A276945 -id:A276945 - cp's OEIS frontend

A286625 Square array A(n,k) = A276945(n,k)/A002110(k-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

1, 1, 3, 1, 4, 4, 1, 6, 6, 5, 1, 8, 10, 7, 7, 1, 12, 14, 11, 16, 9, 1, 14, 22, 15, 36, 19, 10, 1, 18, 26, 23, 78, 41, 21, 11, 1, 20, 34, 27, 144, 85, 45, 22, 13, 1, 24, 38, 35, 222, 155, 91, 46, 31, 15, 1, 30, 46, 39, 324, 235, 165, 92, 71, 34, 16, 1, 32, 58, 47, 438, 341, 247, 166, 155, 76, 36, 17, 1, 38, 62, 59, 668, 457, 357, 248, 287, 162, 80, 37, 18

Offset: 1

Antti Karttunen, Jun 28 2017

			The top left 12 X 12 corner of the array:
   1,  1,  1,   1,   1,   1,   1,   1,    1,    1,    1,    1
   3,  4,  6,   8,  12,  14,  18,  20,   24,   30,   32,   38
   4,  6, 10,  14,  22,  26,  34,  38,   46,   58,   62,   74
   5,  7, 11,  15,  23,  27,  35,  39,   47,   59,   63,   75
   7, 16, 36,  78, 144, 222, 324, 438,  668,  900, 1148, 1518
   9, 19, 41,  85, 155, 235, 341, 457,  691,  929, 1179, 1555
  10, 21, 45,  91, 165, 247, 357, 475,  713,  957, 1209, 1591
  11, 22, 46,  92, 166, 248, 358, 476,  714,  958, 1210, 1592
  13, 31, 71, 155, 287, 443, 647, 875, 1335, 1799, 2295, 3035
  15, 34, 76, 162, 298, 456, 664, 894, 1358, 1828, 2326, 3072
  16, 36, 80, 168, 308, 468, 680, 912, 1380, 1856, 2356, 3108
  17, 37, 81, 169, 309, 469, 681, 913, 1381, 1857, 2357, 3109

Transpose: A286623.
Cf. A002110, A276945.
Column 1: A276155.
Row 1: A000012, Row 2: A008864, Row 3: A100484, Row 4: A072055, Row 5: A023523 (from its second term onward), Row 6: A286624.
Cf. A276617 (analogous array).

(define (A286625 n) (A286625bi (A002260 n) (A004736 n)))
(define (A286625bi row col) (/ (A276945bi row col) (A002110 (- col 1))))

A(n,k) = A276945(n, k) / A002110(k-1).

A276939 Row 2 of A276945: a(n) = A002110(n) + A002110(n+1).

Original entry on oeis.org

3, 8, 36, 240, 2520, 32340, 540540, 10210200, 232792560, 6692786100, 207030183360, 7621298624940, 311671001662020, 13387011595197240, 627972543920161440, 33204048259778536140, 1955349508631402683800, 119211141709561183622340, 7975609932439674026862360, 565799151677779228023294480, 41287621429375723111588738860

Offset: 0

Antti Karttunen, Sep 24 2016

a(n) = number whose primorial base representation (A049345) begins with digits "11", followed by n zeros: 11, 110, 1100, 11000, 110000, ...

Michael De Vlieger, Table of n, a(n) for n = 0..349
Index entries for sequences related to primorial base

Row 2 of A276945 (column 2 of A276943).

Cf. A002110, A049345.

Total /@ Partition[FoldList[Times, 1, Prime@ Range[20]], 2, 1] (* Michael De Vlieger, Jun 24 2023 *)

a(n) = factorback(primes(n)) + factorback(primes(n+1)); \\ Michel Marcus, Nov 23 2022

(define (A276939 n) (+ (A002110 n) (A002110 (+ 1 n))))

a(n) = A002110(n) + A002110(n+1).

