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

6, 336, 333312, 5418319872, 1584858562560

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

Table of n, a(n) for n=1..5.

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

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