OFFSET
1,1
COMMENTS
6 = 6.
336 = 6*28*2.
333312 = 6*28*496*2*2.
5418319872 = 6*28*496*8128*2*2*2.
a(6) > 5*10^14. - Michel Marcus and David A. Corneth, Nov 01 2020
From David A. Corneth, Nov 01 2020: (Start)
sigma(n)/n increases to a record in A004394. This can be used to limit the checked divisors of some candidate m.
For n >= 6, If gcd(a(4), a(5)) | a(n) then a(n) > 1.1*10^17. If (gcd(a(4), a(5)) * 2047) | a(n) then a(n) > 1.8 * 10^20. (End)
a(6) <= 6*28*496*8128*33550336*137438691328*2*2*2*2*2. - Michel Marcus, Nov 01 2020
From David A. Corneth, Nov 01 2020: (Start)
Using the same as above, a(7) <= 1716908124551996896669734276042690920448.
a(8) <= 7917841189233800244470292555938612387093638081493952626688. (End)
Conjecture: a(n) <= 2^(n-1) * Product_{k=1..n} A000396(k). - Daniel Suteu, Nov 01 2020
From Daniel Suteu, Nov 01 2020: (Start)
a(6) <= 7089671638182002688000,
a(7) <= 106345074572730040320,
a(9) <= 1826980530660612389572800675840. (End)
REFERENCES
R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.
LINKS
David A. Corneth, Pari program
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
EXAMPLE
For n=3, 333312 has exactly 3 solutions: sigma(434)*434 = 333312, sigma(372)*372 = 333312, and sigma(336)*336 = 333312; therefore a(3) = 333312.
PROG
(PARI) isok(k, n) = sumdiv(k, d, d*sigma(d) == k) == n;
a(n) = my(k=2); while (! isok(k, n), k++); k; \\ Michel Marcus, Oct 28 2020
(PARI) \\ See Corneth link. David A. Corneth, Nov 01 2020
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Naohiro Nomoto, May 18 2012
EXTENSIONS
a(5) from Donovan Johnson, May 20 2012
STATUS
approved