A358418 - OEIS

%I #18 Nov 15 2022 17:51:32

%S 1,20169691981106018776756331

%N Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

%C Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 7^2*11^2*13*...*67 ~ 2.01697*10^25. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986 are too large to display.

%H Jianing Song, <a href="/A358418/b358418.txt">Table of n, a(n) for n = 1..3</a>

%H Kevin A. Broughan and Qizhi Zhou, <a href="https://hdl.handle.net/10289/1796">Odd multiperfect numbers of abundancy 4</a>, author’s version, Research Commons.

%H Kevin A. Broughan and Qizhi Zhou, <a href="https://doi.org/10.1016/j.jnt.2007.02.001">Odd multiperfect numbers of abundancy 4</a>, Journal of Number Theory 128 (2008) 1566-1575.

%H Mercurial, the Spectre, <a href="http://hi.gher.space/forum/viewtopic.php?f=11&amp;t=2248&amp;sid=cbf9e6743a4ccdcd6cbcadcdf56946db">Abundant numbers coprime to n</a>, Hi.gher. Space.

%e a(2) = A047802(3) = 20169691981106018776756331 is the smallest abundant number coprime to 2, 3, and 5.

%e Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 97, and we have k >= prime(4)^2*...*prime(100)^2 ~ 2.46692*10^436 > A358413(3) ~ 2.54562*10^239. So a(3) = A358413(3).

%e Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 606, and we have k >= prime(4)^2*...*prime(607)^2*prime(608)*prime(609) ~ 6.54355*10^3814 > A358414(3) ~ 1.11116*10^1986. So a(4) = A358414(3).

%Y Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

%Y Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), this sequence (p=7), A358419 (p=11).

%K nonn,bref,hard

%O 1,2

%A _Jianing Song_, Nov 14 2022