A056903 -id:A056903 - cp's OEIS frontend

A153357 Numbers n such that the harmonic number numerator A001008(n) is a semiprime.

4, 6, 11, 14, 15, 17, 19, 20, 23, 25, 31, 33, 34, 35, 37, 39, 49, 53, 55, 59, 61, 68, 90, 93, 94, 101, 116, 117, 121, 124, 145, 155, 158, 163, 169, 170, 186, 193, 194, 199, 205, 211, 214, 245, 258, 259, 264, 267, 283, 311, 315, 328, 340, 347, 359, 365, 371, 385

Offset: 1

Alexander Adamchuk, Dec 24 2008

414, 421, 425, 436, 451, 452, and 480 are in the sequence. 391 and 476 are the remaining candidates below 500. - Daniel M. Jensen, Jun 26 2020

Numerator(H_391) is fully factored and confirmed semiprime with the help of NFS@Home. - Tyler Busby, May 06 2024

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615

Tyler Busby, Table of n, a(n) for n = 1..65
FactorDB, Status of Numerator(H_476) in factordb.com
Hisanori Mishima, Wolstenholme number (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 500, n = 501 to 600).
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem, Harmonic Number.

Cf. A001008 (numerators of harmonic number H(n)=Sum_{i=1..n} 1/i).

Cf. A002805, A056903, A067657.

More terms from Sean A. Irvine, Aug 22 2011
Two missing terms added by D. S. McNeil, Aug 23 2011
More terms from Sean A. Irvine, Apr 01 2013
Two more terms from Daniel M. Jensen, Jun 26 2020

A348017 Numbers k such that the numerator of the fractional part of the k-th harmonic is a prime number.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 19, 21, 24, 29, 34, 39, 45, 46, 54, 65, 84, 86, 116, 128, 161, 177, 248, 254, 274, 297, 349, 352, 412, 422, 475, 493, 636, 747, 793, 811, 855, 864, 1012, 1060, 1074, 1097, 1127, 1139, 1152, 1299, 1371, 1423, 1785, 1847, 1872, 1873, 2072, 2326

Offset: 1

Amiram Eldar, Sep 24 2021

nonn

The corresponding primes are 5, 17, 83, 2089, 2341, 2861, 42503239, 3338549, 276977179, 2239777822987, ...

			3 is a term since 1 + 1/2 + 1/3 = 11/6, the fractional part of 11/6 is 5/6 and its numerator, 5, is prime.
5 is a term since 1 + 1/2 + 1/3 + 1/4 + 1/5 = 137/60, the fractional part of 137/60 is 17/60 and its numerator, 17, is prime.

Amiram Eldar, Table of n, a(n) for n = 1..96

Cf. A056903, A104174.

s = 0; seq = {}; Do[s += 1/n; If[PrimeQ @ Numerator @ FractionalPart[s], AppendTo[seq, n]], {n, 1, 2500}]; seq

from sympy import harmonic, isprime
A348017_list = [k for k in range(10**3) if isprime((lambda x: x.p % x.q)(harmonic(k)))] # Chai Wah Wu, Sep 26 2021

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A153357 Numbers n such that the harmonic number numerator A001008(n) is a semiprime.

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A348017 Numbers k such that the numerator of the fractional part of the k-th harmonic is a prime number.

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