The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A002312

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A002312

Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.
(history; published version)

#49 by Jon E. Schoenfield at Mon Dec 20 20:28:39 EST 2021

STATUS

editing

approved

#48 by Jon E. Schoenfield at Mon Dec 20 20:28:35 EST 2021

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Description and initial term modified Jan 15 1996.

STATUS

approved

editing

#47 by Joerg Arndt at Mon Aug 30 10:53:58 EDT 2021

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reviewed

approved

#46 by Michel Marcus at Mon Aug 30 08:49:26 EDT 2021

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proposed

reviewed

#45 by Michael S. Branicky at Mon Aug 30 08:46:17 EDT 2021

STATUS

editing

proposed

#44 by Michael S. Branicky at Mon Aug 30 08:43:45 EDT 2021

NAME

Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of nk^2 + 1 is less than 2n2*k.

COMMENTS

Also numbers such that nk^2 + 1 has no primitive divisor, hence (by Everest & Harman, Theorem 1.4) 2.138n < a(n) < 10.6n for large enough n. They conjecture that a(n) ~ cn where c = 1/(1 - log 2) = 3.258.... - Charles R Greathouse IV, Nov 15 2014

PROG

(Python)

from sympy import factorint

def ok(n): return max(factorint(n*n + 1)) < 2*n

print(list(filter(ok, range(1, 201)))) # Michael S. Branicky, Aug 30 2021

STATUS

approved

editing

#43 by Susanna Cuyler at Sun Jan 31 08:14:00 EST 2021

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reviewed

approved

#42 by Joerg Arndt at Fri Jan 29 02:14:51 EST 2021

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proposed

reviewed

#41 by Michel Marcus at Fri Jan 29 00:50:46 EST 2021

STATUS

editing

proposed

#40 by Michel Marcus at Fri Jan 29 00:50:43 EST 2021

LINKS

Olga Taussky, <a href="https://doi.org/10.2307/2317016">Sums of Squares</a>, The American Mathematical Monthly, Vol. 77, No. 8 (Oct., 1970), pp. 805-830 (26 pages). See p. 823.

STATUS

approved

editing