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.
LINKS
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 *)
DivisorSigma[1, 2#]/2&/@Prime[Range[2, 60]] (* Harvey P. Dale, Jan 07 2023 *)
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
KEYWORD
nonn,easy
AUTHOR
Hartmut F. W. Hoft, Jul 29 2014
STATUS
approved