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A212490 Least number m > 1 such that A000203(x)*x = m has exactly n solutions.

%I A212490 #41 Sep 03 2024 15:03:25
%S A212490 6,336,333312,5418319872,1584858562560
%N A212490 Least number m > 1 such that A000203(x)*x = m has exactly n solutions.
%C A212490 6 = 6.
%C A212490 336 = 6*28*2.
%C A212490 333312 = 6*28*496*2*2.
%C A212490 5418319872 = 6*28*496*8128*2*2*2.
%C A212490 a(6) > 5*10^14. - _Michel Marcus_ and _David A. Corneth_, Nov 01 2020
%C A212490 From _David A. Corneth_, Nov 01 2020: (Start)
%C A212490 sigma(n)/n increases to a record in A004394. This can be used to limit the checked divisors of some candidate m.
%C A212490 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)
%C A212490 a(6) <= 6*28*496*8128*33550336*137438691328*2*2*2*2*2. - _Michel Marcus_, Nov 01 2020
%C A212490 From _David A. Corneth_, Nov 01 2020: (Start)
%C A212490 Using the same as above, a(7) <= 1716908124551996896669734276042690920448.
%C A212490 a(8) <= 7917841189233800244470292555938612387093638081493952626688. (End)
%C A212490 Conjecture: a(n) <= 2^(n-1) * Product_{k=1..n} A000396(k). - _Daniel Suteu_, Nov 01 2020
%C A212490 From _Daniel Suteu_, Nov 01 2020: (Start)
%C A212490 a(6) <= 7089671638182002688000,
%C A212490 a(7) <= 106345074572730040320,
%C A212490 a(9) <= 1826980530660612389572800675840. (End)
%D A212490 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.
%H A212490 David A. Corneth, <a href="/A212490/a212490.txt">Pari program</a>
%H A212490 Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%e A212490 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.
%o A212490 (PARI) isok(k, n) = sumdiv(k, d, d*sigma(d) == k) == n;
%o A212490 a(n) = my(k=2); while (! isok(k,n), k++); k; \\ _Michel Marcus_, Oct 28 2020
%o A212490 (PARI) \\ See Corneth link. _David A. Corneth_, Nov 01 2020
%Y A212490 Cf. A000203, A004394, A212489, A212327, A212373.
%K A212490 nonn,more
%O A212490 1,1
%A A212490 _Naohiro Nomoto_, May 18 2012
%E A212490 a(5) from _Donovan Johnson_, May 20 2012