A335189 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.
21, 23, 43, 47, 66, 68, 78, 82, 86, 111, 115, 119, 157, 160, 164, 167, 273, 287, 343, 359, 438, 442, 456, 460, 507, 527, 579, 581, 615, 665, 813, 818, 834, 839, 931, 943, 947, 959, 1082, 1090, 1111, 1119, 1140, 1148, 1248, 1288, 1333, 1340, 1346, 1354, 1360, 1367, 1592, 1640, 1641, 1679, 1807, 1847, 2034, 2067, 2069, 2163, 2190
Offset: 1
Keywords
Links
- David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
- Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
- Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
- Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
- Carlo Sanna, On the p-adic valuation of harmonic numbers, Journal of Number Theory, 166 (2016), 41-46.
Programs
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PARI
h(n) = sum(i=1, n, 1/i); is(n) = {forprime(p=1, n, if(valuation((numerator(h(n)-1)), p) > 0, return(1))); return(0)}; for(n=1, 1000, if(is(n)==1, print1(n,",")))
Comments