

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
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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<nexists(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]]


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



