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A005579 Least number of distinct prime factors in even numbers having an abundancy index > n.
(Formerly M0573)
4
2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006
The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - T. D. Noe, May 08 2006
a(n) = smallest k such that A002110(k)/A005867(k) > n. - Artur Jasinski, Nov 06 2008
For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
MATHEMATICA
prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n, 2, 25}] (* T. D. Noe, May 08 2006 *)
prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1), 30]]; If[Min[prod]>n, k, "too few digits"], {n, 2, 38}]
PROG
(PARI) a(n)=my(s=1, k); forprime(p=2, , s*=p/(p-1); k++; if(s>n, return(k))) \\ Charles R Greathouse IV, Aug 20 2015
CROSSREFS
Cf. A005580 (least number of distinct prime factors in odd numbers having an abundancy index > n).
Cf. A023199 (least k such that sigma(k)/k >= n) - T. D. Noe, Sep 18 2008
Sequence in context: A099558 A018140 A256969 * A000381 A350588 A001115
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by T. D. Noe, May 08 2006
a(26) added by T. D. Noe, Sep 18 2008
Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
Comment corrected by T. D. Noe, Apr 04 2010
a(37)-a(39) from T. D. Noe, Nov 16 2010
STATUS
approved

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Last modified March 2 10:43 EST 2024. Contains 370466 sequences. (Running on oeis4.)