A017666 -id:A017666 - cp's OEIS frontend

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.

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

A245775 Numbers k such that A017666(k) = denominator(sigma(k)/k) = 3.

Original entry on oeis.org

3, 12, 84, 234, 270, 1080, 1488, 1638, 6048, 6552, 24384, 35640, 199584, 435708, 2142720, 4713984, 12999168, 18506880, 36197280, 100651008, 208565280, 240589440, 275890944, 299980800, 470564640, 3899750400, 4138364160, 6039429120, 13286744064, 17827568640

Offset: 1

Jaroslav Krizek, Aug 26 2014

nonn

Numbers n such that sigma(n)/n = k + 1/3 with integer k are terms of this sequence (3, 12, 234, 1080, 6048, 6552, 435708, 4713984, ...).
Subsequence of A245774 (numbers n such that n divides 3*sigma(n)).
Union of A160320 (sigma(n)/n = k + 1/3) and A160321 (sigma(n)/n = k + 2/3). - Michel Marcus, Aug 27 2014

			Number 12 is in sequence because A017666(12) = 3.

Jud McCranie, Table of n, a(n) for n = 1..44 (terms <= 10^13).

Cf. A000203, A007691, A160320, A160321, A245774.

[n: n in [1..3000000] | Denominator((SumOfDivisors(n))/n) eq 3]

for(n=1,10^7,if(denominator(sigma(n)/n)==3,print1(n,", "))) \\ Derek Orr, Aug 26 2014

More terms from A160320 and A160321 by Michel Marcus, Aug 27 2014

A341607 Square array A(n,k) = A006530(A017666(A246278(n,k))), read by falling antidiagonals.

Original entry on oeis.org

2, 2, 3, 1, 3, 5, 2, 5, 5, 7, 5, 3, 7, 7, 11, 3, 7, 5, 11, 11, 13, 7, 5, 11, 7, 13, 13, 17, 2, 11, 7, 13, 11, 17, 17, 19, 3, 3, 13, 11, 17, 13, 19, 19, 23, 5, 5, 5, 17, 13, 19, 17, 23, 23, 29, 11, 7, 7, 7, 19, 17, 23, 19, 29, 29, 31, 2, 13, 11, 11, 11, 23, 19, 29, 23, 31, 31, 37, 13, 3, 17, 13, 13, 13, 29, 23, 31, 29, 37, 37, 41

Offset: 1

Antti Karttunen, Feb 16 2021

			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
  2n=   2   4   6   8  10  12  14  16  18  20  22  24   26  28  30  32   34
-----+----------------------------------------------------------------------
   1 |  2,  2,  1,  2,  5,  3,  7,  2,  3,  5, 11,  2,  13,  1,  5,  2,  17,
   2 |  3,  3,  5,  3,  7,  5, 11,  3,  5,  7, 13,  3,  17, 11,  7,  3,  19,
   3 |  5,  5,  7,  5, 11,  7, 13,  5,  7, 11, 17,  7,  19, 13, 11,  5,  23,
   4 |  7,  7, 11,  7, 13, 11, 17,  7, 11, 13, 19, 11,  23, 17, 13,  7,  29,
   5 | 11, 11, 13, 11, 17, 13, 19, 11, 13, 17, 23, 13,  29,*11, 17, 11,  31,
   6 | 13, 13, 17, 13, 19, 17, 23, 13, 17, 19, 29,*13,  31, 23, 19, 13,  37,
   7 | 17, 17, 19, 17, 23, 19, 29, 17, 19, 23, 31, 19,  37, 29, 23, 17,  41,
   8 | 19, 19, 23, 19, 29, 23, 31, 19, 23, 29, 37, 23,  41, 31, 29, 19,  43,
   9 | 23, 23, 29, 23, 31, 29, 37, 23, 29, 31, 41, 29,  43, 37, 31, 23,  47,
  10 | 29, 29, 31, 29, 37, 31, 41, 29, 31, 37, 43, 31,  47, 41, 37, 29,  53,
  11 | 31, 31, 37, 31, 41, 37, 43, 31, 37, 41, 47,*31,  53, 43, 41, 31,  59,
  12 | 37, 37, 41, 37, 43, 41, 47, 37, 41, 43, 53, 41,  59, 47, 43, 37,  61,
  13 | 41, 41, 43, 41, 47, 43, 53, 41, 43, 47, 59, 43,  61, 53, 47, 41,  67,
  14 | 43, 43, 47, 43, 53, 47, 59, 43, 47, 53, 61, 47,  67, 59, 53, 43,  71,
  15 | 47, 47, 53, 47, 59, 53, 61, 47, 53, 59, 67, 53,  71, 47, 59, 47,  73,
  16 | 53, 53, 59, 53, 61, 59, 67, 53, 59, 61, 71, 59,  73, 67, 61, 53,  79,
  17 | 59, 59, 61, 59, 67, 61, 71, 59, 61, 67, 73, 61,  79, 71, 67, 59,  83,
  18 | 61, 61, 67, 61, 71, 67, 73, 61, 67, 71, 79, 67,  83, 73, 71, 61,  89,
  19 | 67, 67, 71, 67, 73, 71, 79, 67, 71, 73, 83, 71,  89, 79, 73, 67,  97,
  20 | 71, 71, 73, 71, 79, 73, 83, 71, 73, 79, 89, 73,  97, 83, 79, 71, 101,
  21 | 73, 73, 79, 73, 83, 79, 89, 73, 79, 83, 97, 79, 101, 89, 83, 73, 103,
etc.
Positions where columns are not strictly monotonic are marked with an asterisk (*). See the example section of A341606 for further elaboration.

