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%I #19 Jul 26 2022 01:35:09
%S 1,14,33,42677635,51,46,155,62,69,46,174,154,285,182,141,62,138,142,
%T 235,158,123,94,213,322,295,94,177,118,159,406,376,266,177,891528365,
%U 321,310,355,248,249,166,213,418,376,602,426,142,570,310,445,248,249,158
%N a(n) = the smallest number k such that k and k + n have the same number and sum of divisors (A000005 and A000203).
%C If a(33) exists, it must be greater than 2*10^8.
%C 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, ...
%C 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
%H Amiram Eldar, <a href="/A276715/b276715.txt">Table of n, a(n) for n = 0..10000</a>
%e a(2) = 33 because 33 is the smallest number such that tau(33) = tau(35) = 4 and simultaneously sigma(33) = sigma(35) = 48.
%t 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 *)
%o (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]]
%o (Python)
%o from itertools import count
%o from sympy import divisor_sigma
%o 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
%Y Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A007365 (smallest k such that sigma(k) = sigma(k+n)).
%Y 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.
%K nonn
%O 0,2
%A _Jaroslav Krizek_, Sep 16 2016
%E a(33) onwards from _Amiram Eldar_, Feb 17 2019
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