A005187 -id:A005187 - cp's OEIS frontend

A283475 a(n) = A019565(A005187(n)).

1, 2, 6, 5, 30, 7, 21, 42, 210, 11, 33, 66, 165, 330, 154, 231, 2310, 13, 39, 78, 195, 390, 182, 273, 1365, 2730, 286, 429, 1430, 2145, 1001, 2002, 30030, 17, 51, 102, 255, 510, 238, 357, 1785, 3570, 374, 561, 1870, 2805, 1309, 2618, 19635, 39270, 442, 663, 2210, 3315, 1547, 3094, 15470, 23205, 2431, 4862, 12155

Offset: 0

Antti Karttunen, Mar 15 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000040, A000051, A001221, A001222, A004198, A005117, A005187, A019565, A046523, A097248, A108951, A280700, A280705, A283477, A283478.

Cf. A283476 (same sequence sorted into ascending order).

Map[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[#, 2] &, Table[2 n - DigitCount[2 n, 2, 1], {n, 0, 60}]] (* Michael De Vlieger, Mar 16 2017 *)

(define (A283475 n) (A019565 (A005187 n)))

a(n) = A019565(A005187(n)).
a(n) = A097248(A283477(n)) = A097248(A108951(A019565(n))) = A283478(A019565(n)).
Other identities:
If A004198(x,y) = 0, then a(x+y) = A097248(a(x)*a(y)).
For all n >= 1, a(A000051(n)) = A000040(n+2).
For all n >= 0, A001221(a(n)) = A001222(a(n)) = A280700(n).
For all n >= 0, A046523(a(n)) = A280705(n).

A256997 Square array A(row,col) read by antidiagonals: A(1,col) = A055938(col), and for row > 1, A(row,col) = A005187(A(row-1,col)).

Original entry on oeis.org

2, 5, 3, 6, 8, 4, 9, 10, 15, 7, 12, 16, 18, 26, 11, 13, 22, 31, 34, 49, 19, 14, 23, 41, 57, 66, 95, 35, 17, 25, 42, 79, 110, 130, 184, 67, 20, 32, 47, 81, 153, 215, 258, 364, 131, 21, 38, 63, 89, 159, 302, 424, 514, 723, 259, 24, 39, 73, 120, 174, 312, 599, 844, 1026, 1440, 515, 27, 46, 74, 143, 236, 343, 620, 1192, 1683, 2050, 2876, 1027

Offset: 2

Antti Karttunen, Apr 14 2015

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 is transpose of array A256995.
If we assume that a(1) = 1 (but which is not explicitly included here because outside of the array proper), then A256998 gives the inverse permutation.

			The top left corner of the array:
    2,    5,    6,    9,   12,   13,   14,   17,   20,   21,    24,    27
    3,    8,   10,   16,   22,   23,   25,   32,   38,   39,    46,    50
    4,   15,   18,   31,   41,   42,   47,   63,   73,   74,    88,    97
    7,   26,   34,   57,   79,   81,   89,  120,  143,  145,   173,   191
   11,   49,   66,  110,  153,  159,  174,  236,  281,  287,   341,   375
   19,   95,  130,  215,  302,  312,  343,  467,  558,  568,   677,   743
   35,  184,  258,  424,  599,  620,  680,  928, 1111, 1132,  1349,  1479
   67,  364,  514,  844, 1192, 1235, 1356, 1852, 2216, 2259,  2693,  2951
  131,  723, 1026, 1683, 2380, 2464, 2707, 3697, 4428, 4512,  5381,  5895
  259, 1440, 2050, 3360, 4755, 4924, 5408, 7387, 8851, 9020, 10757, 11783
  ...

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of square array

Cf. A005187, A055938 (row 1), A256994 (column 1), A256989 (row index), A256990 (column index).
Inverse: A256998.
Transpose: A256995.
Cf. also A254107, A255557 (variants), A246278 (another thematically similar construction).

