Z Wikiźródeł, wolnej biblioteki
p
62
=
293
;
ψ
(
100
000
293
)
=
ψ
(
341
)
=
68
;
{\displaystyle p_{62}=293;\psi \left({\frac {100\,000}{293}}\right)=\psi (341)=68;}
ψ
(
341
,
61
)
=
68
−
60
=
8
{\displaystyle \psi \left(341,61\right)=68\,-60=\,\,8}
p
63
=
307
;
ψ
(
100
000
307
)
=
ψ
(
325
)
=
66
;
{\displaystyle p_{63}=307;\psi \left({\frac {100\,000}{307}}\right)=\psi (325)=66;}
ψ
(
325
,
62
)
=
66
−
61
=
5
{\displaystyle \psi \left(325,62\right)=66\,-61=\,\,5}
p
64
=
311
;
ψ
(
100
000
311
)
=
ψ
(
321
)
=
66
;
{\displaystyle p_{64}=311;\psi \left({\frac {100\,000}{311}}\right)=\psi (321)=66;}
ψ
(
321
,
63
)
=
66
−
62
=
4
{\displaystyle \psi \left(321,63\right)=66\,-62=\,\,4}
p
65
=
313
;
ψ
(
100
000
313
)
=
ψ
(
319
)
=
66
;
{\displaystyle p_{65}=313;\psi \left({\frac {100\,000}{313}}\right)=\psi (319)=66;}
6614
¯
{\displaystyle {\overline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf {6614} }}}
ψ
(
319
,
64
)
=
66
−
63
=
3
{\displaystyle \psi \left(319,64\right)=66\,-63=\,\,3}
4676
¯
{\displaystyle {\overline {\,\,\,\,\,\,\,\,\mathbf {4676} \,\,}}}
A zatem —
∑
s
=
1
s
=
51
ψ
(
100
000
p
14
+
s
)
=
−
6614
{\displaystyle \sum \limits _{s=1}^{s=51}\psi \left({\frac {100\,000}{p_{14+s}}}\right)=-\mathbf {\,6614} }
+
m
(
μ
+
1
)
=
14.
52
=
728
{\displaystyle +m\left(\mu +1\right)=14.\,52=\mathbf {728} }
+
μ
(
μ
−
1
)
2
=
51.
50
2
=
2550
2
=
1275
{\displaystyle +{\frac {\mu (\mu -1)}{2}}={\frac {51.\,50}{2}}={\frac {2550}{2}}=\mathbf {1275} }
ψ
(
100
000
)
=
φ
(
100
000
,
14
)
+
14.
52
+
51.
50
2
−
1
−
∑
s
=
1
s
=
51
ψ
(
100
000
p
14
+
s
)
=
14204
+
728
+
1275
−
1
−
6614
=
16102
−
6615
=
9592
{\displaystyle \psi (100\,000)=\varphi (100\,000,14)+14.\,52+{\frac {51.\,50}{2}}-1-\sum \limits _{s=1}^{s=51}\psi \left({\frac {100\,000}{p_{14+s}}}\right)=14204+728+1275-1-6614=16102-6615=\mathbf {9592} }
Funkcya zaś
φ
(
m
)
{\displaystyle \varphi (m)}
φ
(
m
)
{\displaystyle \varphi (m)}
ψ
(
100
000
)
=
φ
(
100
000
,
14
)
−
φ
(
100
000
,
65
)
+
65
−
1
=
14204
−
4676
−
1
+
65
=
14269
−
4677
=
9592
.
{\displaystyle \psi (100\,000)=\varphi (100\,000,14)-\varphi (100\,000,65)+65-1=14204-4676-1+65=14269-4677=\mathbf {9592} .}
Stąd wnioskuję, że wzór
ψ
(
n
)
=
φ
(
n
,
ψ
n
)
+
ψ
n
−
1
{\displaystyle \psi (n)=\varphi (n,\psi {\sqrt {n}})+\psi {\sqrt {n}}-1}
