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 A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i). 54
 0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002 (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 Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022 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 REFERENCES S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2. D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe) Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. FORMULA 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. a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006 From Antti Karttunen, Jan 31 2024 and Feb 08 2024: (Start) a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n). For n > 0, a(n) = A327860(A143293(n-1)). For n > 0, a(n) = A348301(n) + A002110(n). For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.] (End) EXAMPLE 0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ... MAPLE h:= n-> add(1/(ithprime(i)), i=1..n); t1:=[seq(h(n), n=0..50)]; t1a:=map(numer, t1); # A024451 t1b:=map(denom, t1); # A002110 - N. J. A. Sloane, Apr 25 2014 MATHEMATICA a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a, 18] (* Jean-François Alcover, Apr 11 2011 *) 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 *) (* Clark Kimberling, Dec 29 2011 *) Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *) PROG (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 print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021 (Python) from math import prod from sympy import prime def A024551(n): 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 CROSSREFS Denominators are A002110. See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860. Row sums of A077011 and A258566. Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651. Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12). Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes). Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253. Sequence in context: A261498 A368320 A276312 * A046852 A361408 A056541 Adjacent sequences: A024448 A024449 A024450 * A024452 A024453 A024454 KEYWORD nonn,frac,easy,nice,changed AUTHOR N. J. A. Sloane, Clark Kimberling EXTENSIONS a(0)=0 prepended by Alois P. Heinz, Jun 26 2015 STATUS approved

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Last modified February 20 20:53 EST 2024. Contains 370217 sequences. (Running on oeis4.)