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Weighted quadratic roundness of n. If n=p_1^e_1...p^k_e^k, then a(n) = e_1 + (2^2 * e_2) + ... + (k^2 * e_k). Note that p_i<p_j, i<j, is assumed.
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%I #9 May 31 2017 16:15:16

%S 0,1,1,2,1,5,1,3,2,5,1,6,1,5,5,4,1,9,1,6,5,5,1,7,2,5,3,6,1,14,1,5,5,5,

%T 5,10,1,5,5,7,1,14,1,6,6,5,1,8,2,9,5,6,1,13,5,7,5,5,1,15,1,5,6,6,5,14,

%U 1,6,5,14,1,11,1,5,9,6,5,14,1,8,4,5,1,15,5,5,5,7,1,18,5,6,5,5,5,9,1,9,6,10

%N Weighted quadratic roundness of n. If n=p_1^e_1...p^k_e^k, then a(n) = e_1 + (2^2 * e_2) + ... + (k^2 * e_k). Note that p_i<p_j, i<j, is assumed.

%H Antti Karttunen, <a href="/A079168/b079168.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 0 (an empty sum).

%e a(10) = 5 as 10 = 2*5, therefore a(10) = (1^2)*1 + (2^2)*1 = 5

%e a(45) = 6 as 45 = 3^2 * 5, therefore a(45) = (1^2)*2 + (2^2)*1 = 6.

%e a(50) = 5 as 50 = 2^1 * 5^2, therefore a(50) = (1^2)*1 + (2^2)*2 = 9.

%e a(250) = 7 as 250 = 2^1 * 5^3, therefore a(250) = (1^2)*1 + (2^2)*3 = 13.

%o (PARI) weightedroundness2(n)=local(f,fl,s); f=factor(n); fl=length(f[,1]); s=0; for (i=1,fl,s=s+i^2*f[,2][i]); s alias(wr2, weightedroundness2) for (j=2,500,print1(wr2(j)","))

%Y Cf. A079167, A079169, A001222.

%K nonn

%O 1,4

%A _Jon Perry_, Dec 31 2002

%E a(1) = 0 prepended and more examples by _Antti Karttunen_, May 31 2017