A355925 -id:A355925 - cp's OEIS frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 11, 12, 21, 125, 77, 121, 13, 14, 45, 55, 343, 143, 169, 17, 16, 33, 175, 91, 1331, 221, 289, 19, 18, 81, 65, 539, 187, 2197, 323, 361, 23, 20, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 22, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31

Offset: 2

Antti Karttunen, Aug 21 2014

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
(End)

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)

(define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

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.

Original entry on oeis.org

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

Antti Karttunen, Feb 16 2021

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

			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.

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.

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];

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

A341606 Square array A(n,k) = A017666(A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.

Original entry on oeis.org

2, 4, 3, 1, 9, 5, 8, 5, 25, 7, 5, 27, 35, 49, 11, 3, 21, 125, 77, 121, 13, 7, 15, 55, 343, 143, 169, 17, 16, 11, 175, 13, 1331, 221, 289, 19, 6, 81, 65, 539, 187, 2197, 323, 361, 23, 10, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 11, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31

Offset: 1

Antti Karttunen, Feb 16 2021

See also comments and examples in A341605.

			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
    |
----+--------------------------------------------------------------------------
  1 |  2,   4,   1,     8,   5,     3,   7,     16,     6,    10,  11,      2,
  2 |  3,   9,   5,    27,  21,    15,  11,     81,    75,    63,  39,      9,
  3 |  5,  25,  35,   125,  55,   175,  65,    625,   245,   275,  85,    875,
  4 |  7,  49,  77,   343,  13,   539, 119,   2401,   121,    91, 133,   3773,
  5 | 11, 121, 143,  1331, 187,  1573, 209,  14641,  1859,  2057, 253,  17303,
  6 | 13, 169, 221,  2197, 247,  2873, 299,  28561,  3757,  3211, 377,   2197,
  7 | 17, 289, 323,  4913, 391,  5491, 493,  83521,  6137,  6647, 527,  93347,
  8 | 19, 361, 437,  6859, 551,  8303, 589, 130321, 10051, 10469,  37, 157757,
  9 | 23, 529, 667, 12167, 713, 15341, 851, 279841, 19343, 16399, 943, 352843,
etc.
Arrays A341607 and A341608 give the largest prime factor (A006530) and the number of prime factors with multiplicity (A001222) of these terms. There are nonmonotonicities in both, for example, in columns 11, 12 and 14. This is illustrated below:
For column 11, with successive prime shifts of 22, we obtain:
     n sigma(n)             sigma(n)/n in lowest terms,
                            A017665(n)/A017666(n)
---------------------------------------------------------------------------
    22   36 = (2^2 * 3^2),        18/11  = (2 * 3^2)/11
    39   56 = (2^3 * 7),          56/39  = (2^3 * 7)/(3 * 13)
    85  108 = (2^2 * 3^3),       108/85  = (2^2 * 3^3)/(5 * 17)
   133  160 = (2^5 * 5),         160/133 = (2^5 * 5)/(7 * 19)
   253  288 = (2^5 * 3^2),       288/253 = (2^5 * 3^2)/(11 * 23)
   377  420 = (2^2 * 3 * 5 * 7), 420/377 = (2^2 * 3 * 5 * 7)/(13 * 29)
   527  576 = (2^6 * 3^2),       576/527 = (2^6 * 3^2)/(17 * 31)
   703  760 = (2^3 * 5 * 19),     40/37  = (2^3 * 5)/37 <-- A001222 drops!
   943 1008 = (2^4 * 3^2 * 7),  1008/943 = (2^4 * 3^2 * 7)/(23 * 41)
-
On the second last row, the denominator of 760/703 (= 40/37) has only one prime factor (instead of two), namely 37, because sigma(703) has 19 as its divisor, which otherwise would be present in the denominator.
-
For column 12, with successive prime shifts of 24, we obtain:
      n sigma(n)                        sigma(n)/n
---------------------------------------------------------------------------
     24     60 = (2^2 * 3 * 5),            5/2     = (5)/(2)
    135    240 = (2^4 * 3 * 5),           16/9     = (2^4)/(3^2)
    875   1248 = (2^5 * 3 * 13),        1248/875   = (2^5 * 3 * 13)/(5^3 * 7)
   3773   4800 = (2^6 * 3 * 5^2),       4800/3773  = (2^6 * 3 * 5^2)/(7^3 * 11)
  17303  20496 = (2^4 *3 *7 *61),      20496/17303 = (2^4 *3 *7 *61)/(11^3 * 13)
  37349  42840 = (2^3 *3^2 *5 *7 *17),  2520/2197  = (2^3 * 3^2 *5 *7)/(13^3) !!
  93347 104400 = (2^4 *3^2 *5^2 *29), 104400/93347 = (2^4 *3^2 *5^2 *29)/(17^3 *19)
-
On the second last row, the denominator of 42840/37349 (= 2520/2197) has no prime factor 17 (which would be otherwise present), because sigma(37349) has it as its divisor.
-
For column 14, with successive prime shifts of 28, we obtain:
     n sigma(n)               sigma(n)/n
---------------------------------------------------------------------------
    28   56 = (2^3 * 7),             2/1,
    99  156 = (2^2 * 3 * 13),       52/33   = (2^2 * 13)/(3 * 11)
   325  434 = (2 * 7 * 31),        434/325  = (2 * 7 * 31)/(5^2 * 13)
   833 1026 = (2 * 3^3 * 19),     1026/833  = (2 * 3^3 * 19)/(7^2 * 17)
  2299 2660 = (2^2 * 5 * 7 * 19),  140/121  = (2^2 * 5 * 7)/(11^2) <-- !!
  3887 4392 = (2^3 * 3^2 * 61),   4392/3887 = (2^3 * 3^2 * 61)/(13^2 * 23)
On the second last row, the denominator of 2660/2299 (= 140/121) has no prime factor 19 (which would be otherwise present), because sigma(2299) has it as its divisor.
Note that if A006530 does not grow, then certainly A001222 drops.

