Search: a115963 -id:a115963
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1,1
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COMMENTS
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A115963(n) is the numerator of Sum_{k=1..n} 1/prime(k)^3.
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LINKS
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MATHEMATICA
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f=0; Do[p=Prime[n]; f=f+1/p^3; g=Numerator[f]; If[PrimeQ[g], Print[{n, p, g}]], {n, 1, 50}]
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A024451
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a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).
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+10
61
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0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853
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OFFSET
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0,3
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COMMENTS
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(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.
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LINKS
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FORMULA
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Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
(End)
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EXAMPLE
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0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...
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MAPLE
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h:= n-> add(1/(ithprime(i)), i=1..n);
t1:=[seq(h(n), n=0..50)];
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MATHEMATICA
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f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)
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PROG
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(Magma) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ]; // Bruno Berselli, Apr 11 2011
(PARI) a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018
(Python)
from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
(Python)
from math import prod
from sympy import prime
q = prod(plist:=tuple(prime(i) for i in range(1, n+1)))
return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022
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CROSSREFS
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Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes).
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KEYWORD
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nonn,frac,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A115964
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Denominator of Sum_{i=1..n} 1/prime(i)^3.
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+10
11
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8, 216, 27000, 9261000, 12326391000, 27081081027000, 133049351085651000, 912585499096480209000, 11103427767506874702903000, 270801499821725167129101267000, 8067447481189014453943055845197000
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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a(n) = denominator of Sum_{i=1..n} 1/A000040(i)^3.
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EXAMPLE
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1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.
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MATHEMATICA
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CROSSREFS
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Cf. A024451 (numerator of sum_{i=1..n} 1/prime(i)), A002110 (primorial, also denominator of sum_{i=1..n} 1/prime(i)), A061015 (numerator of sum_{i=1..n} 1/prime(i)^2).
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KEYWORD
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easy,frac,nonn
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AUTHOR
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STATUS
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approved
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A241189
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Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)).
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+10
5
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1, 7, 11, 127, 1693, 29243, 561623, 13019431, 379503437, 11809225121, 438235268123, 18007758091069, 775817745542929, 36524284093223105, 1938403609207158571, 2160165866032831207, 131893095784520401909, 8844093116997411126541, 628373208972323386101329, 45900898298568589325230523
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...
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MAPLE
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g:= n-> add(1/(ithprime(i)*ithprime(i+1)), i=1..n);
t1:=[seq(g(n), n=1..20)];
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MATHEMATICA
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Table[Numerator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Accumulate[1/#&/@(Times@@@Partition[Prime[Range[25]], 2, 1])]//Numerator (* Harvey P. Dale, Mar 14 2023 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A126225
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Least number k > 0 such that the numerator of Sum_{i=1..k} 1/prime(i)^n is a prime.
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+10
0
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OFFSET
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1,1
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COMMENTS
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a(12) > 80, a(13) = 30, a(14) = 16, a(18) = 7, a(19) = 3. - J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
If they exist, a(11) > 1263; a(17) > 954; a(22) > 795; a(23) > 720; a(25) > 570; a(12) = 799, a(15) = 313, a(16) = 780, a(20) = 433, a(21) = 7, a(24) = 4, a(27) = 12, a(29) = 37. - J.W.L. (Jan) Eerland, Jan 26 2023
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LINKS
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EXAMPLE
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a(1) = 2 corresponds to A024451(2) = 5, a prime.
a(2) = 2 corresponds to A061015(2) = 13, a prime.
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MATHEMATICA
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a[n_] := Block[{i = 1, sum = 0}, While[True, sum += 1/Prime[i]^n; If[PrimeQ[Numerator@sum], Return[i]]; i++ ]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Table[y[x_, y_]:=Numerator[FullSimplify[Sum[1/Prime[m]^x, {m, 1, y}]]]; k=1; Monitor[Parallelize[While[True, If[PrimeQ[y[n, k]], Break[]]; k++]; k], k], {n, 1, 10}] (* J.W.L. (Jan) Eerland, Jan 25 2023 *)
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PROG
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(PARI) a(n) = {my(k=1, s=1/prime(k)^n); while (! isprime(numerator(s)), k++; s += 1/prime(k)^n); k; } \\ Michel Marcus, May 27 2019
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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STATUS
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approved
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