cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A272553 Numbers n whose sum of divisors equals the sum of divisors of 2n+1.

Original entry on oeis.org

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
Offset: 1

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Author

Waldemar Puszkarz, May 02 2016

Keywords

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

Examples

			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.
		

Crossrefs

Cf. A000203 (sum of divisors), A074821 (similar sequence for the number of divisors), A005101 (abundant numbers, supersequence), A088831,

Programs

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

Formula

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