1,2

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.

Jianing Song, Table of n, a(n) for n = 1..3

Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.

Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.

Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

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

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).

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).

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

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).

Sequence in context: A095444 A309072 A217416 * A133849 A343357 A324208

Adjacent sequences: A358415 A358416 A358417 * A358419 A358420 A358421

nonn,bref,hard

Jianing Song, Nov 14 2022

approved