A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2*A045763(n) = 2*A073757(n)-n.

1, 2, 3, 4, 5, 4, 7, 6, 7, 4, 11, 6, 13, 4, 7, 8, 17, 4, 19, 6, 9, 4, 23, 6, 19, 4, 15, 6, 29, 0, 31, 10, 13, 4, 19, 4, 37, 4, 15, 6, 41, -4, 43, 6, 13, 4, 47, 2, 39, 0, 19, 6, 53, -4, 31, 6, 21, 4, 59, -6, 61, 4, 19, 12, 37, -12, 67, 6, 25, -8, 71, -2, 73, 4, 15, 6, 49, -16, 79, 2, 35, 4, 83, -14, 49, 4, 31, 6, 89, -20, 59, 6, 33, 4, 55, -10, 97, -4

Offset

1,2

Comments

There are only 2 cases [n=30, n=50] below 10^7 such that a(n) = 0.

No other zeros found up to 10^9. - Michel Marcus, Jul 30 2017

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Formula

a(n) = 2(A000005(n)+A000010(n)-1)-n.

Example

n=37, d=2,r=36,u=0, a(37)=2+36-1-0=37>0; primes are fixed points.
n=42, d=8,r=12,u=23,a(42)=8+12-1-23=-4<0, terms of A083244;
n=30, d=8,r=8,u=15, a(30)=0;
n=50, d=6,r=20,u=25,a(50)=0.

Mathematica

Table[2*(DivisorSigma[0, w]+EulerPhi[w]-1)-w, {w, 1, 1000}]

Prog

(PARI) a(n) = 2*(numdiv(n)+eulerphi(n)-1) - n; \\ Michel Marcus, Jul 30 2017

Crossrefs

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083246, A020488.

Sequence in context: A135682 A287415 A352612 * A111610 A119816 A227538

Adjacent sequences: A083242 A083243 A083244 * A083246 A083247 A083248

Keyword

sign

Author

Labos Elemer, May 07 2003

Status

approved