A246278 -id:A246278 - cp's OEIS frontend

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A250245.
Other similar permutations: A250243, A250248, A250250, A163511, A252756.
Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.
Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

up_to = 16384;
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015  - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018
As a composition of related permutations:
a(n) = A163511(A252756(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].
A001221(a(n)) = A302041(n).
A001222(a(n)) = A253557(n).
A008683(a(n)) = A302050(n).
A000005(a(n)) = A302051(n)
A010052(a(n)) = A302052(n), for n >= 1.
A056239(a(n)) = A302039(n).

A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83, 84, 175, 86, 105

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246278 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A246279 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249817.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A078898(n), namely: A250246, A250248 and A250250.
Other similar or related permutations: A249816.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249822.
Differs from its inverse A249817 for the first time at n=33, where a(33) = 45, while A249817(33) = 39.

lim = 87; a003961[p_?PrimeQ] := a003961[p] = Prime[PrimePi@ p + 1]; a003961[1] = 1; a003961[n_] :=  a003961[n] = Times @@ (a003961[First@ #]^Last@ # &) /@ FactorInteger@ n; a055396[n_] := PrimePi[FactorInteger[n][[1, 1]]]; a078898 = Block[{nn = 90, spfs}, spfs = Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[Count[Take[spfs, i], spfs[[i]]], {i, nn}]]; a246278 = NestList[Map[a003961, #] &, Table[2 k, {k, lim}], lim]; Table[a246278[[a055396@ n, a078898[[n]]]], {n, 2, lim}]
(* Michael De Vlieger, Jan 04 2016, after Harvey P. Dale at A055396 and A078898 *)

a(1) = 1, a(n) = A246278(A055396(n), A078898(n)).
a(1) = 1, a(n) = A246278(A055396(n), A249822(A055396(n), A246277(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249816(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of 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.

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

A249822 Square array of permutations: A(row,col) = A078898(A246278(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 5, 3, 2, 1, 6, 4, 9, 3, 2, 1, 7, 8, 4, 14, 3, 2, 1, 8, 6, 12, 4, 28, 3, 2, 1, 9, 14, 5, 21, 4, 36, 3, 2, 1, 10, 13, 42, 5, 33, 4, 57, 3, 2, 1, 11, 11, 17, 92, 5, 45, 4, 67, 3, 2, 1, 12, 7, 19, 33, 305, 5, 63, 4, 93, 3, 2, 1, 13, 23, 6, 25, 39, 455, 5, 80, 4, 139, 3, 2, 1, 14, 9, 59, 6, 43, 61, 944, 5, 116, 4, 154, 3, 2, 1, 15, 17, 7, 144, 6, 52, 70, 1238, 5, 148, 4, 210, 3, 2, 1

Offset: 1

Antti Karttunen, Nov 06 2014

			The top left corner of the array:
1, 2, 3,  4,  5,   6,   7,    8,    9,   10,  11,   12,  13,   14,   15, ...
1, 2, 3,  5,  4,   8,   6,   14,   13,   11,   7,   23,   9,   17,   18, ...
1, 2, 3,  9,  4,  12,   5,   42,   17,   19,   6,   59,   7,   22,   26, ...
1, 2, 3, 14,  4,  21,   5,   92,   33,   25,   6,  144,   7,   32,   39, ...
1, 2, 3, 28,  4,  33,   5,  305,   39,   43,   6,  360,   7,   48,   50, ...
1, 2, 3, 36,  4,  45,   5,  455,   61,   52,   6,  597,   7,   63,   68, ...
1, 2, 3, 57,  4,  63,   5,  944,   70,   76,   6, 1053,   7,   95,   84, ...
1, 2, 3, 67,  4,  80,   5, 1238,   96,   99,   6, 1502,   7,  106,  121, ...
...

Inverse permutations can be found from table A249821.
Row k+1 is a right-to-left composition of the first k rows of A251722.
Row 1: A000027 (an identity permutation), Row 2: A048673, Row 3: A249824, Row 4: A249826.
Column 4: A250474, Column 6: A250477, Column 8: A250478.
Cf. A002260, A004736, A078898, A246278, A249818.

(define (A249822 n) (A249822bi (A002260 n) (A004736 n)))
(define (A249822bi row col) (A078898 (A246278bi row col))) ;; Code for A246278bi given in A246278.