A286615 Square array A(n,k) = A276945(n,k)-1, read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

0, 1, 2, 5, 7, 3, 29, 35, 11, 4, 209, 239, 59, 13, 6, 2309, 2519, 419, 65, 31, 8, 30029, 32339, 4619, 449, 215, 37, 9, 510509, 540539, 60059, 4829, 2339, 245, 41, 10, 9699689, 10210199, 1021019, 62369, 30239, 2549, 269, 43, 12, 223092869, 232792559, 19399379, 1051049, 512819, 32549, 2729, 275, 61, 14

Offset: 1

Antti Karttunen, Jun 30 2017

A permutation of nonnegative integers.

			The top left 8 X 15 corner of the array:
   0,  1,   5,   29,   209,    2309,    30029,    510509
   2,  7,  35,  239,  2519,   32339,   540539,  10210199
   3, 11,  59,  419,  4619,   60059,  1021019,  19399379
   4, 13,  65,  449,  4829,   62369,  1051049,  19909889
   6, 31, 215, 2339, 30239,  512819,  9729719, 223603379
   8, 37, 245, 2549, 32549,  542849, 10240229, 233303069
   9, 41, 269, 2729, 34649,  570569, 10720709, 242492249
  10, 43, 275, 2759, 34859,  572879, 10750739, 243002759
  12, 61, 425, 4649, 60269, 1023329, 19429409, 446696249
  14, 67, 455, 4859, 62579, 1053359, 19939919, 456395939
  15, 71, 479, 5039, 64679, 1081079, 20420399, 465585119
  16, 73, 485, 5069, 64889, 1083389, 20450429, 466095629
  17, 89, 629, 6929, 90089, 1531529, 29099069, 669278609
  18, 91, 635, 6959, 90299, 1533839, 29129099, 669789119
  19, 95, 659, 7139, 92399, 1561559, 29609579, 678978299

Transpose: A286616.
One less than A276945.
Row 1: A057588.

(define (A286615 n) (A286615bi (A002260 n) (A004736 n)))
(define (A286615bi row col) (+ -1 (A276945bi row col))) ;; For A276945bi see under A276945.

A(n,k) = A276945(n,k)-1.

A276946 Inverse permutation to A276945.

Original entry on oeis.org

1, 2, 3, 6, 10, 4, 15, 5, 21, 28, 36, 9, 45, 14, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 7, 325, 20, 351, 378, 406, 8, 435, 27, 465, 496, 528, 35, 561, 44, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 13, 1225, 54, 1275, 1326, 1378, 19, 1431, 65, 1485, 1540, 1596, 77, 1653, 90

Offset: 1

Antti Karttunen, Sep 24 2016

Inverse: A276945.

Differs from analogous A276956 for the first time at n=24, where a(24)=190, while A276956(24)=7.

A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 24, 30, 12, 5, 120, 144, 48, 14, 7, 720, 840, 240, 54, 26, 9, 5040, 5760, 1440, 264, 126, 32, 10, 40320, 45360, 10080, 1560, 744, 150, 36, 11, 362880, 403200, 80640, 10800, 5160, 864, 168, 38, 13, 3628800, 3991680, 725760, 85680, 41040, 5880, 960, 174, 50, 15, 39916800, 43545600, 7257600, 766080, 367920, 46080, 6480, 984, 246, 56, 16

Offset: 1

Antti Karttunen, Sep 22 2016

The square array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

When viewed in factorial base (A007623) the terms on each row start all with the same prefix, but with an increasing number of zeros appended to the end. For example, for row 8 (A001344 from a(1)=11 onward), the terms written in factorial base look as: 121, 1210, 12100, 121000, ...

