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A322434
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Numbers k that are the numerator of a harmonic number such that k is divisible by the square of a prime >= 5.
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2
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25, 49, 363, 7381, 86021, 2436559, 14274301, 19093197, 315404588903, 9227046511387, 9304682830147, 54801925434709, 2078178381193813, 12309312989335019, 5943339269060627227, 14063600165435720745359, 254381445831833111660789, 15117092380124150817026911
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OFFSET
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1,1
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COMMENTS
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In the data, there are two families of numerators.
1) Numerators of harmonic numbers H_{p-1} which are divisible by p^2 for p >= 5, by Wolstenholme's theorem; these numerators are in A076637.
2) Numerators of harmonic numbers which are also divisible by squares of primes, but not as a result of Wolstenholme's theorem. E.g., the numbers 363, numerator of H_7, and 9227046511387, numerator of H_{29}, are respectively divisible by 11^2 and 43^2. Up through H_{60}, only the two numerators of H_7 and H_{29} belong to this second family.
(End)
The third term in the second family is the numerator of H_{297) ~ 1.153... * 10^129 which is divisible by 1153^2, and the fourth numerator of H_k has k > 335. - Amiram Eldar, Dec 09 2018
Numbers n <= 50000 such that 11^2 divides the numerator of the n-th harmonic number: 7, 10 (by Wolstenholme's theorem), 848, 9328, 9338, 10583, 10591, 102718, 102721, 102728, 116413, 116423. - Jon E. Schoenfield, Dec 09 2018
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LINKS
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EXAMPLE
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49 is a term because the numerator of the harmonic number H_6 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 49/20 is divisible by the square of 7 (by Wolstenholme's Theorem).
363 is a term because the numerator of the harmonic number H_7 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 363/140 is divisible by the square of 11.
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MATHEMATICA
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Select[Numerator@ HarmonicNumber@ Range@ 60, AnyTrue[FactorInteger[#], And[First[#] > 2, Last[#] > 1] &] &] (* Michael De Vlieger, Dec 08 2018 *)
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PROG
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(PARI) isok(nh) = {my(f = factor(nh)); for (i=1, #f~, if ((f[i, 1] >= 5) && (f[i, 2] >= 2), return (1)); ); return (0); }
lista(nn) = {my(h = 0, nh); for (n=1, nn, h += 1/n; nh = numerator(h); if (isok(nh), print1(nh, ", "); ); ); } \\ _Michel Marcus, Dec 08 2018
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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