Cf. A017666, A246278.
Cf. A341605 (numerators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.
Cf. A341607 (the largest prime factor in this array), A341608 (the number of prime factors, with multiplicity).
Cf. also A007691, A341523, A341524.

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));
A017666(n) = denominator(sigma(n)/n);
A341606sq(row,col) = A017666(A246278sq(row,col));
A341606list(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] = A341606sq(col,(a-(col-1))))); (v); };
v341606 = A341606list(up_to);
A341606(n) = v341606[n];

A(n, k) = A017666(A246278(n, k)).

A(n, k) = A246278(n, k) / A355925(n, k). - Antti Karttunen, Jul 22 2022

A355927 Square array A(n, k) = sigma(A246278(n, k)), read by falling antidiagonals.

Original entry on oeis.org

3, 7, 4, 12, 13, 6, 15, 24, 31, 8, 18, 40, 48, 57, 12, 28, 32, 156, 96, 133, 14, 24, 78, 72, 400, 168, 183, 18, 31, 48, 248, 112, 1464, 252, 307, 20, 39, 121, 84, 684, 216, 2380, 360, 381, 24, 42, 124, 781, 144, 1862, 280, 5220, 480, 553, 30, 36, 104, 342, 2801, 240, 3294, 432, 7240, 720, 871, 32, 60, 56, 372, 1064, 16105, 336, 6140, 600, 12720, 960, 993, 38

Offset: 1

Antti Karttunen, Jul 22 2022

Each column is strictly monotonic.

			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
----+--------------------------------------------------------------------------
  1 |  3,   7,  12,    15,  18,    28,  24,     31,    39,    42,   36,     60,
  2 |  4,  13,  24,    40,  32,    78,  48,    121,   124,   104,   56,    240,
  3 |  6,  31,  48,   156,  72,   248,  84,    781,   342,   372,  108,   1248,
  4 |  8,  57,  96,   400, 112,   684, 144,   2801,  1064,   798,  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,  42840,
  7 | 18, 307, 360,  5220, 432,  6140, 540,  88741,  6858,  7368,  576, 104400,
  8 | 20, 381, 480,  7240, 600,  9144, 640, 137561, 11060, 11430,  760, 173760,
  9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
Note: See A355941 for the corresponding numbers in A246278 at which points the value in this array divides the term immediately below.

Cf. A000203, A246278.
Cf. A008864 (column 1), A062731 (row 1).
Cf. also A341605, A355925, A355941.

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));
A355927sq(row,col) = sigma(A246278sq(row,col));
A355927list(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] = A355927sq(col,(a-(col-1))))); (v); };
v355927 = A355927list(up_to);
A355927(n) = v355927[n];

A(n, k) = A000203(A246278(n, k)).

