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A005528
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Størmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.
(Formerly M0950)
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10
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1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 74, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96
(list;
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listen;
history;
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OFFSET
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1,2
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COMMENTS
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Also numbers k such that k^2 + 1 has a primitive divisor, hence (by Everest & Harman, Theorem 1.4) 1.1n < a(n) < 1.88n for large enough n. They conjecture that a(n) ~ cn where c = 1/log 2 = 1.4426.... - Charles R Greathouse IV, Nov 15 2014
Named after the Norwegian mathematician and astrophysicist Carl Størmer (1874-1957). - Amiram Eldar, Jun 08 2021
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REFERENCES
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John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.
Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, in Number Theory and Polynomials (James McKee and Chris Smyth, ed.), London Mathematical Society 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
John Todd, Table of Arctangents, National Bureau of Standards, Washington, DC, 1951, p. 2.
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LINKS
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MATHEMATICA
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PROG
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(Haskell)
a005528 n = a005528_list !! (n-1)
a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..]
(Python)
from sympy import factorint
def ok(n): return max(factorint(n*n + 1)) >= 2*n
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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