			The top left {1..9} x {1..18} corner of the array:
   1,  2,   6,   24,   120,    720,    5040,    40320,    362880
   3,  8,  30,  144,   840,   5760,   45360,   403200,   3991680
   4, 12,  48,  240,  1440,  10080,   80640,   725760,   7257600
   5, 14,  54,  264,  1560,  10800,   85680,   766080,   7620480
   7, 26, 126,  744,  5160,  41040,  367920,  3669120,  40279680
   9, 32, 150,  864,  5880,  46080,  408240,  4032000,  43908480
  10, 36, 168,  960,  6480,  50400,  443520,  4354560,  47174400
  11, 38, 174,  984,  6600,  51120,  448560,  4394880,  47537280
  13, 50, 246, 1464, 10200,  81360,  730800,  7297920,  80196480
  15, 56, 270, 1584, 10920,  86400,  771120,  7660800,  83825280
  16, 60, 288, 1680, 11520,  90720,  806400,  7983360,  87091200
  17, 62, 294, 1704, 11640,  91440,  811440,  8023680,  87454080
  18, 72, 360, 2160, 15120, 120960, 1088640, 10886400, 119750400
  19, 74, 366, 2184, 15240, 121680, 1093680, 10926720, 120113280
  20, 78, 384, 2280, 15840, 126000, 1128960, 11249280, 123379200
  21, 80, 390, 2304, 15960, 126720, 1134000, 11289600, 123742080
  22, 84, 408, 2400, 16560, 131040, 1169280, 11612160, 127008000
  23, 86, 414, 2424, 16680, 131760, 1174320, 11652480, 127370880

Inverse permutation: A276956.
Transpose: A276953.
Cf. A276949 (index of column where n appears), A276951 (index of row).
Cf. A153880.
Columns 1-3: A273670, A276932, A276933.
The following lists some of the rows that have their own entries. Pattern present in the factorial base expansion of the terms on that row is given in double quotes:
Row 1: A000142 (from a(1)=1, "1" onward),
Row 2: A001048 (from a(2)=3, "11" onward),
Row 3: A052849 (from a(2)=4, "20" onward).
Row 4: A052649 (from a(1)=5, "21" onward).
Row 5: A108217 (from a(3)=7, "101" onward).
Row 6: A054119 (from a(3)=9, "111" onward).
Row 7: A052572 (from a(2)=10, "120" onward).
Row 8: A001344 (from a(1)=11, "121" onward).
Row 13: A052560 (from a(3)=18, "300" onward).
Row 16: A225658 (from a(1)=21, "311" onward).
Row 20: A276940 (from a(3) = 27, "1011" onward).
Related or similar permutations: A257505, A275848, A273666.
Cf. also arrays A276617, A276588 & A276945.

(define (A276955 n) (A276955bi (A002260 n) (A004736 n)))
(define (A276955bi row col) (if (= 1 col) (A273670 (- row 1)) (A153880 (A276955bi row (- col 1)))))

A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1))
As a composition of other permutations:
a(n) = A275848(A257505(n)).

A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 30, 32, 36, 38, 42, 44, 60, 62, 66, 68, 72, 74, 90, 92, 96, 98, 102, 104, 120, 122, 126, 128, 132, 134, 210, 212, 216, 218, 222, 224, 240, 242, 246, 248, 252, 254, 270, 272, 276, 278, 282, 284, 300, 302, 306, 308, 312, 314, 330, 332, 336, 338, 342, 344, 420, 422, 426, 428, 432, 434, 450, 452, 456, 458, 462, 464, 480, 482, 486, 488

Offset: 0

Antti Karttunen, Aug 24 2016

			   n   A049345  with one zero           converted back
                appended to the right   to decimal = a(n)
---------------------------------------------------------
   0       0            00                     0
   1       1            10                     2
   2      10           100                     6
   3      11           110                     8
   4      20           200                    12
   5      21           210                    14
   6     100          1000                    30
   7     101          1010                    32
   8     110          1100                    36
   9     111          1110                    38
  10     120          1200                    42
  11     121          1210                    44
  12     200          2000                    60
  13     201          2010                    62
  14     210          2100                    66
  15     211          2110                    68
  16     220          2200                    72

Antti Karttunen, Table of n, a(n) for n = 0..30029
Index entries for sequences related to primorial base

Complement: A276155.
Cf. A002110, A003961, A049345, A276085, A276086, A276151, A276152, A286629 [= a(A061720(n-1))], A324384 [= gcd(n, a(n))], A323879, A328770 (a subsequence).
Cf. also A276156, A328461, A328464.
Dispersion array and its transpose: A276943, A276945, with primorials divided out: A286623, A286625.
Analogous to A153880.

