A206028 a(n) is the sum of distinct values of sigma(d) where d runs over the divisors of n and sigma = A000203.

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 248

Offset: 1

Author

Jaroslav Krizek, Feb 03 2012

Keywords

• nonn

Comments

Sequence is not the same as A007429: a(66) = 248, A007429(66) = 260. Number 66 is the smallest number with at least two divisors d with the same sigma(d); see A206030.

In A007429 all values of sigma(d) of the divisors d of n are included in the sum with repetitions allowed. In this sequence only the distinct values of sigma(d) of the divisors d of n are included in the sum.

If a term is a prime p when n = 2^j then p = 2^(j+2)-(j+3) is also a term of A099440 (primes of the form 2^n-n-1). Greater of twin primes are terms. - Metin Sariyar, Apr 03 2020

Examples

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Sum of k = 1+3+4+12 = 20.
For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Sum of k = 1+3+4+12+36+48+144 = 248 (note that only one twelve is added.).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000217, A007429, A184387, A206029, A206030.

Programs

Mathematica

Table[Total[Union[DivisorSigma[1, Divisors[n]]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

PARI

a(n)={vecsum(Set(apply(sigma, divisors(n))))} \\ Andrew Howroyd, Aug 01 2018

Formula

a(p) = p+2, a(pq) = (p+2)*(q+2) for p, q = distinct primes.

a(n) = A184387(n) - A206029(n) = A000217(A000203(n)) - A206029(n).

a(2^n) = 2^(n+2) - (n+3). - Metin Sariyar, Apr 09 2020

Extensions

Name clarified by David A. Corneth, Aug 01 2018

a(62)-a(66) from Andrew Howroyd, Aug 01 2018