A379367
Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).
Original entry on oeis.org
1, 3, 11, 7, 38, 27, 199, 117, 386, 793, 8933, 1553, 20574, 41863, 127591, 71303, 1227166, 2539417, 48759433, 24864701, 25095646, 50632187, 1174239991, 605711068, 125604071, 252924241, 267797099, 19356010, 564511331, 1891973791, 58959268151, 31867258958, 8730535499
Offset: 1
Fractions begin with 1, 3/2, 11/6, 7/3, 38/15, 27/10, 199/70, 117/35, 386/105, 793/210, 8933/2310, 1553/385, ...
- Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.
- Amiram Eldar, Table of n, a(n) for n = 1..1000
- N. G. de Bruijn, On the number of integers <= x whose prime factors divide n, Illinois Journal of Mathematics, Vol. 6, No. 1 (1962), pp. 137-141.
- Olivier Robert and Gérald Tenenbaum, Sur la répartition du noyau d'un entier, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 802-914.
- László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.6, pp. 24-26.
-
rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]
-
rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(numerator(s), ", "))};
A379370
Denominators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).
Original entry on oeis.org
1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 2002, 1001, 17017, 102102, 1939938, 4849845, 4849845, 9699690, 223092870, 37182145, 37182145, 74364290, 223092870, 111546435, 3234846615, 2156564410, 66853496710, 33426748355, 100280245065, 200560490130
Offset: 1
-
rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Denominator[Accumulate[Table[(-1)^(n+1)/rad[n], {n, 1, 50}]]]
-
rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / rad(k); print1(denominator(s), ", "))};
A379369
Numerators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).
Original entry on oeis.org
1, 1, 5, 1, 8, 11, 107, 1, 12, 17, 257, 193, 3664, 5183, 479, -261, -3436, -37633, -612925, -2017297, -1786352, -4013599, -82613087, -19965872, -12529443, -27919051, -9392863, -12664034, -255710551, -242359181, -5356570201, -19391659278, -55136182529, -116171203003
Offset: 1
Fractions begin with 1, 1/2, 5/6, 1/3, 8/15, 11/30, 107/210, 1/105, 12/35, 17/70, 257/770, 193/1155, ...
-
rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[(-1)^(n+1)/rad[n], {n, 1, 50}]]]
-
rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / rad(k); print1(numerator(s), ", "))};
A380315
Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.
Original entry on oeis.org
1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 30030, 15015, 255255, 170170, 3233230, 1616615, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 14872858, 7436429, 215656441, 6469693230, 200560490130, 100280245065, 100280245065
Offset: 1
0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...
-
Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
-
a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025
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