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

2, 4, 6, 3, 12, 16, 18, 22, 13, 30, 17, 40, 13, 46, 22, 58, 10, 66, 70, 72, 78, 82, 88, 11, 100, 102, 106, 25, 112, 126, 130, 5, 138, 148, 150, 156, 162, 166, 71, 178, 180, 190, 192, 196, 38, 210, 222, 22, 228, 232, 238, 240, 250, 66, 262, 33, 58, 276, 280, 282

Offset: 2

Author

Benoit Cloitre, Aug 21 2002

Keywords

• easy
• nonn

Comments

a(n)<=n for n =2,5,14,18,25,29,33,46,49,...

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.

Links

• T. D. Noe, Table of n, a(n) for n = 2..1000
• David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Feb 10 2016]
• A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Crossrefs

Cf. A092101 (harmonic primes), A092102 (non-harmonic primes), A092103 (size of Jp).

Programs

Mathematica

A072984[n_] := Module[{p, k, sum},
   p = Prime[n]; k = 1; sum = 1/k;
   While[! Divisible[Numerator[sum], p],
    k++; sum += 1/k];
   Return[k]];
Table[A072984[n], {n, 2, 61}] (* Robert Price, May 01 2019 *)

PARI

a(n)=if(n<0,0,s=1; while(numerator(sum(k=1,s,1/k))%prime(n)>0,s++); s)