A341608 -id:A341608 - cp's OEIS frontend

A007691 Multiply-perfect numbers: n divides sigma(n).

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

T. D. Noe, Table of n, a(n) for n=1..1600 (using Flammenkamp's data)
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Anonymous, Multiply Perfect Numbers [broken link]
Eric Bach, Gary Miller, and Jeffrey Shallit, Sums of divisors perfect numbers and factoring, SIAM J. Comput. 15:4 (1986), pp. 1143-1154.
Robert D. Carmichael, A table of multiply perfect numbers, Bull. Amer. Math. Soc. 13 (1907), 383-386.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Florian Luca and John Ferdinands, Problem 11090: Sometimes n divides sigma_k(n), Amer. Math. Monthly 113:4 (2006), pp. 372-373.
Walter Nissen, Abundancy : Some Resources
Kaitlin Rafferty and Judy Holdener, On the form of perfect and multiperfect numbers, Pi Mu Epsilon Journal, Vol. 13, No. 5 (Fall 2011), pp. 291-298.
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See p. 11.
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Hyperperfect Number.
Index entries for sequences where any odd perfect numbers must occur

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012

Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)

for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))

from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

More terms from Jud McCranie and then from David W. Wilson.

Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

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)).

Original entry on oeis.org

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

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

A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 15, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 3, 1, 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:
   k=  1  2  3  4  5  6  7  8  9 10  11  12 13  14 15 16 17 18  19  20 21
  2k=  2  4  6  8 10 12 14 16 18 20  22  24 26  28 30 32 34 36  38  40 42
-----+-----------------------------------------------------------------------
   1 | 1, 1, 6, 1, 2, 4, 2, 1, 3, 2,  2, 12, 2, 28, 6, 1, 2, 1,  2, 10, 6,
   2 | 1, 1, 3, 1, 1, 3, 3, 1, 1, 1,  1, 15, 3,  3, 3, 1, 1, 1,  3,  1, 3,
   3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 5,  1, 1, 1, 1, 1,  5,  1, 7,
   4 | 1, 1, 1, 1, 7, 1, 1, 1, 7, 7,  1,  1, 1,  1, 7, 1, 1, 7,  1,  7, 1,
   5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1, 19, 1, 1, 1, 1,  1,  1, 1,
   6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 17, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1, 1,  1, 1, 1, 1, 1,  1,  1, 1,
   8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19,  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, 1, 1, 1, 1, 1, 1, 1,  1, 37, 1,  1, 1, 1, 1, 1, 31,  1, 1,
  12 | 1, 1, 1, 1, 1, 1, 1, 1, 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, 1,  1,  1, 1, 61, 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, A009194, A246278.

Cf. also A341605, A341606, A341607, A341608, A341626, A341627, A355924, A355927 for related arrays of similar construction.

up_to = 105;
A009194(n) = gcd(n, sigma(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));
A355925sq(row,col) = A009194(A246278sq(row,col));
A355925list(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] = A355925sq(col,(a-(col-1))))); (v); };
v355925 = A355925list(up_to);
A355925(n) = v355925[n];

A(n, k) = A009194(A246278(n, k)).
A(n, k) = gcd(A246278(n,k), A355927(n, k)).
A(n, k) = A355927(n, k) / A341605(n, k).
A(n, k) = A246278(n, k) / A341606(n, k).

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))).

cp's OEIS Frontend

A007691 Multiply-perfect numbers: n divides sigma(n).

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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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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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A355925 Square array A(n, k) = A009194(A246278(n, k)), read by falling antidiagonals.

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

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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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