A272553 - OEIS

A272553

Numbers n whose sum of divisors equals the sum of divisors of 2n+1.

20, 464, 650, 2744, 3980, 5504, 5736, 5922, 7032, 8130, 10472, 18618, 24312, 27654, 38874, 39500, 43032, 45492, 56870, 64410, 71058, 79068, 85158, 89178, 92130, 97014, 109928, 117114, 118902, 127688, 130304, 175554, 180438, 187304, 188292, 208452, 224058, 244674, 249788, 269192, 294380, 305624, 347964

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OFFSET

1,1

COMMENTS

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.

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.

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.

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

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..1000

FORMULA

A000203(n) = A000203(2n+1).

EXAMPLE

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.

MAPLE

select(t -> numtheory:-sigma(t) = numtheory:-sigma(2*t+1), [$1..10^6]); # Robert Israel, May 03 2016

MATHEMATICA

Select[Range@500000, DivisorSigma[1, #]==DivisorSigma[1, 2*#+1]&]

PROG

(PARI) for (n=1, 500000, (sigma(n)==sigma(2*n+1)) && print1(n ", "))