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A007365 Smallest k such that sigma(n+k) = sigma(k).

%I A007365 M4928 #35 Oct 29 2023 21:27:30
%S A007365 1,14,33,382,51,6,20,10,15,14,21,28,35,182,24,26,30,142,40,34,42,20,
%T A007365 57,135,70,30,99,42,66,406,88,56,60,54,93,24,105,248,147,44,63,30,80,
%U A007365 435,114,52,196,310,140,40,105,92,160,66,120,140,105,88,352,154
%N A007365 Smallest k such that sigma(n+k) = sigma(k).
%C A007365 If p > 3 is prime, a(p) <= 14*p. - _Robert Israel_, Feb 21 2020
%D A007365 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A007365 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007365 Donovan Johnson, <a href="/A007365/b007365.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H A007365 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A007365 R. G. Wilson, V, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>.
%p A007365 N:= 1000: # to get all terms before the first with n + a(n) > N
%p A007365 S:= map(numtheory:-sigma, [$1..N]):
%p A007365 Res:= NULL:
%p A007365 found:= true:
%p A007365 for n from 1 while found do
%p A007365 found:= false;
%p A007365 for k from 1 to N-n do
%p A007365    if S[k] = S[k+n] then
%p A007365      Res:= Res, k; found:= true; break;
%p A007365    fi
%p A007365 od;
%p A007365 od:
%p A007365 Res; # _Robert Israel_, Feb 21 2020
%t A007365 sk[n_]:=Module[{k=1},While[DivisorSigma[1,k]!=DivisorSigma[1,n+k], k++];k]; Array[sk,60,0] (* _Harvey P. Dale_, Oct 10 2012 *)
%o A007365 (PARI) A007365(m)= {local(k,n); for(k=1,m,n=1; while(sigma(n)!=sigma(n+k), n++); print1(n,","))} \\ _Klaus Brockhaus_
%Y A007365 Cf. A065932, A065933. sigma(x)=A000203(x) is the sum of the divisors of x.
%K A007365 nonn
%O A007365 0,2
%A A007365 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_