A325968 a(n) is the sum k of such a subset of divisors of n with the largest sum and for which n-k and n-(sigma(n)-k) are relatively prime.

1, 3, 4, 7, 6, 7, 8, 15, 13, 17, 12, 27, 14, 23, 20, 31, 18, 38, 20, 41, 32, 35, 24, 59, 31, 39, 40, 29, 30, 71, 32, 63, 44, 53, 48, 91, 38, 59, 56, 89, 42, 95, 44, 83, 74, 69, 48, 123, 57, 93, 68, 95, 54, 119, 72, 119, 80, 89, 60, 167, 62, 95, 104, 127, 84, 143, 68, 125, 92, 143, 72, 194, 74, 113, 124, 137, 96, 167, 80, 185, 121, 125

Offset: 1

Author

Antti Karttunen, May 29 2019

Keywords

• nonn

Examples

			For n=15, its divisors are [1, 3, 5, 15]. If we take the full set [1, 3, 5, 15] and its complement [], their sums are 24 and 0, but gcd(15-0, 24-15) = gcd(15, 9) = 3 > 1. If we take subsets [1] and [3, 5, 15], then their sums are 1 and 23, but gcd(15-1, 23-15) = gcd(14,8) = 2 > 1. If we take subsets [3] and [1, 5, 15], their sums are 3 and 21, but gcd(15-3, 21-15) = gcd(12, 6) = 6 > 1. Only when we take the subset with the four smallest sum, [1, 3] and its complement [5, 15], we get such sums 4 and 20 for which gcd(15-4, 20-15) = gcd(11, 5) = 1. Thus a(15) = 20, the size of the subset with larger sum.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000040, A000203, A000396, A009194, A325807, A325818, A325967, A325969.

Programs

PARI

A325968(n) = { my(divs=divisors(n), s=sigma(n),r,ms=0); for(b=0,(2^(length(divs)))-1,r=sumbybits(divs,b);if(1==gcd(n-(s-r),n-r),ms=max(r,ms))); (ms); };
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };

Formula

a(n) = A000203(n) - A325967(n).

a(n) = n + A325969(n).

For all n:

a(A000040(n)) = A000040(n)+1.

a(A000396(n)) = A000396(n)+1.

a(n) <= A325818(n).