(define (A256997 n) (if (<= n 1) n (A256997bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A256997bi row col) (if (= 1 row) (A055938 col) (A005187 (A256997bi (- row 1) col))))

A(1,col) = A055938(col), and for row > 1, A(row,col) = A005187(A(row-1,col)).

A213717 Terms of A005187 not found in A179016.

Original entry on oeis.org

10, 18, 22, 25, 34, 38, 41, 47, 50, 54, 56, 66, 70, 73, 79, 82, 86, 88, 95, 98, 102, 105, 110, 113, 117, 119, 130, 134, 137, 143, 146, 150, 152, 159, 162, 166, 169, 174, 177, 181, 183, 191, 194, 198, 201, 206, 208, 212, 213, 216, 222, 224, 228, 229, 232, 237, 239, 243, 244, 246, 258, 262, 265, 271, 274, 278, 280, 287, 290, 294, 297, 302, 305, 309, 311, 319, 322, 326, 329, 334, 336, 340, 341, 344, 350, 352, 356, 357, 360, 365

Offset: 1

Antti Karttunen, Oct 26 2012

nonn

These are numbers i which do not occur on the unique infinite path of A179016, although there exist such k for which A000120(i+k)=k. For example, with the first term 10, we have cases k=2 and 3, as A000120(10+2)=2 and A000120(10+3)=3. Still, it's not possible to proceed further up from either 12 or 13, as they both are members of A055938.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Setwise difference of A005187 and A179016 and also setwise difference of A213713 and A055938.

a(n) = A005187(A213716(n)).

A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 4, 6, 7, 5, 6, 7, 8, 9, 8, 10, 11, 9, 10, 12, 13, 11, 14, 15, 12, 13, 14, 15, 16, 17, 16, 18, 19, 17, 18, 20, 21, 19, 22, 23, 20, 21, 22, 24, 25, 23, 26, 27, 24, 25, 28, 29, 26, 30, 31, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 33, 34, 36, 37, 35, 38, 39, 36, 37, 38, 40, 41, 39, 42, 43, 40, 41, 44, 45, 42

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

In other words, if n = A005187(k) for some k >= 1, then a(n) = k, otherwise it must be that n = A055938(h) for some h, and then a(n) = h.
Each n occurs exactly twice, first at a(A005187(n)), then at a(A055938(n)). Cf. also A257126.
When iterating a(n), a(a(n)), a(a(a(n))), etc, A256993(n) gives the number of steps to reach one, from any starting value n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A055938, A079559, A213714, A234017.

Cf. also A256991 (variant), A256993, A257126.

With[{nn = 92}, Function[{g, h}, Flatten@ Table[If[MemberQ[g, n], First@ Position[g, n] - 1, First@ Position[h, n]], {n, Min[Length /@ {g, h}]}]] @@ {Table[2 n - DigitCount[2 n, 2, 1], {n, 0, nn}], Complement[Range@ Last@ #, #] &@ Table[IntegerExponent[(2 n)!, 2], {n, 0, nn}]} ] (* Michael De Vlieger, Dec 12 2016, after Harvey P. Dale at A005187 and Jean-François Alcover at A055938 *)

(define (A256992 n) (+ (A213714 n) (A234017 n)))
(define (A256992 n) (if (not (zero? (A079559 n))) (A213714 n) (A234017 n)))

a(n) = A213714(n) + A234017(n).
a(n) = A256991(n) + A079559(n).
If A079559(n) = 1, a(n) = A213714(n), otherwise a(n) = A234017(n).

