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.

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, 168, 183, 18, 31, 16, 248, 16, 1464, 252, 307, 20, 13, 121, 84, 684, 216, 2380, 360, 381, 24, 21, 124, 781, 144, 1862, 280, 5220, 480, 553, 30, 18, 104, 342, 2801, 240, 3294, 432, 7240, 720, 871, 32

Offset: 1

Author

Antti Karttunen, Feb 16 2021

Keywords

• nonn
• frac
• tabl
• look

Comments

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:

n = 1 2 3 4 5 6

2n = 2 4 6 8 10 12

----+--------------------------------------------------------------------------

1 | 3/2, 7/4, 2/1, 15/8, 9/5, 7/3,

2 | 4/3, 13/9, 8/5, 40/27, 32/21, 26/15,

3 | 6/5, 31/25, 48/35, 156/125, 72/55, 248/175,

4 | 8/7, 57/49, 96/77, 400/343, 16/13, 684/539,

5 | 12/11, 133/121, 168/143, 1464/1331, 216/187, 1862/1573,

6 | 14/13, 183/169, 252/221, 2380/2197, 280/247, 3294/2873,

7 | 18/17, 307/289, 360/323, 5220/4913, 432/391, 6140/5491,

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.

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:

n = 1 2 3 4 5 6

2n = 2 4 6 8 10 12

----+--------------------------------------------------------------------------

1 | 8/9, 52/63, 4/5, 64/81, 160/189, 26/35,

2 | 9/10, 279/325, 6/7, 1053/1250, 189/220, 372/455,

3 | 20/21, 1425/1519, 10/11, 12500/13377, 110/117, 4275/4774,

4 | 21/22, 343/363, 49/52, 62769/66550, 351/374, 2401/2574,

5 | 77/78, 22143/22477, 33/34, 791945/804102, 6545/6669, 199287/205751,

6 | 117/119, 51883/52887, 130/133, 573417/584647, 13338/13685, 518830/531981,

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.

See also comments and examples in A341606.

By lemma given in A341529, the ratio A341626/A341627 stays in open interval (0.5 .. 1). - Antti Karttunen, Jan 02 2025

Examples

			The top left corner of the array:
  k=   1    2    3      4    5      6    7       8      9     10    11      12
2k =   2    4    6      8   10     12   14      16     18     20    22      24
----+--------------------------------------------------------------------------
n=1 |  3,   7,   2,    15,   9,     7,  12,     31,    13,    21,   18,      5,
  2 |  4,  13,   8,    40,  32,    26,  16,    121,   124,   104,   56,     16,
  3 |  6,  31,  48,   156,  72,   248,  84,    781,   342,   372,  108,   1248,
  4 |  8,  57,  96,   400,  16,   684, 144,   2801,   152,   114,  160,   4800,
  5 | 12, 133, 168,  1464, 216,  1862, 240,  16105,  2196,  2394,  288,  20496,
  6 | 14, 183, 252,  2380, 280,  3294, 336,  30941,  4298,  3660,  420,   2520,
  7 | 18, 307, 360,  5220, 432,  6140, 540,  88741,  6858,  7368,  576, 104400,
  8 | 20, 381, 480,  7240, 600,  9144, 640, 137561, 11060, 11430,   40, 173760,
  9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
etc.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A017665, A246278, A341626, A341627, A378985, A379500.

Cf. A008864 (column 1), A378995 (row 1).

Cf. A341606 (denominators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.

Cf. also A336389, A336834, A336835, A337473, A337474.

Programs

PARI

up_to = 105;
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));
A017665(n) = numerator(sigma(n)/n);
A341605sq(row,col) = A017665(A246278sq(row,col));
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); };
v341605 = A341605list(up_to);
A341605(n) = v341605[n];

Formula

A(n, k) = A017665(A246278(n, k)).

A(n, k) = A355927(n, k) / A355925(n, k). - Antti Karttunen, Jul 22 2022

A(n, k) = A379500(n, k) / A341606(n, k). - Antti Karttunen, Jan 04 2025