A306613 First differences of A063990 (amicable numbers arranged in increasing order).

64, 900, 26, 1410, 304, 2096, 544, 668, 136, 4376, 112, 1429, 2310, 2701, 1120, 44604, 3908, 64, 103, 2520, 1530, 4939, 3666, 7883, 1097, 11755, 21780, 103, 784, 1003, 15660, 1849, 646, 10866, 15554, 3126, 4416, 64, 4512, 4520, 11356, 5720, 988, 77108, 28080, 10930

Offset: 1

Author

Conor Coons, Feb 28 2019

Keywords

• nonn

Comments

a(n) is the difference between the n-th and (n+1)-th amicable numbers when ordered by increasing value.

For 1 <= k <= 8, a(2k-1) is the difference between the larger and the smaller terms of the k-th amicable pair, and for 1 <= k <= 8, a(2k) is the difference between the smaller term of the (k+1)-th pair and the larger term of the k-th pair. Beginning with the 9th pair (63020,76084), the pairs ordered by their first element are no longer adjacent. - Bernard Schott, Mar 09 2019

Examples

			a(2) = amicable(3) - amicable(2) = 1184 - 284 = 900.
From _Bernard Schott_, Mar 10 2019: (Start)
a(1) = 284 - 220 = 64 is the difference between the larger and the smaller terms of the first amicable pair.
a(4) = 2620 - 1210 = 1410 is the difference between the smaller term of the third amicable pair and the larger term of the second amicable pair. (End)
		

Crossrefs

Cf. A063990 (amicable numbers), A306612.

Cf. A002025, A002046.

Cf. A066539 (difference between larger and smaller terms of n-th amicable pair).

Cf. A139228 (first differences of perfect numbers).

Programs

MATLAB

clear
clc
A = zeros(100000,1);
parfor n = 1:1:100000
     f = find(rem(n, 1:floor(sqrt(n))) == 0);
     f = unique([1, n, f, fix(n./f)]);
     A(n) = sum(f) - n;
end
D = [];
d = 1;
for a = 1:1:100000
    for b = 1:1:100000
        if A(a) == b && A(b) == a && a~=b
            D(d) = a;
            d = d+1;
        end
    end
end
D
difference = diff(D)

Formula

a(n) = A063990(n+1) - A063990(n). - Michel Marcus, Apr 08 2019

Extensions

More terms from Michel Marcus, Mar 09 2019