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A072984 Least k such that prime(n) appears in the factorization of A001008(k) (the numerator of the k-th harmonic number).

%I A072984 #23 May 02 2019 03:54:17
%S A072984 2,4,6,3,12,16,18,22,13,30,17,40,13,46,22,58,10,66,70,72,78,82,88,11,
%T A072984 100,102,106,25,112,126,130,5,138,148,150,156,162,166,71,178,180,190,
%U A072984 192,196,38,210,222,22,228,232,238,240,250,66,262,33,58,276,280,282
%N A072984 Least k such that prime(n) appears in the factorization of A001008(k) (the numerator of the k-th harmonic number).
%C A072984 a(n)<=n for n =2,5,14,18,25,29,33,46,49,...
%C A072984 For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the smallest elements of J_p. The largest elements of J_p are given by A177734. The sizes of J_p are given by A092103.
%H A072984 T. D. Noe, <a href="/A072984/b072984.txt">Table of n, a(n) for n = 2..1000</a>
%H A072984 David W. Boyd, <a href="http://www.emis.de/journals/EM/expmath/volumes/3/3.html">A p-adic study of the partial sums of the harmonic series</a>, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Feb 10 2016]
%H A072984 A. Eswarathasan and E. Levine, <a href="http://dx.doi.org/10.1016/0012-365X(90)90234-9">p-integral harmonic sums</a>, Discrete Math. 91 (1991), 249-257.
%t A072984 A072984[n_] := Module[{p, k, sum},
%t A072984    p = Prime[n]; k = 1; sum = 1/k;
%t A072984    While[! Divisible[Numerator[sum], p],
%t A072984     k++; sum += 1/k];
%t A072984    Return[k]];
%t A072984 Table[A072984[n], {n, 2, 61}] (* _Robert Price_, May 01 2019 *)
%o A072984 (PARI) a(n)=if(n<0,0,s=1; while(numerator(sum(k=1,s,1/k))%prime(n)>0,s++); s)
%Y A072984 Cf. A092101 (harmonic primes), A092102 (non-harmonic primes), A092103 (size of Jp).
%K A072984 easy,nonn
%O A072984 2,1
%A A072984 _Benoit Cloitre_, Aug 21 2002