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

Author

Jaroslav Krizek, Sep 16 2016

Keywords

• nonn

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

Examples

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

Links

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

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.

Programs

Magma

A276715:=func; [A276715(n):n in[0..32]]

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 *)

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

Extensions

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