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A335189
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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For more numbers in this list (up to 10^6), see one of the links below by Krattenthaler and Rivoal. The first few numbers L for which v_p(H_L-1) = 2 (rather than 1) for some prime p <= L are 43, 2034 and 2069 with corresponding primes 7, 13 and 7.
The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.
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LINKS
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PROG
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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, ", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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