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

6, 336, 333312, 5418319872, 1584858562560

Offset: 1

Author

Naohiro Nomoto, May 18 2012

Keywords

• nonn
• more

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)

Examples

			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.
		

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.

Crossrefs

Cf. A000203, A004394, A212489, A212327, A212373.

Programs

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

Extensions

a(5) from Donovan Johnson, May 20 2012