A180782 -id:A180782 - cp's OEIS frontend

A180772 Number of distinct solutions to the congruence x(1)*x(2) == 0 (mod n), with x() only in 1..n-1.

0, 0, 0, 1, 0, 2, 0, 3, 3, 4, 0, 9, 0, 6, 8, 10, 0, 15, 0, 17, 12, 10, 0, 27, 10, 12, 15, 25, 0, 38, 0, 26, 20, 16, 24, 51, 0, 18, 24, 51, 0, 56, 0, 41, 51, 22, 0, 74, 21, 50, 32, 49, 0, 69, 40, 75, 36, 28, 0, 121, 0, 30, 75, 68, 48, 92, 0, 65, 44, 106, 0, 141, 0, 36, 90, 73, 60, 110, 0, 138

Offset: 1

R. H. Hardin, formula from Max Alekseyev in the Sequence Fans Mailing List, Sep 20 2010

nonn

Also, number of ordered pairs (a, b) with 0 < a <= b such that a*b = c*n + d and c = d where 0 < a, b, c, d < n. - Naohiro Nomoto, Oct 02 2021

			The a(12)=9 solutions for product of a single 1..11 pair == 0 (mod 12) are 2*6, 3*4, 3*8, 4*6, 4*9, 6*6, 6*8, 6*10, and 8*9.

R. H. Hardin, Table of n, a(n) for n=1..10000

Column 1 of A180782.

Cf. A000188, A018804, A174088.

f1[p_, e_] := (e*(p - 1)/p + 1)*p^e; f2[p_, e_] := p^Floor[e/2]; a[n_] := (Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct)/2 - n; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = A174088(n) - n = ( A018804(n) + A000188(n) )/2 - n.

A180773 Number of distinct solutions of Sum_{i=1..2} (x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 2, 6, 13, 23, 36, 62, 78, 123, 150, 238, 255, 355, 427, 567, 580, 864, 810, 1202, 1232, 1471, 1452, 2310, 1960, 2479, 2712, 3414, 2947, 4649, 3600, 5126, 5022, 5673, 5845, 8457, 6165, 7975, 8405, 11062, 8410, 13133, 9702, 13726, 14475, 14323, 12696

Offset: 1

R. H. Hardin, Sep 20 2010

nonn

			Solutions for sum of products of 2 1..5 pairs = 0 (mod 6) are
(1*1 + 1*5) (1*2 + 1*4) (1*2 + 2*2) (1*2 + 2*5) (1*2 + 4*4) (1*3 + 1*3)
(1*3 + 3*3) (1*3 + 3*5) (1*4 + 2*4) (1*4 + 4*5) (1*5 + 5*5) (2*2 + 2*4)
(2*2 + 4*5) (2*3 + 2*3) (2*3 + 3*4) (2*4 + 2*5) (2*4 + 4*4) (2*5 + 4*5)
(3*3 + 3*3) (3*3 + 3*5) (3*4 + 3*4) (3*5 + 3*5) (4*4 + 4*5)

R. H. Hardin, Table of n, a(n) for n=1..999

Column 2 of A180782.

A180774 Number of distinct solutions of sum{i=1..3}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 5, 14, 42, 119, 253, 526, 944, 1702, 2655, 4510, 6284, 9867, 13519, 19980, 25128, 37286, 44562, 63802, 76850, 102370, 118591, 168208, 183320, 241554, 279852, 359754, 387016, 528861, 543620, 712450, 780810, 953120, 1025355, 1360255

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 3 of A180782

			Solutions for sum of products of 3 1..3 pairs = 0 (mod 4) are
(1*1 + 1*1 + 1*2) (1*1 + 1*1 + 2*3) (1*1 + 1*2 + 3*3) (1*1 + 1*3 + 2*2)
(1*1 + 2*3 + 3*3) (1*2 + 1*2 + 2*2) (1*2 + 1*3 + 1*3) (1*2 + 2*2 + 2*3)
(1*2 + 3*3 + 3*3) (1*3 + 1*3 + 2*3) (1*3 + 2*2 + 3*3) (2*2 + 2*2 + 2*2)
(2*2 + 2*3 + 2*3) (2*3 + 3*3 + 3*3)

R. H. Hardin, Table of n, a(n) for n=1..447

A180775 Number of distinct solutions of sum{i=1..4}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 3, 34, 145, 522, 1518, 4041, 9150, 19970, 38555, 74370, 128040, 224434, 358988, 587014, 876114, 1372578, 1941624, 2912816, 4001868, 5742391, 7599933, 10831065, 13788935, 18946564, 24080514, 32270596, 39619720, 53256875, 63655605, 83580675