A(n, k) = A341605(n, k) * A355925(n, k).

A355924 Square array A(n,k) = A342671(A246278(n,k)), read by falling antidiagonals, where A342671(x) = gcd(sigma(x), A003961(x)).

Original entry on oeis.org

3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 17, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 37, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 21 2022

			The top left corner of the array:
   n=  1  2  3   4  5  6  7   8  9  10 11  12  13  14 15 16 17 18 19  20 21
  2n=  2  4  6   8 10 12 14  16 18  20 22  24  26  28 30 32 34 36 38  40 42
-----+-----------------------------------------------------------------------
   1 | 3, 1, 3,  3, 3, 1, 3,  1, 3, 21, 3, 15,  3,  1, 3, 9, 3, 1, 3,  9, 3,
   2 | 1, 1, 1,  5, 1, 1, 1,  1, 1,  1, 1,  5,  1, 13, 1, 1, 5, 1, 1,  5, 1,
   3 | 1, 1, 1,  1, 1, 1, 7,  1, 1,  1, 1,  1,  1,  7, 1, 7, 1, 1, 1, 13, 7,
   4 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1, 19, 1, 1, 1, 1, 1,  1, 1,
   5 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 19, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   6 | 1, 1, 1, 17, 1, 1, 1,  1, 1,  1, 1, 17,  1,  1, 1, 1, 1, 1, 1, 17, 1,
   7 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1, 19,  1, 1, 1, 1, 1, 1, 29, 1,
   8 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   9 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  10 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  11 | 1, 1, 1, 37, 1, 1, 1,  1, 1,  1, 1, 37,  1,  1, 1, 1, 1, 1, 1, 37, 1,
  12 | 1, 1, 1,  1, 1, 1, 1, 41, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  13 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  14 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  15 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 61, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  16 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  17 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  18 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  19 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  20 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  21 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,

Cf. A000203, A003961, A246278, A342671, A355833.

Cf. also A355925, A355926, A355927 for similarly constructed arrays.

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));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A355924sq(row,col) = A342671(A246278sq(row,col));
A355924list(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] = A355924sq(col,(a-(col-1))))); (v); };
v355924 = A355924list(up_to);
A355924(n) = v355924[n];

A(n,k) = A342671(A246278(n,k)).

A(n, k) = gcd(A246278(1+n,k), A355927(n, k)).

A355926 Square array A(n,k) = A355442(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

3, 9, 1, 5, 5, 1, 3, 5, 1, 1, 3, 125, 7, 1, 1, 5, 5, 343, 1, 1, 1, 3, 175, 7, 11, 1, 1, 1, 9, 1, 49, 1, 1, 1, 1, 1, 25, 125, 7, 121, 1, 1, 1, 1, 1, 3, 245, 2401, 1, 1, 1, 1, 1, 1, 1, 3, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 49, 11, 28561, 1, 1, 1, 1, 1, 1, 1, 3, 175, 7, 121, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 22 2022

			The top left corner of the array:
   n= 1  2  3    4  5    6  7      8    9   10 11     12 13   14  15       16
  2n= 2  4  6    8 10   12 14     16   18   20 22     24 26   28  30       32
----+--------------------------------------------------------------------------
  1 | 3, 9, 5,   3, 3,   5, 3,     9,  25,   3, 3,     5, 3,   9,  7,       3,
  2 | 1, 5, 5, 125, 5, 175, 1,   125, 245,   1, 5,   175, 5,   5, 35,       1,
  3 | 1, 1, 7, 343, 7,  49, 7,  2401,  77,  49, 7,    77, 7,  49, 77,   16807,
  4 | 1, 1, 1,  11, 1, 121, 1,     1,  11, 121, 1, 17303, 1, 121, 11,    1331,
  5 | 1, 1, 1,   1, 1,   1, 1, 28561,   1,   1, 1,  2197, 1,   1, 13,   28561,
  6 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,    17, 1,   1,  1, 1419857,
  7 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,     361,
  8 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,       1,
  9 | 1, 1, 1,   1, 1,   1, 1,     1,   1,   1, 1,     1, 1,   1,  1,       1,

Cf. A003961, A246278, A276086, A355442, A355835.

Cf. also A355924, A355925 for similarly constructed arrays.

up_to = 105;
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
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));
A355926sq(row,col) = A355442(A246278sq(row,col));
A355926list(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] = A355926sq(col,(a-(col-1))))); (v); };
v355926 = A355926list(up_to);
A355926(n) = v355926[n];

A378994 a(n) = gcd(2n, sigma(2n)).