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Cf. A000120, A000203, A005187, A033879, A326131.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326140, A326144.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A326130(n) = gcd(hammingweight(n), A294898(n));

A326130(n) = gcd(hammingweight(n), sigma(n)-A005187(n));

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(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, 3, 5, 6, 4, 12, 13, 9, 7, 27, 28, 20, 14, 8, 58, 59, 43, 29, 17, 10, 121, 122, 90, 60, 36, 21, 11, 248, 249, 185, 123, 75, 44, 24, 15, 503, 504, 376, 250, 154, 91, 51, 30, 16, 1014, 1015, 759, 505, 313, 186, 106, 61, 33, 18, 2037, 2038, 1526, 1016, 632, 377, 217, 124, 68, 37, 19, 4084, 4085, 3061, 2039, 1271, 760, 440, 251, 139, 76, 40, 22

Offset: 1

Antti Karttunen, Jan 26 2015

This sequence is one instance of Clark Kimberling's generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A055938(n), t(n) = A005187(n) [from term A005187(1) onward] and u(n) = A254112(n).
For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...

			The top left corner of the array:
   1,  2,  5,  12,  27,  58,  121,  248,  503,  1014,  2037,  4084
   3,  6, 13,  28,  59, 122,  249,  504, 1015,  2038,  4085,  8180
   4,  9, 20,  43,  90, 185,  376,  759, 1526,  3061,  6132, 12275
   7, 14, 29,  60, 123, 250,  505, 1016, 2039,  4086,  8181, 16372
   8, 17, 36,  75, 154, 313,  632, 1271, 2550,  5109, 10228, 20467
  10, 21, 44,  91, 186, 377,  760, 1527, 3062,  6133, 12276, 24563
  11, 24, 51, 106, 217, 440,  887, 1782, 3573,  7156, 14323, 28658
  15, 30, 61, 124, 251, 506, 1017, 2040, 4087,  8182, 16373, 32756
  16, 33, 68, 139, 282, 569, 1144, 2295, 4598,  9205, 18420, 36851
  18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
etc.

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254106.
Transpose: A254107.
Column 1: A005187.
Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
Columnd index of n: A254111, Row index: A254112.
Cf. A002260, A004736, A055938, A233275-A233278.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.

(define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))

If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).

A255555 Square array A(row,col) read by downwards antidiagonals: A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 10, 6, 31, 16, 19, 11, 9, 63, 32, 38, 22, 18, 12, 127, 64, 74, 42, 35, 23, 13, 255, 128, 146, 82, 70, 46, 25, 14, 511, 256, 290, 162, 138, 89, 49, 26, 17, 1023, 512, 578, 322, 274, 176, 97, 50, 34, 20, 2047, 1024, 1154, 642, 546, 350, 193, 98, 67, 39, 21, 4095, 2048, 2306, 1282, 1090, 695, 385, 194, 134, 78, 41, 24

Offset: 1

Antti Karttunen, Apr 13 2015

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.

Provided that I understand Kimberling's terminology correctly, this array is the dispersion of sequence b(n) = A005187(n+1), for n>=1: A005187[2..] = [3, 4, 7, 8, 10, 11, ...]. The left column is the complement of that sequence, which is {1} followed by A055938. - Antti Karttunen, Apr 17 2015

			The top left corner of the array:
   1,  3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095
   2,  4,  8,  16,  32,  64,  128,  256,  512, 1024,  2048,  4096
   5, 10, 19,  38,  74, 146,  290,  578, 1154, 2306,  4610,  9218
   6, 11, 22,  42,  82, 162,  322,  642, 1282, 2562,  5122, 10242
   9, 18, 35,  70, 138, 274,  546, 1090, 2178, 4354,  8706, 17410
  12, 23, 46,  89, 176, 350,  695, 1387, 2770, 5535, 11067, 22128
  13, 25, 49,  97, 193, 385,  769, 1537, 3073, 6145, 12289, 24577
  14, 26, 50,  98, 194, 386,  770, 1538, 3074, 6146, 12290, 24578
  17, 34, 67, 134, 266, 530, 1058, 2114, 4226, 8450, 16898, 33794
  20, 39, 78, 153, 304, 606, 1207, 2411, 4818, 9631, 19259, 38512
  ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A255556.
Transpose: A255557.
Row 1: A000225.
Cf. A005187, A055938.
Cf. A255559 (column index), A255560 (row index).
Cf. also A254105, A256995 (variants), A233275-A233278.

