A018804 -id:A018804 - cp's OEIS frontend

A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

1, 3, 5, 23, 9, 15, 13, 91, 59, 27, 21, 115, 25, 39, 45, 1451, 33, 177, 37, 207, 65, 63, 45, 455, 179, 75, 353, 299, 57, 135, 61, 5797, 105, 99, 117, 1357, 73, 111, 125, 819, 81, 195, 85, 483, 531, 135, 93, 7255, 363, 537, 165, 575, 105, 1059, 189, 1183, 185, 171, 117, 1035, 121, 183, 767, 46355, 225, 315, 133, 759, 225, 351, 141, 5369

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Because A018804 gets only odd values on primes, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A018804, A046644 (denominators).

Cf. also A318444.

a18804[n_] := Sum[n EulerPhi[d]/d, {d, Divisors[n]}];
f[1] = 1; f[n_] := f[n] = 1/2 (a18804[n] - Sum[f[d] f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
a[n_] := f[n] // Numerator;
Array[a, 72] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318443aux = DirSqrt(vector(up_to, n, A018804(n)));
A318443(n) = numerator(v318443aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, (1-X)^(1/2)/(1-p*X))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A018804(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318443(k) / A046644(k) ~ sqrt(3/2)*n^2/Pi. - Vaclav Kotesovec, May 10 2025

A340078 a(n) = gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 1, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 7, 29, 2, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 14, 43, 1, 5, 2, 47, 1, 1, 2, 1, 1, 53, 2, 5, 1, 3, 2, 59, 1, 61, 2, 1, 1, 1, 2, 67, 1, 1, 2, 71, 1, 73, 2, 1, 1, 1, 2, 79, 1, 1, 2, 83, 1, 1, 2, 1, 1, 89, 2, 1, 1, 3, 2, 1, 3, 97, 2

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A018804, A340079, A340080.

Cf. also A055023, A323071 (similar but different sequences).

A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A340078(n) = gcd(n,1+A018804(n));

a(n) = gcd(n, 1+A018804(n)).

A340079 a(n) = n / gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 1, 1, 4, 1, 3, 1, 8, 9, 5, 1, 12, 1, 7, 15, 16, 1, 9, 1, 20, 7, 11, 1, 24, 25, 13, 27, 4, 1, 15, 1, 32, 33, 17, 35, 36, 1, 19, 13, 40, 1, 3, 1, 44, 9, 23, 1, 48, 49, 25, 51, 52, 1, 27, 11, 56, 19, 29, 1, 60, 1, 31, 63, 64, 65, 33, 1, 68, 69, 35, 1, 72, 1, 37, 75, 76, 77, 39, 1, 80, 81, 41, 1, 84, 85, 43, 87, 88

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

It is conjectured that this is 1 iff n is 1 or a prime. See Thomas Ordowski's Oct 22 2014 comment in A018804.

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A018804, A340078, A340080.

Cf. also A055032, A323072 (similar but different sequences).

A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A340079(n) = (n/gcd(n,1+A018804(n)));

a(n) = n / A340078(n) = n / gcd(n, 1+A018804(n)).

A340080 a(n) = (1+A018804(n)) / gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

2, 2, 2, 9, 2, 8, 2, 21, 22, 14, 2, 41, 2, 20, 46, 49, 2, 32, 2, 73, 22, 32, 2, 101, 66, 38, 82, 15, 2, 68, 2, 113, 106, 50, 118, 169, 2, 56, 42, 181, 2, 14, 2, 169, 38, 68, 2, 241, 134, 98, 166, 201, 2, 122, 38, 261, 62, 86, 2, 361, 2, 92, 274, 257, 226, 158, 2, 265, 226, 176, 2, 421, 2, 110, 326, 297, 274, 188

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A018804, A340078, A340079.

A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A340080(n) = { my(x=1+A018804(n)); x/gcd(n,x); };

a(n) = (1+A018804(n)) / gcd(n, 1+A018804(n)).

A379363 Numerators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 4, 23, 199, 637, 661, 8953, 9187, 65869, 201247, 205927, 26048, 132697, 134272, 135637, 2190667, 24424937, 3513791, 131554667, 132348317, 133227437, 938941259, 947830139, 190366027, 2947643, 74101331, 223443593, 2916305159, 55809797621, 55978686341, 3437499844001

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 4/3, 23/15, 199/120, 637/360, 661/360, 8953/4680, 9187/4680, 65869/32760, 201247/98280, 205927/98280, 26048/12285, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, A survey of gcd-sum functions, Journal of Integer Sequences, Vol. 13 (2010), Article 10.8.1. See pp. 18-19.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.
Shiqin Chen and Wenguang Zhai, Reciprocals of the Gcd-Sum Functions, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.3.

Cf. A018804, A272718, A370895, A379364 (denominators), A379365.

f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/pillai[n], {n, 1, 50}]]]

pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / pillai(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} 1/A018804(k)).

a(n)/A379364(n) = Sum_{j=0..N} K_j/log(n)^(j-1/2) + O(1/log(n)^(N+1/2)), for any integer N >= 1, where K_j are constants, and in particular K_0 = (2/sqrt(Pi)) * Product_{p prime} (sqrt(1-1/p) * Sum_{k>=1} 1/A018804(p^k)) = 1.30088863073811791549... .

