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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

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]]

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

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Last modified November 30 13:47 EST 2021. Contains 349420 sequences. (Running on oeis4.)