A068414 -id:A068414 - cp's OEIS frontend

A068418 Composite numbers k such that k - phi(k) divides sigma(k) - k.

12, 56, 260, 992, 1320, 1976, 2156, 2754, 3696, 5520, 13800, 16256, 19872, 22560, 23688, 25232, 41072, 87000, 89964, 133984, 145888, 366720, 785808, 851760, 1100864, 1235052, 1270208, 1439552, 1470720, 2129400, 2237888, 4729664

Offset: 1

Benoit Cloitre, Mar 03 2002

If 2^p - 1 is prime (a Mersenne prime) then k = 2^p*(2^p - 1) is in the sequence because 3*k - 2*phi(k) = sigma(k) (see Comments at A068414) so sigma(k) - k = 2*(k - phi(k)) hence k - phi(k) divides sigma(k) - k. - Farideh Firoozbakht, Dec 31 2005

Also if 3*2^m - 1 is a prime greater than 5 then k = 15*2^(m+1)*(3*2^m - 1) is in the sequence because 4*k - 3*phi(k) = 4*15*2^(m+1)*(3*2^m - 1) - 3*2^(m+3)*(3*2^m - 2) = 24*(3*2^m)*(2^(m+2) - 1) = sigma(15)*sigma(3*2^m - 1)*sigma(2^(m+1)) = sigma(15*(3*2^m - 1)*2^(m+1)) = sigma(k) hence sigma(k) - k = 3*(k - phi(k)) and k - phi(k) divides sigma(k) - k. - Farideh Firoozbakht, Dec 31 2005

Donovan Johnson, Table of n, a(n) for n = 1..68 (terms < 10^11)

A068414 is the subsequence telling when the quotient is 2.

Cf. A068414, A051953, A000203, A000668.

Do[s=(DivisorSigma[1, n]-n)/(n-EulerPhi[n]); If[ !PrimeQ[n]&&IntegerQ[s], Print[n]], {n, 2, 10000000}]

for(n=1,300000, if((sigma(n)-n)%(n-eulerphi(n))==isprime(n),print1(n,",")))

More terms from Labos Elemer, Apr 02 2002

A069714 Least integer k such that (sigma(k)-k)/(k-phi(k)) = n.

Original entry on oeis.org

2, 12, 1320, 851760

Offset: 1

Labos Elemer, Apr 02 2002

10^11 < a(5) <= 124176229632000. - Donovan Johnson, Oct 22 2011

Cf. A068414, A068418, A001065, A051953, A000010, A000203.

a(n) = Min{x: A001065(x) / A051953(x) = n}.

A069737 Integer quotient defining A068418 is 3.

Original entry on oeis.org

1320, 3696, 5520, 13800, 19872, 22560, 23688, 87000, 89964, 366720, 785808, 1235052, 1470720, 8690112, 45986616, 46862592, 78584640, 142588224, 231718968, 235668528, 377425920, 1598249952, 9518204448, 12026499072, 40964894160, 65368504980

Offset: 1

Labos Elemer, Apr 05 2002

nonn

			40th term in A068418 is m=142588224, sum-of-proper-divisors[m]=294860736, cototient[m]=98286912, quotient=3=294860736/98286912,

Cf. A068418, A068414, A069714.

(Sum of proper divisors[m])/cototient[m]=3, where m is a term of A068418.

More terms from Labos Elemer, Apr 22 2002

a(22)-a(26) from Donovan Johnson, May 27 2011

A069719 Integer quotients arising in A068418: sum of proper divisors is divided by cototient of terms of A068418.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 4, 2, 3, 2, 2, 3, 4, 2, 2, 3, 3, 3, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3

Offset: 1

Labos Elemer, Apr 05 2002

			n=39: m=A068418(39)=130141440, sigma[m]-m=405550080, m-Phi[m]=101387520, quotient=4=a(39).

Cf. A068418, A001065, A000203, A051953, A068414, A069714, A069737.

a(n) = A001065(A068418(n)) - A051953(A068418(n)).

a(47)-a(68) from Donovan Johnson, May 27 2011

cp's OEIS Frontend

A068418 Composite numbers k such that k - phi(k) divides sigma(k) - k.

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A069714 Least integer k such that (sigma(k)-k)/(k-phi(k)) = n.

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A069737 Integer quotient defining A068418 is 3.

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A069719 Integer quotients arising in A068418: sum of proper divisors is divided by cototient of terms of A068418.

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