nn = 75; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[#] <= nn &]]; Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, 75}] (* Michael De Vlieger, Aug 26 2016 *)

A276154(n) = A276085(A003961(A276086(n))); \\ Antti Karttunen, Mar 15 2021

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };
A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n)))); \\ Antti Karttunen, Mar 15 2021

(definec (A276154 n) (if (zero? n) n (+ (A276152 n) (A276154 (A276151 n)))))

a(0) = 0; for n >= 1, a(n) = A276152(n) + a(A276151(n)).

a(n) = A276085(A003961(A276086(n))). - Antti Karttunen, Mar 15 2021

A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 9, 16, 6, 1, 31, 19, 36, 8, 1, 33, 106, 41, 78, 12, 1, 37, 109, 386, 85, 144, 14, 1, 39, 121, 391, 1002, 155, 222, 18, 1, 211, 124, 421, 1009, 2432, 235, 324, 20, 1, 213, 1156, 426, 1079, 2443, 4200, 341, 438, 24, 1, 217, 1159, 5006, 1086, 2575, 4213, 7430, 457, 668, 30, 1, 219, 1171, 5011, 17018, 2586, 4421, 7447, 12674, 691, 900, 32, 1

Offset: 1

Antti Karttunen, Oct 16 2019

Array is read by falling antidiagonals with n (row) and k (column) ranging as: (n,k) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Row n contains all such sums of distinct primorials whose least significant summand is A002110(n-1), with each sum divided by that least significant primorial, which is also the largest primorial which divides that sum.

			Top left 9 X 11 corner of the array:
1: | 1,  3,   7,   9,    31,    33,    37,    39,    211,    213,    217
2: | 1,  4,  16,  19,   106,   109,   121,   124,   1156,   1159,   1171
3: | 1,  6,  36,  41,   386,   391,   421,   426,   5006,   5011,   5041
4: | 1,  8,  78,  85,  1002,  1009,  1079,  1086,  17018,  17025,  17095
5: | 1, 12, 144, 155,  2432,  2443,  2575,  2586,  46190,  46201,  46333
6: | 1, 14, 222, 235,  4200,  4213,  4421,  4434,  96578,  96591,  96799
7: | 1, 18, 324, 341,  7430,  7447,  7753,  7770, 215442, 215459, 215765
8: | 1, 20, 438, 457, 12674, 12693, 13111, 13130, 392864, 392883, 393301
9: | 1, 24, 668, 691, 20678, 20701, 21345, 21368, 765050, 765073, 765717

Cf. A328463 (transpose).
Cf. A000265, A002110, A007814, A135764, A276154, A276156,
Rows 1 - 5: A328462, A328465, A328466, A328467, A328468.
Column 2: A008864.
Column 3: A023523 (after its initial term).
Column 4: A286624.
Cf. also arrays A276945, A286625.

up_to = 105;
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328464sq(n,k) = (A276156((2^(n-1)) * (k+k-1)) / A002110(n-1));
A328464list(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] = A328464sq(col,(a-(col-1))))); (v); };
v328464 = A328464list(up_to);
A328464(n) = v328464[n];

A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1).

a(n) = A328461(A135764(n)). [When all sequences are considered as one-dimensional]

A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 12, 36, 30, 7, 14, 60, 240, 210, 9, 32, 66, 420, 2520, 2310, 10, 38, 216, 450, 4620, 32340, 30030, 11, 42, 246, 2340, 4830, 60060, 540540, 510510, 13, 44, 270, 2550, 30240, 62370, 1021020, 10210200, 9699690, 15, 62, 276, 2730, 32550, 512820, 1051050, 19399380, 232792560, 223092870

Offset: 1

Antti Karttunen, Sep 24 2016

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Entries on row n are all multiples of A002110(n-1).

