A358412
Least number k coprime to 2 and 3 such that sigma(k)/k >= n.
Original entry on oeis.org
1, 5391411025, 5164037398437051798923642083026622326955987448536772329145127064375
Offset: 1
a(2) = A047802(2) = 5391411025 is the smallest abundant number coprime to 2 and 3.
Even if there is a number k coprime to 2 and 3 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=3..m+2} (prime(i)/(prime(i)-1)) => m >= 33, and we have k >= prime(3)^2*...*prime(35)^2 ~ 6.18502*10^112 > A358413(2) ~ 5.16403*10^66. So a(3) = A358413(2).
Even if there is a number k coprime to 2 and 3 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=3..m+2} (prime(i)/(prime(i)-1)) => m >= 140, and we have k >= prime(3)^2*...*prime(140)^2*prime(141)*prime(142) ~ 2.65585*10^669 > A358414(2) ~ 1.83947*10^370. So a(4) = A358414(2).
- Jianing Song, Table of n, a(n) for n = 1..4
- 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.
Smallest k-abundant number which is not divisible by any of the first n primes:
A047802 (k=2),
A358413 (k=3),
A358414 (k=4).
A358413
Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.
Original entry on oeis.org
180, 1018976683725, 5164037398437051798923642083026622326955987448536772329145127064375
Offset: 0
a(1) = A119240(3) = 1018976683725 is the smallest 3-abundant odd number.
a(2) = A358412(3) = 5164037398437051798923642083026622326955987448536772329145127064375 is the smallest 3-abundant number that is coprime to 2 and 3.
Smallest k-abundant number which is not divisible by any of the first n primes:
A047802 (k=2), this sequence (k=3),
A358414 (k=4).
A358414
Smallest 4-abundant number (sigma(x) > 4x) which is not divisible by any of the first n primes.
Original entry on oeis.org
27720, 1853070540093840001956842537745897243375
Offset: 0
a(1) = A119240(4) = 1853070540093840001956842537745897243375 is the smallest 4-abundant odd number.
a(2) = A358412(4) ~ 1.83947*10^370 is the smallest 4-abundant number that is coprime to 2 and 3.
Smallest k-abundant number which is not divisible by any of the first n primes:
A047802 (k=2),
A358413 (k=3), this sequence (k=4).
A358418
Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.
Original entry on oeis.org
1, 20169691981106018776756331
Offset: 1
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).
- 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.
Smallest k-abundant number which is not divisible by any of the first n primes:
A047802 (k=2),
A358413 (k=3),
A358414 (k=4).
Showing 1-4 of 4 results.
Comments