Cf. A006530, A017666, A246278, A341606, A341608, A341628.

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));
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A017666(n) = denominator(sigma(n)/n);
A341607sq(row,col) = A006530(A017666(A246278sq(row,col)));
A341607list(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] = A341607sq(col,(a-(col-1))))); (v); };
v341607 = A341607list(up_to);
A341607(n) = v341607[n];

A(n,k) = A006530(A341606(n, k)) = A006530(A017666(A246278(n,k))).

A249670 a(n) = A017665(n)*A017666(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 2, 56, 120, 117, 45, 132, 21, 182, 84, 40, 496, 306, 78, 380, 210, 672, 198, 552, 10, 775, 273, 1080, 2, 870, 60, 992, 2016, 176, 459, 1680, 3276, 1406, 570, 2184, 36, 1722, 112, 1892, 231, 390, 828, 2256, 372, 2793, 4650, 408, 1274, 2862

Offset: 1

Michel Marcus, Nov 03 2014

nonn

If n is a k-multiperfect, then a(n) = k.

Allan C. Wechsler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)).
Cf. A017665/A017666 (abundancy of n).
Cf. A009194 (gcd(n, sigma(n))), A064987 (n*sigma(n)).
Cf. A000396, A005820, A027687, A046060, A046061.

a249670 n = div (n * s) (gcd n s ^ 2)
 where s = sum (filter (\k -> mod n k == 0) [1..n])
-- Allan C. Wechsler, Mar 31 2023

a249670[n_Integer] := Numerator[DivisorSigma[-1, n]]*Denominator[DivisorSigma[-1, n]]; a249670 /@ Range[80] (* Michael De Vlieger, Nov 10 2014 *)

a(n) = my(ab = sigma(n)/n); numerator(ab)*denominator(ab);

a(n) = A064987(n)/A009194(n)^2.
a(A000396(n)) = 2 (perfect).
a(A005820(n)) = 3 (tri-perfect).
For p prime, a(p) = p*(p+1).

A341524 Number of prime factors in A017666(n), counted with multiplicity: a(n) = bigomega(n) - bigomega(gcd(n, sigma(n))).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 0, 1, 1, 1, 5, 1, 1, 2, 4, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 6, 2, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 0

Offset: 1

Antti Karttunen, Feb 17 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A001222, A009194, A017666, A341523.
Cf. A007691 (positions of zeros).
Cf. A341608 (applied onto prime shift array A246278).

Table[PrimeOmega[n] - PrimeOmega[GCD[n, DivisorSigma[1, n]]], {n, 1, 100}] (* Amiram Eldar, Feb 17 2021 *)

A341524(n) = (bigomega(n) - bigomega(gcd(n, sigma(n))));

a(n) = A001222(A017666(n)).

a(n) = A001222(n) - A341523(n).

A374196 a(n) is the minimum value of A017666 that it obtains among divisors of n larger than 1. By convention a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 5, 2, 3, 1, 29, 1, 31, 2, 3, 2, 5, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 7, 2, 3, 2, 53, 1, 5, 1, 3, 2, 59, 1, 61, 2, 3, 2, 5, 1, 67, 2, 3, 2, 71, 1, 73, 2, 3, 2, 7, 1, 79, 2, 3, 2, 83, 1, 5, 2, 3, 2, 89, 1, 7, 2, 3, 2, 5, 1, 97, 2, 3, 2

Offset: 1

Antti Karttunen, Jul 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to sigma(n)

Cf. A000203, A017666, A374198 (indices of 1's in this sequence).

