A370895 -id:A370895 - cp's OEIS frontend

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

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)).

A379714 Partial alternating sums of the number of exponential divisors function (A049419).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -2, 0, -1, 0, -2, -1, -2, -1, -4, -3, -5, -4, -6, -5, -6, -5, -7, -5, -6, -4, -6, -5, -6, -5, -7, -6, -7, -6, -10, -9, -10, -9, -11, -10, -11, -10, -12, -10, -11, -10, -13, -11, -13, -12, -14, -13, -15, -14, -16, -15, -16, -15, -17, -16

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.15, p. 36.

Cf. A049419, A065442, A145353, A327837.

Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]

ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).

Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .

cp's OEIS Frontend

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

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PARI

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A379714 Partial alternating sums of the number of exponential divisors function (A049419).

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Mathematica

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