A023199 a(n) is the least k with sigma(k) >= n*k.

1, 6, 120, 27720, 122522400, 130429015516800, 1970992304700453905270400, 1897544233056092162003806758651798777216000, 4368924363354820808981210203132513655327781713900627249499856876120704000

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• changed

Comments

Following a suggestion from Ed Pegg Jr, the sequence can be written in a more readable form as: 1!, 3!, 5!, 11# * 3! * 2, 17# * 5! * 2, 29# * 7! * 4, 53# * 7! * 12, 89# * 11! * 2, 157# * 17# * 8! * 6, 271# * 23# * 10!, 487# * 29# * 10!, 857# * 37# * 11! * 42, 1487# * 53# * 15! * 2, ..., where p# = primorial(p) = A034386.

From T. D. Noe, Jul 06 2005: (Start)

Let c(p) be the smallest colossally-abundant number having the prime factor p. See A073751 for info about computing these numbers.

Then the terms of this sequence can be expressed as

a(2) = c(3)

a(3) = c(5) * 2

a(4) = c(11) / 2

a(5) = c(17) / 3

a(6) = c(29) * 14

a(7) = c(53)

a(8) = c(89) * 4

a(9) = c(157) * 34

a(10) = c(271) * 23

a(11) = c(487) / 2

a(12) = c(857) / 2

a(13) = c(1487) * 212

a(14) = c(2621) * 710

a(15) = c(4567) * 2/21

a(16) = c(8011) / 2

a(17) = c(13999) * 1630. (End)

Initially, each term is divisible by the previous one. Is there a reason this should always be true? - Santi Spadaro, Aug 13 2002

The conjecture a(n)|a(n+1) holds out to n=10. - Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003

The conjecture a(n)|a(n+1) fails for n=15. - T. D. Noe, Jul 08 2005

We have a(n) = min{A007539(n), A134716(n)}, and clearly A007539(n) != A134716(n) for every n. For what values of n is the former less than the latter? - Jeppe Stig Nielsen, Jun 16 2015

Links

• Amiram Eldar, Table of n, a(n) for n = 1..13
• Walter Nissen, Abundancy : Some Resources, 2008-2010.
• T. D. Noe, An algorithm for finding the least k with sigma(k) >= nk, 2005-2009.

Crossrefs

A subsequence of A004394.

The dominating primes are in A108402.

Cf. A000203 (sigma), A007539, A034386, A073751, A134716.

Programs

PARI

a(n) = my(k=1); while (sigma(k)/k < n, k++); k; \\ Michel Marcus, Oct 07 2019

Extensions

More terms from Walter Nissen, Apr 15 1997

Further terms from Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003

The term a(10) = 271#23#10! was apparently found independently by Bodo Zinser and Don Reble, circa Jul 05 2005

The next term, a(11) = 487#29#10!, was corrected by Don Reble, Jul 06 2005

a(12) = 857#37#11!42 from Don Reble, Jul 06 2005

a(13) = 1487#53#15!2 found by T. D. Noe and confirmed by Don Reble, Jul 07 2005

a(14)-a(17) found by T. D. Noe and rechecked by him Oct 11 2005

a(15) corrected. The conjecture still fails at n=15. - T. D. Noe, Oct 13 2009