A116020 Numbers n such that sigma(n) - phi(n) is a repdigit greater than 2.

4, 8, 9, 18, 25, 27, 28, 57, 62, 85, 123, 192, 218, 258, 259, 261, 322, 403, 632, 662, 693, 1127, 2195, 2218, 2321, 2658, 3548, 4577, 4763, 5597, 5603, 5921, 6662, 7421, 7697, 9617, 9683, 10721, 10877, 11537, 12317, 13323, 17243, 18659, 23363, 26483

Offset: 1

Author

Giovanni Resta, Feb 13 2006

Keywords

• nonn
• base

Comments

For every prime p sigma(p)-phi(p) is 2, so that case is trivial.

(I). If both numbers p=4*10^n+1 & q=(4*10^n-13)/9 are primes then m=p*q is in the sequence because sigma(m)-phi(m)=8*(10^(n+1)-1)/9 is a repdigit number. Conjecture: 123, 17243 & 1772443 are all such terms. - Farideh Firoozbakht, Aug 24 2006

(II). If p=(10^n-7)/3 is prime then m=2p is in the sequence because sigma(m)-phi(m)=2p+4=6*(10^n-1)/9 is a repdigit number. 62 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

(III). If p=(4*10^n-31)/9 is prime then m=3p is in the sequence because sigma(m)-phi(m)=2p+6=8*(10^n-1)/9 is a repdigit number. 123 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

(IV). If p=(8*10^n-17)/9 is a prime then both numbers 4p & 46p are in the sequence because sigma(4p)-phi(4p)=5p+9=4*(10^(n+1)-1)/9 & sigma(46p)-phi(46p)=50p+94=4*(10^(n+2)-1)/9 are repdigit numbers. 28 & 322 are the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

(V). If p=(4*10^n-13)/9 is a prime greater than 3 then m=6p is in the sequence because sigma(m)-phi(m)=10p+14=4*(10^(n+1)-1)/9 is a repdigit number. 258 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

(VI). If p=(8*10^(2n+1)-179)/99 is prime then m=8p is in the sequence because sigma(m)-phi(m)=11p+19=8*(10^(2n+1)-1)/9 is a repdigit number. 632 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

(VII). If p=(10^(3n+1)-37)/27 is prime then m=12p is in the sequence because sigma(m)-phi(m)=24p+32=8*(10^(3n+1)-1)/9 is a repdigit number. 4444444428 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

Examples

			sigma(662) - phi(662) = 666.
		

Crossrefs

Cf. A116017, A116018, A116019.