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, 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.
-
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), ", "))};
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
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, ...
-
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), ", "))};
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
-
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), ", "))};
A178881
Sum of all pairs of greatest common divisors for (i,j) where 1 <= i < j <= n.
Original entry on oeis.org
0, 1, 3, 7, 11, 20, 26, 38, 50, 67, 77, 105, 117, 142, 172, 204, 220, 265, 283, 335, 379, 420, 442, 518, 558, 607, 661, 737, 765, 870, 900, 980, 1052, 1117, 1199, 1331, 1367, 1440, 1526, 1666, 1706, 1859, 1901, 2025, 2169, 2258, 2304, 2496, 2580, 2725
Offset: 1
Enric Cusell (cusell(AT)gmail.com), Jun 20 2010
Denote gcd(i,j) by (i,j), then a(6) = (1,2) + (1,3) + (1,4) + (1,5) + (1,6) + (2,3) + (2,4) + (2,5) + (2,6) + (3,4) + (3,5) + (3,6) + (4,5) + (4,6) + (5,6) = 20. - _Jianing Song_, Feb 07 2021
-
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Accumulate[Array[s, 100]] (* Amiram Eldar, Dec 10 2024 *)
-
a(n)=sum(k=1, n, eulerphi(k)*(n\k)^2)/2-n*(n+1)/4 \\ Charles R Greathouse IV, Apr 11 2016
-
first(n)=my(v=vector(n),t); for(k=1,n, t=eulerphi(k); for(m=k,n, v[m] += t*(m\k)^2)); v/2-vector(n,k,k*(k+1)/4) \\ Charles R Greathouse IV, Apr 11 2016
Comments