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A335210
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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 alternating harmonic number.
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0
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16, 19, 81, 211, 231, 232, 242, 243, 267, 274, 340, 357, 559, 637, 644, 898, 1121, 1391, 1399, 1412, 1433, 1436, 1439, 1470, 1474, 1501, 1892, 2304, 2336, 2477, 2496, 2520, 2768, 2948, 2992, 3351, 3367, 3563, 3953, 3966, 4431, 4505, 4587, 4596, 4626, 5061, 6058, 6781, 6847, 6861
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OFFSET
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1,1
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COMMENTS
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This sequence was inspired by the database of Krattenthaler and Rivoal (see the link below) about 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. Here v_p(x) is the p-adic valuation of x and H(L) is the L-th harmonic number. See also the sequences A268112, A335189, and A335207.
Here we tabulate the numbers L >= 1 for which there is a prime p <= L such that v_p(H'(L) - 1) >= 1, where H'(L) = Sum_{k=1..L} (-1)^(k+1)/k. The first few numbers L for which v_p(H'(L) - 1) = 2 (rather than 1) for some p <= L are 1501, 4596, and 9367 with corresponding p equal to 7, 19, and 37, respectively.
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LINKS
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PROG
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(PARI) listaa(nn) = {my(h=0, s=1, nh); for (n=1, nn, h += s/n; nh = numerator(h-1); forprime(p=1, n-1, if(valuation(nh, p) > 0, print1(n, ", "); break)); s = -s; ); }
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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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