A355926 -id:A355926 - 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

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)).

A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 5, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 25, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 33, 3, 34, 17, 35, 36, 37, 3, 38, 39, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 56, 57, 3, 58, 59, 60, 3, 61, 62, 63, 64, 65, 3, 66, 67, 68, 69, 70, 71, 72, 3, 73, 74

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355442(n)].
For all i, j: a(i) = a(j) => A355836(i) = A355836(j).
Terms that occur in positions given by A355822 may occur only a finite number of times in this sequence. Most of these seem to be in the singular equivalence classes, i.e., have unique values, apart from exceptions like pairs {6, 15}, {273, 1729}, (see the examples and the array A355926). In a coarser variant A355836 multiple such finite equivalence classes may coalesce together into several infinite equivalence classes.

			a(6) = a(15) [= 5 as allotted by the rgs-transform] because 15 = A003961(6) [i.e., 15 is in the same column in prime shift array A246278 as 6 is], and because A355442(6) = A355442(15) = 5.
a(138) = a(435) [= 103 as allotted by the rgs-transform] because 435 = A003961(138), and A355442(138) = A355442(435) = 5.
a(273) = a(1729) [= 205 as allotted by the rgs-transform] because 1729 = A003961(A003961(273)) [i.e., 273 and 1729 are in the same column of A246278], and A355442(273) = A355442(1729) = 11.

Cf. A003961, A276086, A348717, A355442, A355821, A355822, A355926.

Cf. also A246278, A305801, A355833, A355834.

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; };
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));
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));
Aux355835(n) = [A348717(n), A355442(n)];
v355835 = rgs_transform(vector(up_to,n,Aux355835(n)));
A355835(n) = v355835[n];

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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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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A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

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