A244448 a(n) is the smallest integer m such that m-n is composite and phi(m-n) + sigma(m+n) = phi(m+n) + sigma(m-n).

4, 153, 442, 213, 179, 120, 46, 37, 47, 264, 145416, 1101, 107, 79, 71, 78, 716, 637, 98, 249, 71, 126, 13258, 1243, 119, 163, 119, 131, 140497, 381, 191, 156, 101, 169, 1574, 315, 151, 193, 167, 2158, 148, 104, 202, 289, 1969, 882, 2572, 428, 251, 357, 314, 283

Offset: 0

Author

Jahangeer Kholdi and Farideh Firoozbakht, Aug 30 2014

Keywords

• nonn

Comments

For each n, a(n) > n and like a(n)-n, a(n)+n is also composite.

If both numbers p and p + 2n are primes then x = p+n is a solution to the equation phi(x-n) + sigma(x+n) = phi(x+n) + sigma(x-n). But for these many solutions x, both numbers x-n and x+n are primes.

a(n) is the smallest integer m such that m-n is composite and A051612(m+n) = A051612(m-n) where A051612(n) = sigma(n) - phi(n). - Michel Marcus, Mar 20 2020

Examples

			a(1)=153 because 153-1 is composite, phi(153-1)+sigma(153+1) = phi(153+1)+sigma(153-1) and there is no such number less than 153.
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..4501 (terms 0..1000 from Jinyuan Wang)

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A051612 (sigma - phi).

Cf. A244446, A244447, A246628.

Programs

Mathematica

a[0]=4;a[n_]:=a[n]=(For[m=n+1,PrimeQ[m-n]||EulerPhi[m-n]+DivisorSigma[1,m+n]!=EulerPhi[m+n]+DivisorSigma[1,m-n],m++];m);
Table[a[n],{n,0,70}]

PARI

a(n) = {my(m=n+4); while(isprime(m-n) || eulerphi(m+n)+sigma(m-n)!=eulerphi(m-n)+sigma(m+n), m++); m; }
vector(100,n,a(n)) \\ Derek Orr, Aug 30 2014