(define (A255555 n) (A255555bi (A002260 n) (A004736 n)))
(define (A255555bi row col) (if (= 1 col) (if (= 1 row) 1 (A055938 (- row 1))) (A005187 (+ 1 (A255555bi row (- col 1))))))

A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

A255557 Square array A(row,col): A(1,1) = 1, A(1,col) = A055938(col-1), and for row > 1: A(row,col) = A005187(1+A(row-1,col)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 10, 8, 15, 9, 11, 19, 16, 31, 12, 18, 22, 38, 32, 63, 13, 23, 35, 42, 74, 64, 127, 14, 25, 46, 70, 82, 146, 128, 255, 17, 26, 49, 89, 138, 162, 290, 256, 511, 20, 34, 50, 97, 176, 274, 322, 578, 512, 1023, 21, 39, 67, 98, 193, 350, 546, 642, 1154, 1024, 2047, 24, 41, 78, 134, 194, 385, 695, 1090, 1282, 2306, 2048, 4095

Offset: 1

Antti Karttunen, Apr 13 2015

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 is transpose of array A255555, see comments and links given there.

			The top left corner of the array:
     1,    2,    5,    6,    9,   12,   13,   14,   17,   20,    21,    24
     3,    4,   10,   11,   18,   23,   25,   26,   34,   39,    41,    47
     7,    8,   19,   22,   35,   46,   49,   50,   67,   78,    81,    94
    15,   16,   38,   42,   70,   89,   97,   98,  134,  153,   161,   184
    31,   32,   74,   82,  138,  176,  193,  194,  266,  304,   321,   365
    63,   64,  146,  162,  274,  350,  385,  386,  530,  606,   641,   726
   127,  128,  290,  322,  546,  695,  769,  770, 1058, 1207,  1281,  1447
   255,  256,  578,  642, 1090, 1387, 1537, 1538, 2114, 2411,  2561,  2891
   511,  512, 1154, 1282, 2178, 2770, 3073, 3074, 4226, 4818,  5121,  5778
  1023, 1024, 2306, 2562, 4354, 5535, 6145, 6146, 8450, 9631, 10241, 11551
  ...

Inverse permutation: A255558.
Transpose: A255555.
Column 1: A000225.
Cf. A005187, A055938.
Cf. A255559 (row index), A255560 (column index).
Cf. also A254107, A256997 (variants).

(define (A255557 n) (A255555bi (A004736 n) (A002260 n)))
(define (A255555bi row col) (if (= 1 col) (if (= 1 row) 1 (A055938 (- row 1))) (A005187 (+ 1 (A255555bi row (- col 1))))))

A(row,col): A(1,1) = 1, and for the rest of topmost row: A(1,col) = A055938(col-1), and for any row > 1: A(row,col) = A005187(1+A(row-1,col)).

A294899 a(n) = A000203(n) XOR A005187(n), where XOR is bitwise-XOR, A003987.

Original entry on oeis.org

0, 0, 0, 0, 14, 6, 3, 0, 29, 0, 31, 10, 25, 1, 2, 0, 50, 5, 55, 12, 7, 13, 50, 18, 48, 27, 26, 13, 40, 112, 25, 0, 112, 116, 115, 29, 97, 117, 114, 20, 101, 49, 126, 1, 24, 16, 105, 34, 102, 60, 42, 7, 80, 16, 33, 21, 62, 42, 77, 220, 75, 23, 16, 0, 212, 18, 199, 248, 231, 25, 194, 77, 197, 227, 238, 25, 246, 48, 201, 36

Offset: 1

Antti Karttunen, Nov 25 2017

nonn

Cf. A000203, A003987, A005187, A294898, A295296 (positions of zeros), A295297 (parity of a(n)).

