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%I #21 May 06 2016 02:39:49
%S 20,464,650,2744,3980,5504,5736,5922,7032,8130,10472,18618,24312,
%T 27654,38874,39500,43032,45492,56870,64410,71058,79068,85158,89178,
%U 92130,97014,109928,117114,118902,127688,130304,175554,180438,187304,188292,208452,224058,244674,249788,269192,294380,305624,347964
%N Numbers n whose sum of divisors equals the sum of divisors of 2n+1.
%C Most terms are even; the first three odd ones are 1264545, 8770125, and 9346995, and these are the only odd terms among the first 10^7 numbers that include 135 terms.
%C For some n, 2n+1 is prime; for example, this is so for the first three terms, but this happens rarely with only 4 cases among the first 10^7 numbers.
%C All terms are abundant numbers (A005101): since sigma(x)>x for x>1, sigma(2n+1)>2n+1>2n for n>0, the defining formula, sigma(n)=sigma(2n+1), implies sigma(n)>2n, which proves that n is an abundant number.
%C Up to 6*10^9 there are 1151 terms, 46 of which are odd. All these odd terms are multiple of 3 and all are multiple of 5, except 1501989489 and 4242679749. The values n for which 2n+1 is a prime number are a subset of A088831, thus it is easy to verify that up to 10^13 there are only 4 such values (20, 464, 650, and 130304). - _Giovanni Resta_, May 03 2016
%H Giovanni Resta, <a href="/A272553/b272553.txt">Table of n, a(n) for n = 1..1000</a>
%F A000203(n) = A000203(2n+1).
%e 20 is a term as its sum of divisors, 42=1+2+4+5+10+20, is the same as the sum of divisors of 41=2*20+1; 41 has only two divisors 1 and 41.
%p select(t -> numtheory:-sigma(t) = numtheory:-sigma(2*t+1), [$1..10^6]); # _Robert Israel_, May 03 2016
%t Select[Range@500000, DivisorSigma[1, #]==DivisorSigma[1, 2*#+1]&]
%o (PARI) for (n=1, 500000, (sigma(n)==sigma(2*n+1)) && print1(n ", "))
%Y Cf. A000203 (sum of divisors), A074821 (similar sequence for the number of divisors), A005101 (abundant numbers, supersequence), A088831,
%K nonn
%O 1,1
%A _Waldemar Puszkarz_, May 02 2016
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