Original entry on oeis.org

1, 1, 6, 1, 2, 4, 2, 1, 3, 2, 2, 12, 2, 28, 6, 1, 2, 1, 2, 10, 6, 4, 2, 4, 1, 2, 6, 8, 2, 12, 2, 1, 6, 2, 2, 3, 2, 4, 6, 2, 2, 28, 2, 4, 18, 4, 2, 12, 1, 1, 6, 2, 2, 4, 2, 8, 6, 2, 2, 120, 2, 4, 6, 1, 2, 12, 2, 2, 6, 28, 2, 1, 2, 2, 6, 4, 2, 4, 2, 2, 3, 2, 2, 24, 2, 4, 6, 4, 2, 6, 14, 8, 6, 4, 10, 4, 2, 7, 18, 5

Offset: 1

Antti Karttunen, Dec 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences related to sigma(n).

Even bisection of A009194.
Topmost row of array A355925.
Cf. A000203, A062731, A378995, A378996.
Cf. also A336850.

Table[GCD[2n,DivisorSigma[1,2n]],{n,100}] (* James C. McMahon, Dec 14 2024 *)

PARI
```
A378994(n) = gcd(2*n, sigma(2*n));
```

a(n) = A009194(2*n) = gcd(2*n, A062731(n)).

a(n) = 2*n / A378996(n).

A379500 Square array A(n, k) = A249670(A246278(n, k)), read by falling antidiagonals; A249670(n) = A017665(n)*A017666(n), applied to the prime shift array.

Original entry on oeis.org

6, 28, 12, 2, 117, 30, 120, 40, 775, 56, 45, 1080, 1680, 2793, 132, 21, 672, 19500, 7392, 16093, 182, 84, 390, 3960, 137200, 24024, 30927, 306, 496, 176, 43400, 208, 1948584, 55692, 88723, 380, 78, 9801, 5460, 368676, 40392, 5228860, 116280, 137541, 552, 210, 9300, 488125, 17136, 2928926, 69160, 25645860, 209760, 292537, 870

Offset: 1

Antti Karttunen, Jan 02 2025

			The top left corner of the array:
k=|   1      2      3        4      5        6      7          8        9       10
2k|   2      4      6        8     10       12     14         16       18       20
--+---------------------------------------------------------------------------------
1 |   6,    28,     2,     120,    45,      21,    84,       496,      78,     210,
2 |  12,   117,    40,    1080,   672,     390,   176,      9801,    9300,    6552,
3 |  30,   775,  1680,   19500,  3960,   43400,  5460,    488125,   83790,  102300,
4 |  56,  2793,  7392,  137200,   208,  368676, 17136,   6725201,   18392,   10374,
5 | 132, 16093, 24024, 1948584, 40392, 2928926, 50160, 235793305, 4082364, 4924458,

Elementwise product of arrays A341605 and A341606.
Cf. A246278, A249670, A355925, A379499.
Cf. A036690 (leftmost column), A361468 (even bisection gives row 2).

up_to = 55;
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
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));
A379500sq(row,col) = A249670(A246278sq(row,col));
A379500list(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] = A379500sq(col,(a-(col-1))))); (v); };
v379500 = A379500list(up_to);
A379500(n) = v379500[n];

A(n, k) = A341605(n, k) * A341606(n, k).

A(n, k) = A379499(n, k) / (A355925(n, k)^2).

cp's OEIS Frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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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.

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A341606 Square array A(n,k) = A017666(A246278(n,k)), read by falling antidiagonals; denominator of abundancy index as applied onto prime shift array A246278.

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A355927 Square array A(n, k) = sigma(A246278(n, k)), read by falling antidiagonals.

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A355924 Square array A(n,k) = A342671(A246278(n,k)), read by falling antidiagonals, where A342671(x) = gcd(sigma(x), A003961(x)).

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A355926 Square array A(n,k) = A355442(A246278(n,k)), read by falling antidiagonals.

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A378994 a(n) = gcd(2*n, sigma(2*n)).

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A379500 Square array A(n, k) = A249670(A246278(n, k)), read by falling antidiagonals; A249670(n) = A017665(n)*A017666(n), applied to the prime shift array.

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A378994 a(n) = gcd(2n, sigma(2n)).