k)
    A054525(n,k)*A018804(k) ;
end proc: # R. J. Mathar, Oct 21 2012

T(n,1) = A008683(n).

T(n,n) = A018804(n).

A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 21, 25, 54, 0, 35, 0, 78, 90, 49, 0, 51, 0, 63, 130, 126, 0, 95, 81, 150, 85, 91, 0, 0, 0, 113, 210, 198, 234, 172, 0, 222, 250, 171, 0, 0, 0, 147, 153, 270, 0, 235, 169, 147, 330, 175, 0, 231, 378, 247, 370, 342, 0, 405, 0, 366, 221, 257, 450, 0, 0, 231, 450, 0, 0, 424, 0, 438, 245, 259, 546

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A018804, A030059, A030229, A101035.

Cf. also A347094.

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA018804(n)));
A101035(n) = v101035[n];
A347095(n) = (A018804(n)+A101035(n));

a(n) = A018804(n) + A101035(n).
For n > 1, a(n) = -Sum_{d|n, 1A018804(d) * A101035(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A018804(A030229(n)).

A349692 Dirichlet convolution of the gcd-sum function (A018804) with itself.

Original entry on oeis.org

1, 6, 10, 25, 18, 60, 26, 88, 67, 108, 42, 250, 50, 156, 180, 280, 66, 402, 74, 450, 260, 252, 90, 880, 211, 300, 372, 650, 114, 1080, 122, 832, 420, 396, 468, 1675, 146, 444, 500, 1584, 162, 1560, 170, 1050, 1206, 540, 186, 2800, 435, 1266, 660, 1250, 210, 2232, 756

Offset: 1

Ilya Gutkovskiy, Nov 25 2021

Vaclav Kotesovec, Asymptotics of Sum_{k=1..n} a(k) with a graph

Cf. A000203, A018804, A029935, A038040, A344683.

A018804[n_] := Sum[GCD[n,k], {k, 1, n}]; a[n_] := Sum[A018804[d] A018804[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
f[p_, e_] := (e + 1)*p^(e - 2)*((e + 2)*(e + 3)*p^2 - 2*e*(e + 2)*p + e*(e - 1))/6; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 25 2021 *)

A029935(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ From A029935.
A349692(n) = sumdiv(n, d, A029935(n/d)*d*numdiv(d)); \\ Antti Karttunen, Nov 25 2021

Dirichlet g.f.: zeta(s-1)^4 / zeta(s)^2.
a(n) = Sum_{d|n} A018804(d) * A018804(n/d).
a(n) = Sum_{d|n} A029935(d) * A038040(n/d).
a(n) = Sum_{d|n} A344683(d) * A000203(n/d).
Multiplicative with a(p^e) = (e+1) * p^(e-2) * ((e+2)*(e+3)*p^2 - 2*e*(e+2)*p + e*(e-1))/6. - Amiram Eldar, Nov 25 2021

A349724 Numbers k >= 1 such that A000217(k) divided by A018804(k) is an integer.

Original entry on oeis.org

1, 2, 24, 25, 77, 153, 729, 1183, 1875, 6174, 7502, 14819, 15066, 18225, 19683, 21384, 26411, 26624, 28160, 37179, 146334, 155000, 157464, 194579, 236313, 336091, 399854, 418950, 632709, 701519, 818741, 1572864, 1605632, 2001824, 2067624, 2142075, 3670016, 3746287

Offset: 1

Ctibor O. Zizka, Nov 27 2021

nonn

			k = 24: A000217(24) = 300, A018804(24) = 100, 300/100 = 3 thus 24 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..92

Cf. A000217, A018804.

A018804[n_]:=Apply[Times,Apply[((#1-1)#2/#1+1)#1^#2&,FactorInteger[n],{1}]]; (* After Amiram Eldar in A018804 *)
upto=10^5;Reap[Do[If[Divisible[k(k+1)/2,A018804[k]],Sow[k]],{k,upto}]][[-1,-1]] (* Paolo Xausa, Aug 19 2022 *)

isok(k) = !(k*(k+1)/2 % sumdiv(k, d, k*eulerphi(d)/d)); \\ Michel Marcus, Nov 27 2021

from itertools import islice, count
from sympy import factorint
from math import prod
def A349724(): # generator of terms
    for k in count(1):
        if not k*(k+1)//2 % prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(k).items()):
            yield k
A349724_list = list(islice(A349724(),20)) # Chai Wah Wu, Nov 29 2021

a(12)-a(20) from Paolo Xausa, Nov 27 2021

More terms from Amiram Eldar, Nov 27 2021

A349725 Numbers k >= 1 such that A018804(k) divided by A000010(k) is an integer.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 32, 36, 48, 64, 100, 108, 112, 128, 132, 144, 192, 256, 320, 324, 432, 500, 512, 576, 756, 768, 784, 960, 972, 1024, 1296, 1452, 1600, 1728, 1892, 2048, 2052, 2112, 2240, 2304, 2500, 2816, 2880, 2916, 3072, 3888, 4096, 4800, 5120, 5184, 5292, 5488

Offset: 1

Ctibor O. Zizka, Nov 27 2021

nonn

			A018804(20) = 72, A000010(20) = 8, 72/8 = 9 thus 20 is a term.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000010, A018804.

