This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A341605 #34 Jan 04 2025 16:11:01
%S A341605 3,7,4,2,13,6,15,8,31,8,9,40,48,57,12,7,32,156,96,133,14,12,26,72,400,
%T A341605 168,183,18,31,16,248,16,1464,252,307,20,13,121,84,684,216,2380,360,
%U A341605 381,24,21,124,781,144,1862,280,5220,480,553,30,18,104,342,2801,240,3294,432,7240,720,871,32
%N A341605 Square array A(n,k) = A017665(A246278(n,k)), read by falling antidiagonals; numerator of the abundancy index as applied onto prime shift array A246278.
%C A341605 Ratio A341605(row, col)/A341606(row, col) shows the abundancy index when applied to the natural numbers > 1 as ordered in the prime shift array A246278:
%C A341605 n = 1 2 3 4 5 6
%C A341605 2n = 2 4 6 8 10 12
%C A341605 ----+--------------------------------------------------------------------------
%C A341605 1 | 3/2, 7/4, 2/1, 15/8, 9/5, 7/3,
%C A341605 2 | 4/3, 13/9, 8/5, 40/27, 32/21, 26/15,
%C A341605 3 | 6/5, 31/25, 48/35, 156/125, 72/55, 248/175,
%C A341605 4 | 8/7, 57/49, 96/77, 400/343, 16/13, 684/539,
%C A341605 5 | 12/11, 133/121, 168/143, 1464/1331, 216/187, 1862/1573,
%C A341605 6 | 14/13, 183/169, 252/221, 2380/2197, 280/247, 3294/2873,
%C A341605 7 | 18/17, 307/289, 360/323, 5220/4913, 432/391, 6140/5491,
%C A341605 we see that when going down in each column, the magnitude of the ratio decreases monotonically, which follows because the abundancy index of prime(i+1)^e is less than that of prime(i)^e (see A336389). The first ratio that is < 2 (corresponding to the first deficient number obtained when 2*n is successively prime shifted) is found at row number 1+A336835(2*n) = 1+A378985(n) for column n.
%C A341605 Each ratio r at row n and column k is a product of the topmost ratio (on row 1), and the product of all ratios on rows 1..(row-1) given in arrays A341626/A341627:
%C A341605 n = 1 2 3 4 5 6
%C A341605 2n = 2 4 6 8 10 12
%C A341605 ----+--------------------------------------------------------------------------
%C A341605 1 | 8/9, 52/63, 4/5, 64/81, 160/189, 26/35,
%C A341605 2 | 9/10, 279/325, 6/7, 1053/1250, 189/220, 372/455,
%C A341605 3 | 20/21, 1425/1519, 10/11, 12500/13377, 110/117, 4275/4774,
%C A341605 4 | 21/22, 343/363, 49/52, 62769/66550, 351/374, 2401/2574,
%C A341605 5 | 77/78, 22143/22477, 33/34, 791945/804102, 6545/6669, 199287/205751,
%C A341605 6 | 117/119, 51883/52887, 130/133, 573417/584647, 13338/13685, 518830/531981,
%C A341605 In other words, if r(row,col) = A341605(row,col)/A341606(row,col) and d(row,col) = A341626(row,col)/A341627(row,col), then r(row+1,col) = r(row,col)*d(row,col), that is, each column in the latter arrays of ratios gives the first quotients of ratios in the corresponding columns in the former array, and they are all < 1.
%C A341605 See also comments and examples in A341606.
%C A341605 By lemma given in A341529, the ratio A341626/A341627 stays in open interval (0.5 .. 1). - _Antti Karttunen_, Jan 02 2025
%H A341605 Antti Karttunen, <a href="/A341605/b341605.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals</a>
%H A341605 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A341605 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A341605 A(n, k) = A017665(A246278(n, k)).
%F A341605 A(n, k) = A355927(n, k) / A355925(n, k). - _Antti Karttunen_, Jul 22 2022
%F A341605 A(n, k) = A379500(n, k) / A341606(n, k). - _Antti Karttunen_, Jan 04 2025
%e A341605 The top left corner of the array:
%e A341605 k= 1 2 3 4 5 6 7 8 9 10 11 12
%e A341605 2k = 2 4 6 8 10 12 14 16 18 20 22 24
%e A341605 ----+--------------------------------------------------------------------------
%e A341605 n=1 | 3, 7, 2, 15, 9, 7, 12, 31, 13, 21, 18, 5,
%e A341605 2 | 4, 13, 8, 40, 32, 26, 16, 121, 124, 104, 56, 16,
%e A341605 3 | 6, 31, 48, 156, 72, 248, 84, 781, 342, 372, 108, 1248,
%e A341605 4 | 8, 57, 96, 400, 16, 684, 144, 2801, 152, 114, 160, 4800,
%e A341605 5 | 12, 133, 168, 1464, 216, 1862, 240, 16105, 2196, 2394, 288, 20496,
%e A341605 6 | 14, 183, 252, 2380, 280, 3294, 336, 30941, 4298, 3660, 420, 2520,
%e A341605 7 | 18, 307, 360, 5220, 432, 6140, 540, 88741, 6858, 7368, 576, 104400,
%e A341605 8 | 20, 381, 480, 7240, 600, 9144, 640, 137561, 11060, 11430, 40, 173760,
%e A341605 9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
%e A341605 etc.
%o A341605 (PARI)
%o A341605 up_to = 105;
%o A341605 A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
%o A341605 A017665(n) = numerator(sigma(n)/n);
%o A341605 A341605sq(row,col) = A017665(A246278sq(row,col));
%o A341605 A341605list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341605sq(col,(a-(col-1))))); (v); };
%o A341605 v341605 = A341605list(up_to);
%o A341605 A341605(n) = v341605[n];
%Y A341605 Cf. A017665, A246278, A341626, A341627, A378985, A379500.
%Y A341605 Cf. A008864 (column 1), A378995 (row 1).
%Y A341605 Cf. A341606 (denominators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.
%Y A341605 Cf. also A336389, A336834, A336835, A337473, A337474.
%K A341605 nonn,frac,tabl,look
%O A341605 1,1
%A A341605 _Antti Karttunen_, Feb 16 2021