A355924 -id:A355924 - 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

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

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));

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

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).

A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 4, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 4, 17, 18, 19, 3, 20, 3, 21, 22, 23, 10, 24, 3, 25, 26, 27, 3, 28, 3, 29, 8, 30, 3, 31, 4, 32, 33, 34, 3, 35, 14, 36, 37, 38, 3, 39, 3, 40, 41, 42, 43, 44, 3, 45, 46, 47, 3, 48, 3, 49, 50, 51, 10, 52, 3, 53, 11, 54, 3, 55, 26, 56, 57, 58, 3, 59, 14, 60, 61, 62, 33, 63, 3, 64, 65, 66

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].

Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.

Cf. also A305801, A305897, A355834, A355835.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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));
A348717(n) = if(1==n, 1, 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));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];

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];

A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

0, 2, 1, 3, 12, 1, 12, 11, 18, 3, 3, 85, 29, 64, 1, 17, 23, 187, 47, 36, 3, 9, 97, 19, 931, 53, 106, 1, 50, 17, 291, 75, 733, 71, 54, 3, 36, 504, 35, 889, 31, 2533, 77, 148, 5, 21, 121, 1620, 65, 1011, 111, 1639, 187, 288, 1, 3, 171, 505, 11840, 59, 2197, 119, 4927, 179, 90, 5

Offset: 1

Antti Karttunen, May 21 2024

			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  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,

Cf. A000203, A003961, A246278, A286385, A355927.
Cf. A046933 (column 1).
Cf. also A355924, A372562.

up_to = 66;
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); };
A286385(n) = (A003961(n)-sigma(n));
A372563sq(row,col) = A286385(A246278sq(row,col));
A372563list(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] = A372563sq(col,(a-(col-1))))); (v); };
v372563 = A372563list(up_to);
A372563(n) = v372563[n];

A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).

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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A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

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

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A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

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

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A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

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