A018804[n_]:=Apply[Times,Apply[((#1-1)#2/#1+1)#1^#2&,FactorInteger[n],{1}]]; (* After Amiram Eldar in A018804 *)
upto=10^5;Join[{1,2},Reap[Do[If[Divisible[A018804[k],EulerPhi[k]],Sow[k]],{k,4,upto,4}]][[-1,-1]]] (* Paolo Xausa, Jul 25 2022 *)

isok(k) = !(sumdiv(k, d, k*eulerphi(d)/d) % eulerphi(k)); \\ Michel Marcus, Nov 27 2021

For n >= 3, a(n) mod 4 = 0. - Paolo Xausa, Jul 25 2022

A360425 Indices of records in A018804.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 105, 108, 120, 140, 144, 168, 180, 210, 240, 252, 270, 280, 288, 300, 330, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 990, 1008, 1080, 1200, 1260, 1440

Offset: 1

Matthew Russell Downey, Feb 08 2023

nonn

a(n) seems to be divisible by any positive integer as n approaches infinity. Example: There seem to be 6 terms without a 2 (i.e., 1, 3, 5, 9, 15, 105), 14 terms without a 3 (as large as 280), 22 terms without a factor of 4 (as large as 6930), and 29 terms without a 5 (as large as 3276). True for a(n) < 10^8.
For any term a(n) > 1, it seems that there exists at least one term a(x) such that a(x) * prime(y) = a(n) where prime(y) <= (7/2) * A053669(a(x)). True for a(n) < 10^8.
The ratio of prime(n)/a(n) for indices {1..8, 11, 15, 17} is almost 2 (with an error of at most 1 on the numerator). The exact ratios are 2/1, 3/2, 5/3, 7/4, 11/5, 13/6, 17/8, 19/9, 31/15, 47/24, 59/30. - Matthew Russell Downey, Jul 25 2023

			A018804(36) = 168 is the largest value among the first 36 terms of A018804, so 36 is a term here; since it is the 18th value that sets a new record, a(18) = 36.

David A. Corneth, Table of n, a(n) for n = 1..286
Matthew Russell Downey, On Bifurcations And Beauty (2023): pages 40-43.

Indices of records in A018804. Cf. A000040.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ (f @@@ FactorInteger[n]); seq[nmax_] := Module[{p, pm = 0, s = {}}, Do[If[(p = pil[n]) > pm, pm = p; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[1200] (* Amiram Eldar, Feb 13 2023 *)

f(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ A018804
lista(nn) = my(r=0, list=List()); for (n=1, nn, my(m=f(n)); if (m > r, listput(list, n); r = m);); Vec(list); \\ Michel Marcus, Feb 26 2023

from sympy import factorint
from math import prod
def A018804(m): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(m).items())
record = 0
for m in range(1, 2000):
  value = A018804(m)
  if value > record:
    record = value
    print(m, end=", ")

Conjecture: a(n) ~ exp(4*(n-1)/21). - Matthew Russell Downey, Jul 25 2023

A344456 Composite numbers k such that (k-1) divides (A018804(k)-1).

Original entry on oeis.org

41124, 230867, 358267, 37539572, 148025049, 235167249, 242788284, 1085464188, 142772845653, 202728626748

Offset: 1

Max Alekseyev, May 19 2021

For any prime p, (p-1) divides (A018804(p)-1) = 2(p-1).

Some larger terms: 62763888399737.

Pruthviraj et al., Are there infinitely many composite a such that Sum_{k=1..a} (k,a) == 1 (mod a-1)?, MathOverflow, 2021.

Cf. A018804.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 400000], CompositeQ[#] && Divisible[pil[#] - 1, # - 1] &] (* Amiram Eldar, May 19 2021 *)

a(9)-a(10) confirmed by Martin Ehrenstein, May 27 2021

A349676 Iterated Pillai's arithmetical function: a(1) = 2, for n >= 2, a(n) = A018804(a(n-1)).

Original entry on oeis.org

2, 3, 5, 9, 21, 65, 225, 1365, 14625, 223125, 5630625, 376228125, 38759765625, 3471767578125, 934835712890625, 497199051298828125, 613613372156103515625, 2538878964828409423828125, 25280374785655551910400390625, 243473923053903449878692626953125, 9202204429788304689688510894775390625

Offset: 1

Ctibor O. Zizka, Nov 24 2021

nonn

			a(5) = f(f(f(f(a(1))))) = 21. Here f represents A018804.

Cf. A018804.

f[p_, e_] := (e*(p - 1)/p + 1) * p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; NestList[pil, 2, 20] (* Amiram Eldar, Nov 24 2021 *)

f(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ A018804
lista(nn) = my(a = 2); for (n=1, nn, print1(a, ", "); a = f(a);); \\ Michel Marcus, Nov 24 2021

a(1) = 2, a(n) = A018804(a(n-1)) for n >= 2.

More terms from Michel Marcus, Nov 24 2021

cp's OEIS Frontend

A071645 a(n) = A051193(A072109(n))/A018804(A072109(n)).

Views

Author

Keywords

Links

Crossrefs

Extensions

A127626 Triangle T(n,k) = A018804(k) if k|n, else T(n,k)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A127627 Triangle T(n,k) = A054525(n,k)*A018804(k), read by rows 1<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A347095 Sum of Pillai's arithmetical function (A018804) and its Dirichlet inverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A349692 Dirichlet convolution of the gcd-sum function (A018804) with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349724 Numbers k >= 1 such that A000217(k) divided by A018804(k) is an integer.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A349725 Numbers k >= 1 such that A018804(k) divided by A000010(k) is an integer.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360425 Indices of records in A018804.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica