A005579 Least number of distinct prime factors in even numbers having an abundancy index > n. (Formerly M0573)

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

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

Table of n, a(n) for n=2..39.

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

Adjacent sequences: A005576 A005577 A005578 * A005580 A005581 A005582

Keyword

nonn

Author

N. J. A. Sloane, R. K. Guy

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