A379364 Denominators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 3, 15, 120, 360, 360, 4680, 4680, 32760, 98280, 98280, 12285, 61425, 61425, 61425, 982800, 10810800, 1544400, 57142800, 57142800, 57142800, 399999600, 399999600, 79999920, 1230768, 30769200, 92307600, 1199998800, 22799977200, 22799977200, 1390798609200, 695399304600

Offset: 1

Amiram Eldar, Dec 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, A survey of gcd-sum functions, Journal of Integer Sequences, Vol. 13 (2010), Article 10.8.1. See pp. 18-19.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.
Shiqin Chen and Wenguang Zhai, Reciprocals of the Gcd-Sum Functions, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.3.

Cf. A018804, A272718, A370895, A379363 (numerators), A379366.

f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/pillai[n], {n, 1, 50}]]]

pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / pillai(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A018804(k)).

A379365 Numerators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 2, 13, 89, 307, 283, 4039, 761, 5639, 16189, 17125, 10396, 54437, 52862, 54227, 847157, 9646327, 9474727, 361375699, 355820149, 27844153, 27355753, 28039513, 27731821, 366667513, 72266837, 219763471, 217455781, 4211659759, 835576403, 51882159671, 25692722941

Offset: 1

Amiram Eldar, Dec 21 2024

			Fractions begin with 1, 2/3, 13/15, 89/120, 307/360, 283/360, 4039/4680, 761/936, 5639/6552, 16189/19656, 17125/19656, 10396/12285, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.

Cf. A018804, A272718, A370895, A379363, A379366 (denominators).

f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/pillai[n], {n, 1, 50}]]]

pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / pillai(k); print1(numerator(s), ", "))};

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A018804(k)).

a(n)/A379366(n) = Sum_{j=0..N} D_j/log(n)^(j-1/2) + O(1/log(n)^(N+1/2), for any integer N >= 1, where D_j are constants, and in particular D_0 = (1/(4*log(2)-2)-1) * (2/sqrt(Pi)) * Product_{p prime} (sqrt(1-1/p) * Sum_{k>=1} 1/A018804(p^k)) = 0.38291621042855537524... .

A379366 Denominators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 3, 15, 120, 360, 360, 4680, 936, 6552, 19656, 19656, 12285, 61425, 61425, 61425, 982800, 10810800, 10810800, 399999600, 399999600, 30769200, 30769200, 30769200, 30769200, 399999600, 79999920, 239999760, 239999760, 4559995440, 911999088, 55631944368, 27815972184

Offset: 1

Amiram Eldar, Dec 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.

Cf. A018804, A272718, A370895, A379364, A379365 (numerators).

f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/pillai[n], {n, 1, 50}]]]

pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / pillai(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A018804(k)).

A348493 a(n) = A003415(n) / gcd(A003415(n), A018804(n)), where A003415 is the arithmetic derivative and A018804 is Pillai's arithmetical function.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 2, 7, 1, 2, 1, 3, 8, 2, 1, 1, 1, 1, 2, 13, 1, 11, 2, 1, 1, 4, 1, 31, 1, 5, 2, 19, 4, 5, 1, 7, 16, 17, 1, 41, 1, 2, 13, 5, 1, 7, 2, 3, 4, 7, 1, 1, 16, 23, 22, 31, 1, 23, 1, 11, 17, 3, 2, 61, 1, 3, 26, 59, 1, 13, 1, 13, 11, 10, 6, 71, 1, 11, 4, 43, 1, 31, 2, 3, 32, 1, 1, 41, 4, 4, 34, 49, 8

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A018804, A348492, A348494.

Array[#2/GCD[#1, #2] & @@ {Total@ GCD[#, Range[#]], If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]]} &, 95] (* Michael De Vlieger, Oct 21 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348493(n) = { my(u=A003415(n)); (u/gcd(u,A018804(n))); };

a(n) = A003415(n) / A348492(n).

A382872 For n >= 1, a(n) is the number of divisors of the Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n) (A018804).

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 2, 6, 4, 4, 4, 8, 3, 4, 6, 10, 4, 6, 2, 12, 4, 6, 6, 9, 4, 6, 5, 8, 4, 8, 2, 10, 8, 6, 6, 16, 2, 4, 4, 18, 5, 8, 4, 16, 8, 8, 4, 20, 4, 8, 8, 12, 8, 6, 8, 12, 4, 6, 6, 24, 3, 4, 8, 9, 9, 12, 4, 16, 9, 8, 4, 24, 4, 4, 6, 8, 8, 8, 2, 20

Offset: 1

Ctibor O. Zizka, Apr 07 2025

nonn

a(n) is from A005408 for n from {1, 5, 13, 24, 27, 41, 61, 64, 65, 69, 99, 113, ...}.

a(n) is from A065091 for n from {5, 13, 27, 41, 61, 135, 181, 205, 313, 421, ...}.

			For n = 5, a(5) = A000005(A018804(5)) = A000005(9) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A005382, A005408, A018804, A065091, A067756, A277201.

f:= proc(n) local i; numtheory:-tau(add(igcd(i,n),i=1..n)) end proc:
map(f, [$1..100]); # Robert Israel, May 07 2025

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := DivisorSigma[0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 07 2025 *)

a(n) = numdiv(sumdiv(n, d, n*eulerphi(d)/d)); \\ Michel Marcus, Apr 07 2025

a(n) = A000005(A018804(n)).
a(A005382(n)) = 2.
a(A067756(n)) = 3.
a(A277201(n)) = 5.

cp's OEIS Frontend

A318443 Numerators of the sequence whose Dirichlet convolution with itself yields A018804, Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

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A340078 a(n) = gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

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A340079 a(n) = n / gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

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A340080 a(n) = (1+A018804(n)) / gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n).

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A379363 Numerators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

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A379364 Denominators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

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A379365 Numerators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).

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Author

Keywords

Examples

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Programs

Mathematica

PARI

Formula

A379366 Denominators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).

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