cp's OEIS Frontend

A115414 Odd abundant numbers not divisible by 3.

%I A115414 #50 Feb 16 2025 08:33:00
%S A115414 5391411025,26957055125,28816162375,33426748355,34393484125,
%T A115414 37739877175,40342627325,48150877775,50866790975,53356378075,
%U A115414 55959128225,59305521275,60711976325,61164628525,63395557225,64899009175,67275433225,68972878975,70088343325,74922022175,75665665075
%N A115414 Odd abundant numbers not divisible by 3.
%C A115414 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
%C A115414 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
%H A115414 David A. Corneth, <a href="/A115414/b115414.txt">Table of n, a(n) for n = 1..42320</a> (terms < 10^14, terms 1..394 from Donovan Johnson, terms 395..4343 from Giovanni Resta)
%H A115414 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%e A115414 a(1)=5391411025 because it is the smallest abundant number (sigma(n)/n =~ 2.003) that is not divisible by 2 or 3.
%o A115414 (PARI) is(n)=gcd(n,6)==1 && sigma(n,-1)>2 \\ _Charles R Greathouse IV_, Jul 28 2016
%Y A115414 Cf. A005101, A005231, A064001, A112640, A112644, A248150, A325311.
%K A115414 nonn
%O A115414 1,1
%A A115414 _Sergio Pimentel_, Mar 08 2006
%E A115414 Added missing term 55959128225 and a(14)-a(16) from _Donovan Johnson_, Dec 29 2008
%E A115414 a(17)-a(20) from _Donovan Johnson_, Dec 01 2011
%E A115414 More terms from _M. F. Hasler_, Jul 28 2016