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).

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249818.

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A246275 Square array A246278 minus 1.

Original entry on oeis.org

1, 3, 2, 5, 8, 4, 7, 14, 24, 6, 9, 26, 34, 48, 10, 11, 20, 124, 76, 120, 12, 13, 44, 54, 342, 142, 168, 16, 15, 32, 174, 90, 1330, 220, 288, 18, 17, 80, 64, 538, 186, 2196, 322, 360, 22, 19, 74, 624, 118, 1572, 246, 4912, 436, 528, 28, 21, 62, 244, 2400, 208, 2872, 390, 6858, 666, 840, 30

Offset: 1

Antti Karttunen, Aug 21 2014

			The top-left corner of the array:
   1,     3,     5,     7,     9,    11,    13,    15,    17,   ...
   2,     8,    14,    26,    20,    44,    32,    80,    74,   ...
   4,    24,    34,   124,    54,   174,    64,   624,   244,   ...
   6,    48,    76,   342,    90,   538,   118,  2400,   846,   ...
  10,   120,   142,  1330,   186,  1572,   208, 14640,  1858,   ...
  12,   168,   220,  2196,   246,  2872,   298, 28560,  3756,   ...
  ...

Inverse permutation: A246276.
Transpose: A246273.
One less than A246278.
Related permutations: A038722, A246675, A246676.
Cf. also A003961.

(define (A246275 n) (- (A246278 (+ 1 n)) 1))

a(n) = A246278(n+1)-1.
As a composition of related permutations:
a(n) = A246273(A038722(n)).

Formula edited slightly because of changed starting offset of A246278. - Antti Karttunen, Jan 27 2015

A341627 Square array A(n,k) = A341527(A246278(n,k)), read by falling antidiagonals; denominators of the columnwise first quotients of A341605/A341606.

Original entry on oeis.org

9, 63, 10, 5, 325, 21, 81, 7, 1519, 22, 189, 1250, 11, 363, 78, 35, 220, 13377, 52, 22477, 119, 33, 455, 117, 66550, 34, 52887, 171, 2511, 260, 4774, 374, 804102, 133, 110827, 115, 325, 6875, 833, 2574, 6669, 584647, 69, 201549, 116, 1323, 3038, 1875181, 627, 205751, 13685, 1790199, 58, 465073, 465

Offset: 1

Antti Karttunen, Feb 16 2021

			The top left corner of the array:
   n =  1       2    3        4      5        6      7             8        9
  2n =  2       4    6        8     10       12     14            16       18
----+--------------------------------------------------------------------------
  1 |   9,     63,   5,      81,   189,      35,    33,         2511,     325,
  2 |  10,    325,   7,    1250,   220,     455,   260,         6875,    3038,
  3 |  21,   1519,  11,   13377,   117,    4774,   833,      1875181,    1089,
  4 |  22,    363,  52,   66550,   374,    2574,   627,     41009441,    6422,
  5 |  78,  22477,  34,  804102,  6669,  205751,  1495,    459974905,  317322,
  6 | 119,  52887, 133,  584647, 13685,  531981, 13804,   2584223261,  775789,
  7 | 171, 110827,  69, 1790199,  9918,  670795, 15903,  11564815861, 1813941,
  8 | 115, 201549,  58, 2202227, 17825, 1016508, 34040,  38495207801, 2325365,
  9 | 116, 465073,  93, 5170468, 68672, 7457205, 90364, 206922836641, 3348124,
etc.

Cf. A246278, A341605, A341606, A341527.

Cf. A341626 (numerators), A341628 (the greatest prime factor of these terms).

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));
A341627sq(row,col) = A341527(A246278sq(row,col));
A341627list(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] = A341627sq(col,(a-(col-1))))); (v); };
v341627 = A341627list(up_to);
A341627(n) = v341627[n];

A(n,k) = A341527(A246278(n,k)), where A341527(n) is the denominator of the ratio (n * sigma(A003961(n))) / (sigma(n) * A003961(n)), i.e., of A341528(n)/A341529(n).

For all n, k, A(n,k) > A341626(n, k).

cp's OEIS Frontend

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

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A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

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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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A249822 Square array of permutations: A(row,col) = A078898(A246278(row,col)), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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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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A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

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