|
| |
|
|
A276715
|
|
a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
