A276714 -id:A276714 - cp's OEIS frontend

A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

2, 35, 55, 62, 74, 82, 91, 102, 115, 119, 122, 135, 142, 143, 155, 158, 172, 186, 202, 203, 206, 214, 215, 218, 242, 255, 259, 262, 282, 295, 298, 299, 302, 323, 326, 343, 351, 354, 355, 362, 391, 395, 399, 425, 426, 435, 451, 466, 478, 482, 492, 502, 511, 514

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

			35 is in sequence because tau(35) = tau(38) = 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A015861 (sigma(n) = sigma(n+3)), A276714.

Cf. Similar sequences with numbers n such that n and n+k have the same number of divisors for k = 1: A005237, for k = 2: A062832.

[n: n in [1..10000] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3)]

with(numtheory): A276713:=n->`if`(tau(n) = tau(n+3), n, NULL): seq(A276713(n), n=1..10^3); # Wesley Ivan Hurt, May 02 2017

SequencePosition[DivisorSigma[0,Range[600]],{x_,,,x_}][[All,1]] (* Harvey P. Dale, Nov 12 2022 *)

isok(n) = numdiv(n) == numdiv(n+3); \\ Michel Marcus, May 03 2017

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

Original entry on oeis.org

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

Jaroslav Krizek, Sep 16 2016

nonn

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

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

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

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.

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

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 *)

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

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

cp's OEIS Frontend

A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

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