A276715 a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).

1, 14, 33, 42677635, 51, 46, 155, 62, 69, 46, 174, 154, 285, 182, 141, 62, 138, 142, 235, 158, 123, 94, 213, 322, 295, 94, 177, 118, 159, 406, 376, 266, 177, 891528365, 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158

Offset

0,2

Comments

If a(33) exists, it must be greater than 2*10^8.

a(n) for n >= 34: 321, 310, 355, 248, 249, 166, 213, 418, 376, 602, 426, 142, 570, 310, 445, 248, 249, 158, 267, 406, 632, 166, 267, ...

The records occur at indices 0, 1, 2, 3, 33, 207, 471, ... with values 1, 14, 33, 42677635, 891528365, 2944756815, 3659575815, ... - Amiram Eldar, Feb 17 2019

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Example

a(2) = 33 because 33 is the smallest number such that tau(33) = tau(35) = 4 and simultaneously sigma(33) = sigma(35) = 48.

Mathematica

a[k_] := Module[{n=1}, While[DivisorSigma[0, n] != DivisorSigma[0, n+k] || DivisorSigma[1, n] != DivisorSigma[1, n+k], n++]; n]; Array[a, 50, 0] (* Amiram Eldar, Feb 17 2019 *)

Prog

(Magma) A276715:=func<n|exists(r){k:k in[1..1000000] | NumberOfDivisors(k) eq NumberOfDivisors(k+n) and SumOfDivisors(k) eq SumOfDivisors(k+n)}select r else 0>; [A276715(n):n in[0..32]]
(Python)
from itertools import count
from sympy import divisor_sigma
def A276715(n): return next(k for k in count(1) if all(divisor_sigma(k, i)==divisor_sigma(n+k, i) for i in (0, 1))) # Chai Wah Wu, Jul 25 2022

Crossrefs

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A007365 (smallest k such that sigma(k) = sigma(k+n)).

Cf. Sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254, k=3: A276714.

Sequence in context: A018949 A007365 A065933 * A115670 A182180 A112878

Adjacent sequences: A276712 A276713 A276714 * A276716 A276717 A276718

Keyword

nonn

Author

Jaroslav Krizek, Sep 16 2016

Extensions

a(33) onwards from Amiram Eldar, Feb 17 2019

Status

approved