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%I A245685 #47 Jan 07 2023 17:02:25
%S A245685 6,9,12,18,21,27,30,36,45,48,57,63,66,72,81,90,93,102,108,111,120,126,
%T A245685 135,147,153,156,162,165,171,192,198,207,210,225,228,237,246,252,261,
%U A245685 270,273,288,291,297,300,318,336,342,345,351,360,363,378,387,396,405
%N A245685 Sigma(2p)/2, for odd primes p.
%C A245685 The symmetric representation of sigma(2*p), p > 3 prime, consists of two sections each with three contiguous legs of width one (for a proof see the link).
%C A245685 The two ratios of successive legs in the symmetric representation of sigma(2*p) are integers 3 and 2, respectively, for all primes p > 3 satisfying p = -1(mod 6); see also A003627.  If one ratio is an integer then so is the other.
%C A245685 The sequence 2*p for primes p > 3 is a subsequence of A239929, numbers n whose symmetric representation of sigma(n) has two parts.
%C A245685 Since sigma(2*p) = 3*(p+1), each element of the sequence is a multiple of 3; furthermore, a(n)/3 = A006254(n) = A111333(n+1).
%H A245685 Hartmut F. W. Hoft, <a href="/A245685/a245685.pdf">Proofs of properties of sigma(2p)</a>
%H A245685 Hartmut F. W. Hoft, <a href="/A245685/a245685_1.pdf">Visualization of sigma(2p)</a>
%F A245685 a(n) = T(2*prime(n+1), 1) - T(2*prime(n+1), 4) = 3*(prime(n+1)+1)/2 = sigma(2*prime(n+1))/2 where T(n,k) is defined in A235791.
%F A245685 a(n)=A247159(n+1)/2. - _Omar E. Pol_, Nov 22 2014
%e A245685 a(4) = T(22, 1) - T(22, 4) = 22 - 4 = 18 = sigma(22)/2
%e A245685 The last image in the Example section of A237593 includes the first four symmetric representations for this sequence, i.e., when 2*p = 10, 14, 22 & 26; see also the link for an image of the first 10 symmetric representations.
%t A245685 a[n_]:=3(Prime[n+1]+1)/2
%t A245685 Map[a,Range[55]] (* data *)
%t A245685 DivisorSigma[1,2#]/2&/@Prime[Range[2,60]] (* _Harvey P. Dale_, Jan 07 2023 *)
%o A245685 (PARI) vector(100,n,3*(prime(n+1)+1)/2) \\ _Derek Orr_, Sep 19 2014
%o A245685 (Magma) [3*(NthPrime(n+1)+1)/2: n in [1..60]]; // _Vincenzo Librandi_, Sep 19 2014
%o A245685 (PARI) vector(60, n, sigma(2*prime(n+1))/2) \\ _Michel Marcus_, Nov 25 2014
%Y A245685 Cf. A000203, A006254, A111333, A237048, A237270, A237271, A237591, A237593, A239929.
%K A245685 nonn,easy
%O A245685 1,1
%A A245685 _Hartmut F. W. Hoft_, Jul 29 2014

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