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 A212490 Least number m > 1 such that A000203(x)*x = m has exactly n solutions. 6
 6, 336, 333312, 5418319872, 1584858562560 (list; graph; refs; listen; history; text; internal format)
 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 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 Cf. A000203, A004394, A212489, A212327, A212373. Sequence in context: A210769 A003031 A144849 * A047941 A229501 A289738 Adjacent sequences:  A212487 A212488 A212489 * A212491 A212492 A212493 KEYWORD nonn,more AUTHOR Naohiro Nomoto, May 18 2012 EXTENSIONS a(5) from Donovan Johnson, May 20 2012 STATUS approved

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Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)