Cf. also A374204.

A374196(n) = { my(m=0,x); fordiv(n,d,if(d>1, x = denominator(sigma(d)/d); if(!m || x

a(1) = 1, and for n > 1, a(n) = Min_{d|n, d>1} A017666(d).

A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1

Offset: 1

Michel Marcus, Sep 22 2015

The sum of terms of the n-th row is n.
T(n, n) = 1 when n is in A014567.
T(n, n) = 0 when n is in A069059.
T(n, 1) increases when n is a multiperfect number A007691.
For a given k, the first index n for which T(n,k) = 1 is A162657(k).

			The first 6 terms of A017666 are 1, 2, 3, 4, 5, 1 where 1 appears twice, 2 to 5 appear once and 6 is absent; giving the 6th row: 2, 1, 1, 1, 1, 0.
Triangle starts
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
2, 1, 1, 1, 1, 0;
2, 1, 1, 1, 1, 0, 1;
2, 1, 1, 1, 1, 0, 1, 1;
2, 1, 1, 1, 1, 0, 1, 1, 1;
2, 1, 1, 1, 2, 0, 1, 1, 1, 0;
...

Michel Marcus, Table of n, a(n) for n = 1..5050

Cf. A007691, A014567, A017666, A069059, A162657.

Table[Length@ Select[Range@ n, Denominator[DivisorSigma[-1, #]] == k &], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Sep 22 2015 *)

tabl(nn) = {vds = vector(nn, n, denominator(sigma(n,-1))); for (n=1, nn, vin = vector(n, k, vds[k]); rown = vector(n, k, #select(x->x==k, vin)); for(k=1, n, print1(rown[k], ", ")); print(););}

A324395 a(n) = A017666(A005940(1+n)), where A005940 is the Doudna sequence and A017666(n) = n/gcd(n,sigma(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 9, 8, 7, 5, 5, 3, 25, 6, 27, 16, 11, 7, 21, 10, 35, 5, 15, 2, 49, 50, 75, 36, 125, 9, 81, 32, 13, 11, 11, 1, 55, 7, 63, 4, 77, 35, 35, 5, 175, 5, 9, 12, 121, 98, 49, 100, 245, 25, 225, 24, 343, 125, 125, 27, 625, 54, 243, 64, 17, 13, 39, 11, 65, 11, 33, 7, 13, 55, 55, 3, 275, 21, 189, 40, 143, 77, 77, 5, 385, 35, 105, 1, 539

Offset: 0

Antti Karttunen, Mar 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16500
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A005940, A017666, A324054, A324055, A324058, A324349, A324394.

A324395(n) = { my(m1=1,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); m1/gcd(m1,m2); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A017666(n) = (n/gcd(n, sigma(n)));
A324395(n) = A017666(A005940(1+n));

a(n) = A017666(A005940(1+n)) = A005940(1+n) / A324394(n).

A330746 Number of values of k, 1 <= k <= n, with A017666(k) = A017666(n), where A017666(n) = n/gcd(n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 6, 4, 4, 1, 2, 4, 2, 1, 1, 1, 1, 1, 4, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A017666.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A014567, A017666.
A left inverse of following sequences: A007691, A159907, A245775.
Cf. also A331175.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A017666(n) = (n/gcd(n, sigma(n)));
v330746 = ordinal_transform(vector(up_to, n, A017666(n)));
A330746(n) = v330746[n];

For all n >= 1, a(A014567(n)) = 1.

For all n >= 1, a(A007691(n)) = a(A159907(n)) = a(A245775(n)) = 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

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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A245775 Numbers k such that A017666(k) = denominator(sigma(k)/k) = 3.

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A341607 Square array A(n,k) = A006530(A017666(A246278(n,k))), read by falling antidiagonals.

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A249670 a(n) = A017665(n)*A017666(n).

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A341524 Number of prime factors in A017666(n), counted with multiplicity: a(n) = bigomega(n) - bigomega(gcd(n, sigma(n))).

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A374196 a(n) is the minimum value of A017666 that it obtains among divisors of n larger than 1. By convention a(1) = 1.

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A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

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A324395 a(n) = A017666(A005940(1+n)), where A005940 is the Doudna sequence and A017666(n) = n/gcd(n,sigma(n)).

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