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 4 of A180782

			Solutions for sum of products of 4 1..3 pairs = 0 (mod 4) are
(1*1 + 1*1 + 1*1 + 1*1) (1*1 + 1*1 + 1*1 + 3*3) (1*1 + 1*1 + 1*2 + 2*2)
(1*1 + 1*1 + 1*3 + 1*3) (1*1 + 1*1 + 2*2 + 2*3) (1*1 + 1*1 + 3*3 + 3*3)
(1*1 + 1*2 + 1*2 + 1*3) (1*1 + 1*2 + 1*3 + 2*3) (1*1 + 1*2 + 2*2 + 3*3)
(1*1 + 1*3 + 1*3 + 3*3) (1*1 + 1*3 + 2*2 + 2*2) (1*1 + 1*3 + 2*3 + 2*3)
(1*1 + 2*2 + 2*3 + 3*3) (1*1 + 3*3 + 3*3 + 3*3) (1*2 + 1*2 + 1*2 + 1*2)
(1*2 + 1*2 + 1*2 + 2*3) (1*2 + 1*2 + 1*3 + 3*3) (1*2 + 1*2 + 2*2 + 2*2)
(1*2 + 1*2 + 2*3 + 2*3) (1*2 + 1*3 + 1*3 + 2*2) (1*2 + 1*3 + 2*3 + 3*3)
(1*2 + 2*2 + 2*2 + 2*3) (1*2 + 2*2 + 3*3 + 3*3) (1*2 + 2*3 + 2*3 + 2*3)
(1*3 + 1*3 + 1*3 + 1*3) (1*3 + 1*3 + 2*2 + 2*3) (1*3 + 1*3 + 3*3 + 3*3)
(1*3 + 2*2 + 2*2 + 3*3) (1*3 + 2*3 + 2*3 + 3*3) (2*2 + 2*2 + 2*2 + 2*2)
(2*2 + 2*2 + 2*3 + 2*3) (2*2 + 2*3 + 3*3 + 3*3) (2*3 + 2*3 + 2*3 + 2*3)
(3*3 + 3*3 + 3*3 + 3*3)

R. H. Hardin, Table of n, a(n) for n=1..380

A180776 Number of distinct solutions of sum{i=1..5}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 7, 62, 402, 1960, 7587, 25470, 73146, 193036, 455161, 1022770, 2098782, 4196208, 7797217, 14286908, 24527232, 42125055, 67955391, 110571248, 170260213, 264585538, 390623343, 588725404, 837466268, 1220597598, 1696535256

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 5 of A180782

			Solutions for sum of products of 5 1..2 pairs = 0 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*2) (1*1 + 1*1 + 1*1 + 1*2 + 2*2)
(1*1 + 1*1 + 1*2 + 2*2 + 2*2) (1*1 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*2 + 2*2 + 2*2 + 2*2) (1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*2 + 2*2 + 2*2 + 2*2 + 2*2)

R. H. Hardin, Table of n, a(n) for n=1..249

A180777 Number of distinct solutions of sum{i=1..6}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 12, 118, 995, 6518, 32893, 139369, 499659, 1599064, 4551225, 12014040, 29034510, 66658219, 142828285, 295240250, 576393604, 1098913338, 1993354080, 3560486370, 6091808112, 10312711744, 16796812703, 27247744649, 42564714900

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 6 of A180782

			Solutions for sum of products of 6 1..2 pairs = 0 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1) (1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2) (1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2) (1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2) (1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2) (1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2) (2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

R. H. Hardin, Table of n, a(n) for n=1..183

A180778 Number of distinct solutions of sum{i=1..7}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 9, 198, 2294, 19467, 126864, 674922, 2997717, 11615754, 39660995, 123162204, 348412638, 920581314, 2264397902, 5294830516, 11692554408, 24858751383, 50403391227, 99282023930, 187902783369, 347689331443, 621482071507

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 7 of A180782

			Solutions for sum of products of 7 1..2 pairs = 0 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

R. H. Hardin, Table of n, a(n) for n=1..183

A180779 Number of distinct solutions of sum{i=1..8}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 15, 327, 4856, 53455, 444003, 2948305, 16112289, 75389116, 307372600, 1122069080, 3701885580, 11258893954, 31699979961, 83910860201, 209004408715, 496246703439, 1121475446118, 2440154664350, 5096346969372, 10323359668079

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 8 of A180782

			Solutions for sum of products of 8 1..2 pairs = 0 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

R. H. Hardin, Table of n, a(n) for n=1..183

A180780 Number of distinct solutions of sum{i=1..9}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 22, 499, 9730, 136491, 1430727, 11783122, 78770456, 443607864, 2151608155, 9218591346, 35373572400, 123749340262, 398005623516, 1192411118090, 3344070542568, 8869510553867, 22304900540593, 53635016669434

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 9 of A180782

			Solutions for sum of products of 9 1..2 pairs = 0 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

R. H. Hardin, Table of n, a(n) for n=1..183

A180781 Number of distinct solutions of sum{i=1..10}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 18, 756, 18478, 327338, 4292145, 43576141, 354466005, 2394458182, 13770292261, 69111228938, 307750080438, 1236998118274, 4537247867399, 15375840213938, 48489022864504, 143625325278096, 401488209736317

Offset: 1

R. H. Hardin Sep 20 2010

nonn

Column 10 of A180782

R. H. Hardin, Table of n, a(n) for n=1..183

cp's OEIS Frontend

A180772 Number of distinct solutions to the congruence x(1)*x(2) == 0 (mod n), with x() only in 1..n-1.

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A180773 Number of distinct solutions of Sum_{i=1..2} (x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180774 Number of distinct solutions of sum{i=1..3}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180775 Number of distinct solutions of sum{i=1..4}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180776 Number of distinct solutions of sum{i=1..5}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180777 Number of distinct solutions of sum{i=1..6}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180778 Number of distinct solutions of sum{i=1..7}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180779 Number of distinct solutions of sum{i=1..8}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180780 Number of distinct solutions of sum{i=1..9}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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A180781 Number of distinct solutions of sum{i=1..10}(x(2i-1)*x(2i)) = 0 (mod n), with x() only in 1..n-1.

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