A068390 Numbers k such that sigma(k) = 4*phi(k).

14, 105, 248, 418, 1485, 3135, 3596, 3956, 4064, 5396, 8636, 20026, 23374, 25714, 35074, 35343, 39105, 41656, 55154, 56134, 56536, 71145, 74613, 87087, 124605, 150195, 175305, 192855, 263055, 393104, 413655, 421005, 474548, 604012, 697851, 711988, 819772

Offset: 1

Author

Benoit Cloitre, Mar 03 2002

Keywords

• easy
• nonn

Comments

If 2^p-1 is a prime (Mersenne prime) greater than 3 then 2^(p-2)*(2^p-1) is in the sequence. So for n>1, 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Feb 23 2005

Theorem: If m>0, k is an integer and p=2^(m+2)+k-1 is a prime number then n=2^m*p is a solution to the equation sigma(x) = 4*phi(x)-k. The previous comment is the special case k=0. - Farideh Firoozbakht, Oct 01 2014

References

• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 88.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
• Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
• Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
• Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1

Crossrefs

Cf. A000010, A000203, A079546, A000043, A292422.

Subsequence of A248150 (sigma(k) is divisible by 4).

Programs

Magma

[n: n in [1..10^6] | SumOfDivisors(n) eq 4*EulerPhi(n)]; // Vincenzo Librandi, Sep 25 2017

Mathematica

Select[Range[900000],DivisorSigma[1,#]==4EulerPhi[#]&] (* Harvey P. Dale, Nov 29 2013 *)

PARI

for(n=1,300000, if(sigma(n)==4*eulerphi(n),print1(n,",")))

Extensions

More terms from Carl Najafi, Aug 16 2011