			The top left corner of the array:
    1,    3,    4,    5,     7,     9,    10,    11,    13,    15,    16
    2,    8,   12,   14,    32,    38,    42,    44,    62,    68,    72
    6,   36,   60,   66,   216,   246,   270,   276,   426,   456,   480
   30,  240,  420,  450,  2340,  2550,  2730,  2760,  4650,  4860,  5040
  210, 2520, 4620, 4830, 30240, 32550, 34650, 34860, 60270, 62580, 64680

Inverse permutation: A276944.
Transpose: A276945.
Column 1: A002110, Row 1: A276155.
Cf. A276154.
Cf. also array A276953.

(define (A276943 n) (A276943bi (A002260 n) (A004736 n)))
(define (A276943bi row col) (if (= 1 row) (A276155 col) (A276154 (A276943bi (- row 1) col))))

A(1,col) = A276155(col); for row > 1, A(row,col) = A276154(A(row-1,col)).

A276155 Complement of A276154; numbers that cannot be obtained by shifting left the primorial base representation (A049345) of some number.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 108, 109

Offset: 1

Antti Karttunen, Aug 24 2016

The first 25 terms, when viewed in primorial base (A049345) look as: 1, 11, 20, 21, 101, 111, 120, 121, 201, 211, 220, 221, 300, 301, 310, 311, 320, 321, 400, 401, 410, 411, 420, 421, 1001.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Complement: A276154.
Row 1 of A276943 and A286623. Column 1 of A276945 and A286625.
Cf. A002110, A049345.
Cf. A005408, A057588, A061720, A143293, A286630 (subsequences).
For the first 17 terms coincides with A273670.

nn = 109; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Complement[Range@ nn, Table[FromDigits[#, b] &@ Append[IntegerDigits[n, b], 0], {n, 0, nn}]] (* Version 10.2, or *)
nn = 109; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Complement[Range@ nn, Table[Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ # - 1]], Reverse@ #}] &@ Append[f@ n, 0], {n, 0, nn}]] (* Michael De Vlieger, Aug 26 2016 *)

A097250 Smallest m such that A097249(m) = n; from n=1 onwards, twice the primorials, 2*A002110(n).

Original entry on oeis.org

1, 4, 12, 60, 420, 4620, 60060, 1021020, 19399380, 446185740, 12939386460, 401120980260, 14841476269620, 608500527054420, 26165522663340060, 1229779565176982820, 65178316954380089460, 3845520700308425278140, 234576762718813941966540, 15716643102160534111758180

Offset: 0

Reinhard Zumkeller, Aug 03 2004

nonn

A097249(a(n))=n and A097249(m)

a(n) = A088860(n) for n>=1. - G. C. Greubel, Apr 23 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to primorial numbers

Cf. A002110, A088860, A097249.

From a(1)=4 onwards, row 3 of A276945.

Join[{1}, 2 Denominator[Accumulate[1/Prime[Range[20]]]]] (* Vincenzo Librandi, Mar 25 2017 *)
Join[{1}, 2*FoldList[Times, 1, Prime[Range[50]]]] (* G. C. Greubel, Apr 23 2017 *)

a(n) = if n=0 then 1 else 2*A002110(n).

Name amended by Antti Karttunen, Sep 24 2016

a(18)-a(19) from Vincenzo Librandi, Mar 25 2017

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A286625 Square array A(n,k) = A276945(n,k)/A002110(k-1), read by descending antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A276939 Row 2 of A276945: a(n) = A002110(n) + A002110(n+1).

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A286615 Square array A(n,k) = A276945(n,k)-1, read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A276946 Inverse permutation to A276945.

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A276955 Square array A(row,col): A(row,1) = A273670(row-1), and for col > 1, A(row,col) = A153880(A(row,col-1)); Dispersion of factorial base left shift A153880.

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A276154 a(n) = Shift primorial base representation (A049345) of n left by one digit (append one zero to the right, then convert back to decimal).

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A328464 Square array A(n,k) = A276156((2^(n-1)) * (2k-1)) / A002110(n-1), read by descending antidiagonals.

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A276943 Square array A(row,col) read by antidiagonals: A(1,col) = A276155(col), and for row > 1, A(row,col) = A276154(A(row-1,col)); Dispersion of primorial base left shift A276154 (array transposed).

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