A281140 Least k such that k is the product of n distinct primes and sigma(k) is an n-th power.

2, 22, 102, 510, 90510, 995610, 11616990, 130258590, 1483974030, 18404105922510, 428454465915630, 10195374973815570, 240871269907008510, 94467020965716904490370, 4580445736068712946096430, 7027383957579235221501981990, 419420669769073022876839238610, 24967450935148397377034326845390

Offset: 1

Author

Altug Alkan, Jan 15 2017

Keywords

• nonn

Comments

Freiberg (Theorem 1.2) shows that there are >> (n*x^(1/n))/(log x)^(n+2) such values of k up to x. He calls the set of such numbers B*(x;+1;n). In particular, a(n) exists for each n.

Corresponding values of sigma(k) are 3 = 3^1, 36 = 6^2, 216 = 6^3, 1296 = 6^4, 248832 = 12^5, 2985984 = 12^6, 12^7, 12^8, 12^9, 24^10, 24^11, 24^12, 24^13, 48^14, 48^15, 60^16, 60^17, 60^18, 60^19, 84^20, 84^21, 84^22, 84^23, ...

a(14) <= 94467020965716904490370. - Daniel Suteu, Mar 28 2021

Examples

			a(3) = 102 because 102 = 2 * 3 * 17 and (2 + 1)*(3 + 1)*(17 + 1) = 6^3.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..23
• Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, arXiv:1008.1978 [math.NT], 2010.

Crossrefs

Cf. A000203, A001597, A005117, A065496, A281069.

Programs

PARI

a(n) = my(k=2); while(!issquarefree(k) || !ispower(sigma(k), n) || omega(k)!=n, k++); k \\ Felix Fröhlich, Jan 17 2017

Extensions

a(10)-a(13) from Jinyuan Wang, Nov 08 2020

a(14) from Daniel Suteu and David A. Corneth, Mar 28 2021

a(15)-a(18) from David A. Corneth, Mar 29 2021