A247159 -id:A247159 - cp's OEIS frontend

A245685 Sigma(2p)/2, for odd primes p.

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

Offset: 1

Hartmut F. W. Hoft, Jul 29 2014

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).

			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.

Hartmut F. W. Hoft, Proofs of properties of sigma(2p)
Hartmut F. W. Hoft, Visualization of sigma(2p)

Cf. A000203, A006254, A111333, A237048, A237270, A237271, A237591, A237593, A239929.

[3*(NthPrime(n+1)+1)/2: n in [1..60]]; // Vincenzo Librandi, Sep 19 2014

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 *)

vector(100,n,3*(prime(n+1)+1)/2) \\ Derek Orr, Sep 19 2014

vector(60, n, sigma(2*prime(n+1))/2) \\ Michel Marcus, Nov 25 2014

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

A289055 Triangle read by rows: T(n,k) = (k+1)*A028815(n) for 0 <= k <= n.

Original entry on oeis.org

2, 3, 6, 4, 8, 12, 6, 12, 18, 24, 8, 16, 24, 32, 40, 12, 24, 36, 48, 60, 72, 14, 28, 42, 56, 70, 84, 98, 18, 36, 54, 72, 90, 108, 126, 144, 20, 40, 60, 80, 100, 120, 140, 160, 180, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330

Offset: 0

Vincenzo Librandi, Sep 02 2017

			Triangle begins:
   2;
   3,   6;
   4,   8,  12;
   6,  12,  18,  24;
   8,  16,  24,  32,  40;
  12,  24,  36,  48,  60,  72;
  14,  28,  42,  56,  70,  84,  98;
  18,  36,  54,  72,  90, 108, 126, 144;
  20,  40,  60,  80, 100, 120, 140, 160, 180;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A289108.

Columns k: A028815 (k=0), A089241 (k=1), A247159 (k=2), A273801 (k=3).

/* As triangle (here NthPrime(0)=1) */ [[(k+1)*(NthPrime(n)+1): k in [0..n]]: n in [0.. 15]];

Join[{2}, t[n_, k_] := (k + 1) (Prime[n] + 1); Table[t[n, k], {n, 10}, {k, 0, n}] //Flatten]

def A289055(n,k): return 2 if n==0 else (k+1)*(nth_prime(n) +1)
flatten([[A289055(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 05 2024

a(n) = A289108(n) + 1.

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A245685 Sigma(2p)/2, for odd primes p.

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