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

Author

Sergio Pimentel, Mar 08 2006

Keywords

• nonn

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

Examples

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

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

Crossrefs

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

Programs

PARI

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

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