A115414 Odd abundant numbers not divisible by 3.

5391411025, 26957055125, 28816162375, 33426748355, 34393484125, 37739877175, 40342627325, 48150877775, 50866790975, 53356378075, 55959128225, 59305521275, 60711976325, 61164628525, 63395557225, 64899009175, 67275433225, 68972878975, 70088343325, 74922022175, 75665665075

Offset

1,1

Comments

An odd abundant number (A005231) not divisible by 3 must have at least 7 distinct prime factors (e.g., 5^4*7^2*11^2*13*17*19*23) and be >= 5*29#/3# = 5^2*7*11*13*17*19*23*29 = 5391411025 = A047802(2) = a(1). This is most easily seen by writing the relative abundancy A(N) = sigma(N)/2N = sigma[-1](N) as A(Product p_i^e_i) = (1/2)*Product (p_i-1/p_i^e_i)/(p_i-1) < (1/2)*Product p_i/(p_i-1). See A064001 for odd abundant numbers not divisible by 5. - M. F. Hasler, Jul 27 2016

This is not a subsequence of A248150. For example, 81324229811825 and 37182145^2 = 1382511906801025 are terms, with sigma(.) == 2 (mod 4) and sigma(.) == 3 (mod 4) respectively. - Amiram Eldar, Aug 24 2020

Links

David A. Corneth, Table of n, a(n) for n = 1..42320 (terms < 10^14, (terms 1..394 from Donovan Johnson, terms 395..4343 from Giovanni Resta)

Eric Weisstein's World of Mathematics, Abundant Number

Example

a(1)=5391411025 because it is the smallest abundant number (sigma(n)/n =~ 2.003) that is not divisible by 2 or 3.

Prog

(PARI) is(n)=gcd(n, 6)==1 && sigma(n, -1)>2 \\ Charles R Greathouse IV, Jul 28 2016

Crossrefs

Cf. A005101, A005231, A064001, A112640, A112644, A248150, A325311.

Sequence in context: A335084 A133688 A306497 * A358412 A112430 A272714

Adjacent sequences: A115411 A115412 A115413 * A115415 A115416 A115417

Keyword

nonn

Author

Sergio Pimentel, Mar 08 2006

Extensions

Added missing term 55959128225 and a(14)-a(16) from Donovan Johnson, Dec 29 2008

a(17)-a(20) from Donovan Johnson, Dec 01 2011

More terms from M. F. Hasler, Jul 28 2016

Status

approved