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%I #6 Aug 19 2014 16:33:35
%S 6,8,10,22,46,58,82,106,166,178,188,226,262,285,346,358,382,466,478,
%T 502,562,586,718,838,862,886,902,982,1018,1186,1282,1306,1318,1366,
%U 1438,1486,1522,1618,1822,1906,2013,2026,2038,2062,2098,2206,2446,2458,2578
%N Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).
%C If n/2 is an odd prime and n+1 is prime then n is in the sequence, the proof is easy. 8,188,285,902,2013,... are terms of the sequence which they aren't of such form. This sequence is a subsequence of A066198.
%C If p is an odd Sophie Germain prime then 2*p is in the sequence. There is no term of the sequence which is of the form 2*p where p is prime and p isn't Sophie Germain prime. A244438 gives terms of the sequence which isn't of the form 2*p where p is prime. - _Farideh Firoozbakht_, Aug 14 2014
%H Jens Kruse Andersen, <a href="/A145749/b145749.txt">Table of n, a(n) for n = 1..10000</a>
%F {n: A065387(n)=A065387(n+1)}.
%e 10 is in the sequence because phi(10) + sigma(10) = 4 + 18 = 22 and phi(11) + sigma(11) = 10 + 12 = 22 also.
%e 12 is not in the sequence because phi(12) + sigma(12) = 4 + 28 = 32 but phi(13) + sigma(13) = 12 + 14 = 26.
%t Select[Range[2600],DivisorSigma[1,# ]+EulerPhi[ # ]==DivisorSigma[1,#+1]+EulerPhi[ #+1]&]
%o (PARI)
%o for(n=1,10^4, s=eulerphi(n)+sigma(n); if(s==eulerphi(n+1)+sigma(n+1), print1(n,", "))) /* _Derek Orr_, Aug 14 2014*/
%Y Cf. A065387, A066198, A145748, A005384.
%K easy,nonn
%O 1,1
%A _Farideh Firoozbakht_, Nov 01 2008