

A245685


Sigma(2p)/2, for odd primes p.


6



6, 9, 12, 18, 21, 27, 30, 36, 45, 48, 57, 63, 66, 72, 81, 90, 93, 102, 108, 111, 120, 126, 135, 147, 153, 156, 162, 165, 171, 192, 198, 207, 210, 225, 228, 237, 246, 252, 261, 270, 273, 288, 291, 297, 300, 318, 336, 342, 345, 351, 360, 363, 378, 387, 396, 405
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OFFSET

1,1


COMMENTS

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).
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.
The sequence 2*p for primes p > 3 is a subsequence of A239929, numbers n whose symmetric representation of sigma(n) has two parts.
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).


LINKS

Table of n, a(n) for n=1..56.
Hartmut F. W. Hoft, Proofs of properties of sigma(2p)
Hartmut F. W. Hoft, Visualization of sigma(2p)


FORMULA

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.
a(n)=A247159(n+1)/2.  Omar E. Pol, Nov 22 2014


EXAMPLE

a(4) = T(22, 1)  T(22, 4) = 22  4 = 18 = sigma(22)/2
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.


MATHEMATICA

a[n_]:=3(Prime[n+1]+1)/2
Map[a, Range[55]] (* data *)


PROG

(PARI) vector(100, n, 3*(prime(n+1)+1)/2) \\ Derek Orr, Sep 19 2014
(MAGMA) [3*(NthPrime(n+1)+1)/2: n in [1..60]]; // Vincenzo Librandi, Sep 19 2014
(PARI) vector(60, n, sigma(2*prime(n+1))/2) \\ Michel Marcus, Nov 25 2014


CROSSREFS

Cf. A000203, A006254, A111333, A237048, A237270, A237271, A237591, A237593, A239929.
Sequence in context: A023042 A128245 A117714 * A315957 A114554 A315958
Adjacent sequences: A245682 A245683 A245684 * A245686 A245687 A245688


KEYWORD

nonn,easy


AUTHOR

Hartmut F. W. Hoft, Jul 29 2014


STATUS

approved