Cf. also A169813, A279357, A283997.

Array[BitXor[DivisorSigma[1, #], IntegerExponent[(2 #)!, 2]] &, 80] (* Michael De Vlieger, Nov 26 2017 *)

(define (A294899 n) (A003987bi (A000203 n) (A005187 n))) ;; Where A003987bi implements the bitwise-XOR, A003987.

a(n) = A000203(n) XOR A005187(n).

A256995 Square array A(row,col) read by antidiagonals: A(row,1) = A055938(row), and for col > 1, A(row,col) = A005187(A(row,col-1)).

Original entry on oeis.org

2, 3, 5, 4, 8, 6, 7, 15, 10, 9, 11, 26, 18, 16, 12, 19, 49, 34, 31, 22, 13, 35, 95, 66, 57, 41, 23, 14, 67, 184, 130, 110, 79, 42, 25, 17, 131, 364, 258, 215, 153, 81, 47, 32, 20, 259, 723, 514, 424, 302, 159, 89, 63, 38, 21, 515, 1440, 1026, 844, 599, 312, 174, 120, 73, 39, 24, 1027, 2876, 2050, 1683, 1192, 620, 343, 236, 143, 74, 46, 27

Offset: 2

Antti Karttunen, Apr 14 2015

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 is transpose of array A256997.
If we assume that a(1) = 1 (but which is not explicitly included here because outside of the array proper), then A256996 gives the inverse permutation.

			The top left corner of the array:
   2,  3,  4,   7,  11,  19,   35,   67,  131,  259,   515,  1027
   5,  8, 15,  26,  49,  95,  184,  364,  723, 1440,  2876,  5745
   6, 10, 18,  34,  66, 130,  258,  514, 1026, 2050,  4098,  8194
   9, 16, 31,  57, 110, 215,  424,  844, 1683, 3360,  6716, 13425
  12, 22, 41,  79, 153, 302,  599, 1192, 2380, 4755,  9504, 19004
  13, 23, 42,  81, 159, 312,  620, 1235, 2464, 4924,  9841, 19675
  14, 25, 47,  89, 174, 343,  680, 1356, 2707, 5408, 10812, 21617
  17, 32, 63, 120, 236, 467,  928, 1852, 3697, 7387, 14765, 29521
  20, 38, 73, 143, 281, 558, 1111, 2216, 4428, 8851, 17696, 35388
  21, 39, 74, 145, 287, 568, 1132, 2259, 4512, 9020, 18033, 36059
  ...

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of square array

Inverse permutation: A256996.
Transpose: A256997.
Cf. A005187, A055938 (column 1), A256994 (row 1), A256989 (column index), A256990 (row index).
Cf. also A254105, A255555 (variants), A114537, A246279 (other thematically similar constructions).

(define (A256995 n) (if (<= n 1) n (A256995bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A256995bi row col) (if (= 1 col) (A055938 row) (A005187 (A256995bi row (- col 1)))))

A(row,1) = A055938(row), and for col > 1, A(row,col) = A005187(A(row,col-1)).

cp's OEIS Frontend

A283475 a(n) = A019565(A005187(n)).

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A256997 Square array A(row,col) read by antidiagonals: A(1,col) = A055938(col), and for row > 1, A(row,col) = A005187(A(row-1,col)).

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A213717 Terms of A005187 not found in A179016.

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A256992 Position of n in either of the complementary sequences, A005187 or A055938: a(n) = A213714(n) + A234017(n).

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Mathematica

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Formula

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

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PARI

PARI

Formula

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(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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Formula

A255555 Square array A(row,col) read by downwards antidiagonals: A(1,1) = 1, A(row,1) = A055938(row-1), and for col > 1, A(row,col) = A005187(1+A(row,col-1)).

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Formula

A255557 Square array A(row,col): A(1,1) = 1, A(1,col) = A055938(col-1), and for row > 1: A(row,col) = A005187(1+A(row-1,col)).

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