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

1,1

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

Links

Table of n, a(n) for n=1..46.

Example

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

Crossrefs

Cf. A116017, A116018, A116019.

Sequence in context: A071592 A089765 A116030 * A354869 A378172 A213015

Adjacent sequences: A116017 A116018 A116019 * A116021 A116022 A116023

Keyword

nonn,base

Author

Giovanni Resta, Feb 13 2006

Status

approved