A144711 Numbers n such that [sum_i=1..r (p(i)^2)]/r = c^2. p(i) prime divisors of n, c integer.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 119, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Author

Ctibor O. Zizka, Sep 19 2008

Keywords

• easy
• nonn

Comments

A005063(n)/A001221(n) = c^2.

Also numbers n such that the root mean square (quadratic mean) of the prime divisors of n is an integer.

These numbers are power of primes (p^k with k>=1) (A000961) or in A255580. - Daniel Lignon, Feb 26 2015

Links

• Daniel Lignon, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005063, A001221, A140480.

Programs

Maple

A005063 := proc(n) add(p^2,p=numtheory[factorset](n)) ; end: A001221 := proc(n) nops(numtheory[factorset](n)) ; end: isA144711 := proc(n) local sofpr ; sofpr := A001221(n) ; if sofpr <> 0 then issqr(A005063(n)/sofpr) ; else false ; fi; end: for n from 1 to 500 do if isA144711(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 20 2008

Mathematica

Select[Range[2,1000],IntegerQ[RootMeanSquare[Select[Divisors[#],PrimeQ]]]&] (* Daniel Lignon, Feb 26 2015 *)

Extensions

More terms from R. J. Mathar, Sep 20 2008