|
%I #52 Aug 10 2019 17:53:04
%S 2,4,3,8,4,2,4,16,12,9,6,14,6,12,8,32,10,26,8,21,14,18,12,20,30,16,18,
%T 2,14,24,10,64,16,24,22,88,14,30,26,36,18,32,14,42,26,28,24,54,56,80,
%U 20,32,26,40,36,60,38,42,30,56,18,42,48,128,42,48,22,50,28,72,26,122,26,54,58,46,48,56,26,86,120,60,42,96,54
%N Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.
%C Number of ways to form the sum sigma(n) = x+y so that n-x and n-y are coprime, with x and y in range 0..sigma(n).
%C From _Antti Karttunen_, May 28 - Jun 08 2019: (Start)
%C Empirically, it seems that a(n) >= A034444(n) and also that a(n) >= A034444(A000203(n)) unless n is in A000396.
%C Specifically, if it could be proved that a(n) >= A034444(n)/2 for n >= 2, which in turn would imply that a(n) >= A001221(n) for all n, then we would know that no odd perfect numbers could exist. Note that a(n) must be 2 on all perfect numbers, whether even or odd. See also A325819.
%C (End)
%H Antti Karttunen, <a href="/A324213/b324213.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = Sum_{i=0..sigma(n)} [1 == gcd(n-i,n-(sigma(n)-i))], where [ ] is the Iverson bracket and sigma(n) is A000203(n).
%F a(A000396(n)) = 2.
%F a(n) = A325815(n) + A034444(n).
%F a(n) = 1+A000203(n) - A325816(n).
%F a(A228058(n)) = A325819(n).
%e For n=1, sigma(1) = 1, both gcd(1-0, 1-(1-0)) = gcd(1,0) = 1 and gcd(1-1, 1-(1-1)) = gcd(0,1) = 1, thus a(1) = 2.
%e --
%e For n=3, sigma(3) = 4, we have 5 cases to consider:
%e gcd(3-0, 3-(4-0)) = 1 = gcd(3-4, 3-(4-4)),
%e gcd(3-1, 3-(4-1)) = 2 = gcd(3-3, 3-(4-3)),
%e gcd(3-2, 3-(4-2)) = 1,
%e of which three cases give 1 as a result, thus a(3) = 3.
%e --
%e For n=6, sigma(6) = 12, we have 13 cases to consider:
%e gcd(6-0, 6-(12-0)) = 6 = gcd(6-12, 6-(12-12)),
%e gcd(6-1, 6-(12-1)) = 5 = gcd(6-11, 6-(12-11)),
%e gcd(6-2, 6-(12-2)) = 4 = gcd(6-10, 6-(12-10)),
%e gcd(6-3, 6-(12-3)) = 3 = gcd(6-9, 6-(12-9)),
%e gcd(6-4, 6-(12-4)) = 2 = gcd(6-8, 6-(12-8))
%e gcd(6-5, 6-(12-5)) = 1 = gcd(6-7, 6-(12-7)),
%e gcd(6-6, 6-(12-6)) = 0,
%e of which only two give 1 as a result, thus a(6) = 2.
%e --
%e For n=10, sigma(10) = 18, we have 19 cases to consider:
%e gcd(10-0, 10-(18-0)) = 2 = gcd(10-18, 10-(18-18)),
%e gcd(10-1, 10-(18-1)) = 1 = gcd(10-17, 10-(18-17)),
%e gcd(10-2, 10-(18-2)) = 2 = gcd(10-16, 10-(18-16)),
%e gcd(10-3, 10-(18-3)) = 1 = gcd(10-15, 10-(18-15)),
%e gcd(10-4, 10-(18-4)) = 2 = gcd(10-14, 10-(18-14)),
%e gcd(10-5, 10-(18-5)) = 1 = gcd(10-13, 10-(18-13)),
%e gcd(10-6, 10-(18-6)) = 2 = gcd(10-12, 10-(18-12)),
%e gcd(10-7, 10-(18-7)) = 1 = gcd(10-11, 10-(18-11)),
%e gcd(10-8, 10-(18-8)) = 2 = gcd(10-10, 10-(18-10)),
%e gcd(10-9, 10-(18-9)) = 1,
%e of which 9 cases give 1 as a result, thus a(10) = 9.
%e --
%e For n=15, sigma(15) = 24, we have 25 cases to consider:
%e gcd(15-0, 15-(24-0)) = 3 = gcd(15-24, 15-(24-24)),
%e gcd(15-1, 15-(24-1)) = 2 = gcd(15-23, 15-(24-23)),
%e gcd(15-2, 15-(24-2)) = 1 = gcd(15-22, 15-(24-22)),
%e gcd(15-3, 15-(24-3)) = 6 = gcd(15-21, 15-(24-21)),
%e gcd(15-4, 15-(24-4)) = 1 = gcd(15-20, 15-(24-20)),
%e gcd(15-5, 15-(24-5)) = 2 = gcd(15-19, 15-(24-19)),
%e gcd(15-6, 15-(24-6)) = 3 = gcd(15-18, 15-(24-18)),
%e gcd(15-7, 15-(24-7)) = 2 = gcd(15-17, 15-(24-17)),
%e gcd(15-8, 15-(24-8)) = 1 = gcd(15-16, 15-(24-16)),
%e gcd(15-9, 15-(24-9)) = 6 = gcd(15-15, 15-(24-15)),
%e gcd(15-10, 15-(24-10)) = 1 = gcd(15-14, 15-(24-14)),
%e gcd(15-11, 15-(24-11)) = 2 = gcd(15-13, 15-(24-13)),
%e gcd(15-12, 15-(24-12)) = 3,
%e of which 2*4 = 8 cases give 1 as a result, thus a(15) = 8.
%t Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* _Michael De Vlieger_, Jun 09 2019 *)
%o (PARI) A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };
%Y Cf. A000010, A000203, A000396, A001221, A014567, A034444, A058062, A062401, A228058, A325807, A325815, A325816, A325817, A325818, A325819, A325960, A325961, A325962, A325965, A325966, A325967, A325968, A325970, A325971, A325972.
%K nonn
%O 1,1
%A _Antti Karttunen_ and _David A. Corneth_, May 26 2019, with better name from _Charlie Neder_, Jun 02 2019
|
