A079167 Weighted roundness of n. If n = p_1^e_1...p_k^e_k, then a(n) = e_1 + 2*e_2 + ... + k*e_k. Note that p_i < p_j, i < j is assumed.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 5, 1, 4, 3, 3, 1, 5, 2, 3, 3, 4, 1, 6, 1, 5, 3, 3, 3, 6, 1, 3, 3, 5, 1, 6, 1, 4, 4, 3, 1, 6, 2, 5, 3, 4, 1, 7, 3, 5, 3, 3, 1, 7, 1, 3, 4, 6, 3, 6, 1, 4, 3, 6, 1, 7, 1, 3, 5, 4, 3, 6, 1, 6, 4, 3, 1, 7, 3, 3, 3, 5, 1, 8, 3, 4, 3, 3, 3, 7, 1, 5, 4, 6, 1, 6, 1, 5, 6, 3

Offset

1,4

Comments

Note that in contrast to A056239 the factors i, j, ..., k in the sum are not the absolute indices (A000720) of those primes p_i, p_j, ..., p_k that divide n, but their relative order among all distinct primes dividing n, so that k = A001221(n). - Antti Karttunen, May 31 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Example

a(1) = 0 (an empty sum).
a(10) = 3 as 10 = 2 * 5, therefore a(10) = 1*1 + 2*1 = 1+2 = 3.
a(45) = 4 as 45 = 3^2 * 5, therefore a(45) = 1*2 + 1*2 = 4.
a(50) = 5 as 50 = 2^1 * 5^2, therefore a(50) = 1*1 + 2*2 = 5.
a(250) = 7 as 250 = 2 * 5^3, therefore a(250) = 1*2 + 3*2 = 7.

Prog

(PARI) weightedroundness(n) = local(f, fl, s); f=factor(n); fl=length(f[, 1]); s=0; for (i=1, fl, s=s+i*f[, 2][i]); s alias(wr, weightedroundness) for (j=2, 500, print1(wr(j)", "))
(PARI) A079167(n) = { my(f, s); f=factor(n); s=0; for(i=1, #f~, s += (i*f[i, 2])); s; }; \\ Modified from the above function by Antti Karttunen, May 31 2017

Crossrefs

Cf. A001222, A056239, A079168, A079169.

Sequence in context: A356159 A377247 A294926 * A304793 A199570 A350338

Adjacent sequences: A079164 A079165 A079166 * A079168 A079169 A079170

Keyword

nonn

Author

Jon Perry, Dec 31 2002

Extensions

a(1) = 0 prepended and more examples by Antti Karttunen, May 31 2017

Status

approved