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%I #132 Feb 10 2024 00:07:44
%S 0,1,5,31,247,2927,40361,716167,14117683,334406399,9920878441,
%T 314016924901,11819186711467,492007393304957,21460568175640361,
%U 1021729465586766997,54766551458687142251,3263815694539731437539,201015517717077830328949,13585328068403621603022853
%N a(n) is the numerator of Sum_{i = 1..n} 1/prime(i).
%C Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - _Reinhard Zumkeller_, Feb 25 2002
%C (n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - _Clark Kimberling_, Dec 29 2011
%C Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - _Colin Barker_, Nov 14 2014
%C 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
%C Conjecture: All terms are squarefree numbers. - _Nicolas Bělohoubek_, Apr 13 2022
%C 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
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.
%H Alois P. Heinz, <a href="/A024451/b024451.txt">Table of n, a(n) for n = 0..350</a> (terms n = 1..100 from T. D. Noe)
%H Victor Ufnarovski and Bo Åhlander, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html">How to Differentiate a Number</a>, J. Integer Seqs., Vol. 6, 2003.
%F Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
%F a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - _Benoit Cloitre_, Jan 30 2002
%F (n+1)-st elementary symmetric function of the first n primes.
%F a(n) = a(n-1)*A000040(n) + A002110(n-1). - _Henry Bottomley_, Sep 27 2006
%F From _Antti Karttunen_, Jan 31 2024 and Feb 08 2024: (Start)
%F a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).
%F For n > 0, a(n) = A327860(A143293(n-1)).
%F For n > 0, a(n) = A348301(n) + A002110(n).
%F For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]
%F (End)
%e 0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...
%p h:= n-> add(1/(ithprime(i)),i=1..n);
%p t1:=[seq(h(n),n=0..50)];
%p t1a:=map(numer,t1); # A024451
%p t1b:=map(denom,t1); # A002110 - _N. J. A. Sloane_, Apr 25 2014
%t a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a,18] (* _Jean-François Alcover_, Apr 11 2011 *)
%t f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
%t a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t Table[a[n], {n, 1, 16}] (* A024451 *)
%t (* _Clark Kimberling_, Dec 29 2011 *)
%t Numerator[Accumulate[1/Prime[Range[20]]]] (* _Harvey P. Dale_, Apr 11 2012 *)
%o (Magma) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ]; // _Bruno Berselli_, Apr 11 2011
%o (PARI) a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ _Michel Marcus_, Sep 18 2018
%o (Python)
%o from sympy import prime
%o from fractions import Fraction
%o def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
%o print([a(n) for n in range(20)]) # _Michael S. Branicky_, Feb 12 2021
%o (Python)
%o from math import prod
%o from sympy import prime
%o def A024551(n):
%o q = prod(plist:=tuple(prime(i) for i in range(1,n+1)))
%o return sum(q//p for p in plist) # _Chai Wah Wu_, Nov 03 2022
%Y Denominators are A002110.
%Y See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.
%Y Row sums of A077011 and A258566.
%Y Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.
%Y Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).
%Y Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes).
%Y Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.
%K nonn,frac,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _Clark Kimberling_
%E a(0)=0 prepended by _Alois P. Heinz_, Jun 26 2015
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