/
new.sf
166 lines (126 loc) · 2.28 KB
/
new.sf
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#!/usr/bin/ruby
func f(n) {
(1i)**(n**2)
}
func g(n) {
1i**bigomega(n)
}
func F(n, m) {
sum(1..n, {|k|
f(k) * faulhaber(floor(n/k), m)
})
}
func G(n, m) {
sum(1..n, {|k|
g(k) * faulhaber(floor(n/k), m)
})
}
say 20.of { F(_, 0) }.map{.re}
say 20.of { F(_, 0) }.map{.im}
#say 20.of { G(_, 0) }.map{.im}
__END__
func PF(m) { # https://oeis.org/A065469
1e3.primes.prod {|p|
(1 - 1/(p**m - 1))
}
}
func PG(m) { # https://oeis.org/A182448 Pi^2/15 = zeta(2*2) / zeta(2)
#~ 1e3.primes.prod {|p|
#~ #(1 - 1/(p**m - 1))
#~ p**m / (p**m + 1)
#~ }
zeta(2*m) / zeta(m)
}
for k in (1..10) {
#say (G(1e5, k) / faulhaber(1e5, k))
#say PG(k+1)
say F(1e4, k)
say (faulhaber(1e4, k) * PF(k+1))
say ''
}
#
#say PG(4)
__END__
say 26.of {|n|
n.divisors.sum{|d|
f(d)
} != 0 ? 1 : 0
}.accumulate
say 26.of {|n|
n.divisors.sum{|d|
g(d)
}
}.accumulate
__END__
for k in (1..100) {
#k.is_powerful && say [k.factor, k.divisors.sum{|d| f(d) }]
if (k.divisors.sum{|d| g(d) }) {
say [k.sqrt]
}
}
__END__
say 20.of {|n|
n.divisors.sum{|d|
g(d)
}
}
__END__
func foo(n) {
sum(1..n, {|k|
k.divisors.sum{|d|
f(k/d) * d
}
})
}
func bar(n) {
n.divisors.sum {|d|
f(n/d) * d
}
}
func z(n) {
n.divisors.sum {|d|
d.divisors.sum {|d|
(-1)**omega(d)
} * euler_phi(n/d)
}
}
func conj(n) {
n.factor_prod{|p,e|
p**e - (p**e - 1)/(p - 1)
#sum(0..e.dec, {|k|
# -p**k
#})
}
}
func f2(n) {
sum(1..n, {|k|
(-1)**omega(k) * (n//k) * (1 + n//k)
}) / 2
}
#say 20.of(bar)
for k in (1..500) {
#say [bar(k), euler_phi(k), euler_phi(k) - bar(k)]
print(euler_phi(k) - bar(k), ", ")
}
__END__
for n in (1..10000) {
assert_eq(bar(n), conj(n))
#var t = 1e30.irand
#say [z(t), conj(t)]
#say conj(n.pn_primorial ** 2)
}
#say z(23*7)
__END__
say 20.of(bar).accumulate
say 20.of(f2)
for n in (1..10) {
say (f2(10**n) / faulhaber(10**n, 1))
}
# sqrt(n)
#say 20.of(foo)
#say 20.of(bar).slice(1)
#say 20.of(foo).map_cons(2, {|a,b| b-a })
#say 800.of(bar)
#~ for k in (1..1000) {
#~ say bar(k)
#~ }