A077664 -id:A077664 - cp's OEIS frontend

A077665 Final term of n-th row of A077664.

2, 5, 7, 11, 11, 23, 15, 23, 22, 33, 23, 47, 27, 45, 43, 47, 35, 71, 39, 69, 58, 69, 47, 95, 56, 81, 67, 93, 59, 139, 63, 95, 86, 105, 86, 143, 75, 117, 101, 139, 83, 187, 87, 139, 128, 141, 95, 191, 106, 173, 131, 163, 107, 215, 129, 185, 146, 177, 119, 283, 123, 189

Offset: 1

Amarnath Murthy, Nov 14 2002

nonn

a(p) = 2p+1, p is a prime. a(2^n) = 3*2^n -1.

a(n) also equals the (n+phi(n))th integer from among those positive integers coprime to n, where phi(n) = A000010(n). a(n) also equals n + (the n-th integer from among those positive integers coprime to n) = n + A069213(n). - Leroy Quet, Apr 12 2007

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A077664, A077666.

a077665 = last . a077664_row  -- Reinhard Zumkeller, Aug 03 2015

More terms from Sascha Kurz, Jan 03 2003

A077666 Sum of terms of n-th row of A077664.

Original entry on oeis.org

2, 8, 16, 32, 41, 90, 78, 128, 142, 224, 188, 360, 261, 424, 436, 512, 443, 810, 552, 900, 826, 1016, 806, 1440, 1016, 1408, 1276, 1698, 1277, 2588, 1458, 2048, 1984, 2384, 2115, 3240, 2073, 2968, 2753, 3600, 2543, 4854, 2796, 4064, 3922, 4328, 3338, 5760

Offset: 1

Amarnath Murthy, Nov 14 2002

nonn

a(p) = p(2p+1)- p(p+1)/2 + 1, p is a prime. a(2^n) = 2^n + (2^n)^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A077664, A077665.

a077666 = sum . a077664_row  -- Reinhard Zumkeller, Aug 03 2015

More terms from Sascha Kurz, Jan 03 2003

A260910 Triangle read by rows: Fresenius numbers of n and A077664(n,k), k = 1..n.

Original entry on oeis.org

-1, 1, 3, 5, 7, 11, 11, 17, 23, 29, 19, 23, 27, 31, 39, 29, 49, 59, 79, 89, 109, 41, 47, 53, 59, 65, 71, 83, 55, 69, 83, 97, 111, 125, 139, 153, 71, 79, 95, 103, 119, 127, 143, 151, 167, 89, 107, 143, 161, 179, 197, 233, 251, 269, 287, 109, 119, 129, 139

Offset: 1

Reinhard Zumkeller, Aug 04 2015

For n > 1: T(n,1) = A028387(n-2).

			.   1:   -1
.   2:    1    3
.   3:    5    7   11
.   4:   11   17   23   29
.   5:   19   23   27   31   39
.   6:   29   49   59   79   89  109
.   7:   41   47   53   59   65   71   83
.   8:   55   69   83   97  111  125  139  153
.   9:   71   79   95  103  119  127  143  151  167
.  10:   89  107  143  161  179  197  233  251  269  287
.  11:  109  119  129  139  149  159  169  179  189  199  219
.  12:  131  175  197  241  263  307  329  373  395  439  461  505 .

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Frobenius Number.
Wikipedia, Coin problem

Cf. A077664, A028387.

a260910 n k = a260910_tabl !! (n - 1) !! (k-1)
a260910_row n = a260910_tabl !! (n-1)
a260910_tabl = zipWith (map . sylvester) [1..] a077664_tabl where
   sylvester u v = u * v - u - v

row[n_] := Module[{j, k}, Reap[For[j = n+1; k = 1, k <= n, j++, If[CoprimeQ[n, j], Sow[j]; k++]]][[2, 1]]];
T[n_, k_] := (n-1) row[n][[k]] - n;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2021 *)

T(n,k) = (n-1) * A077664(n,k) - n.

A260895 Number of primes in n-th row of triangle A077664.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 5, 4, 7, 4, 10, 3, 8, 8, 9, 4, 13, 4, 11, 8, 11, 6, 15, 7, 13, 10, 15, 7, 24, 7, 13, 12, 16, 12, 23, 9, 18, 14, 22, 10, 29, 9, 20, 17, 20, 9, 28, 12, 25, 17, 23, 12, 31, 15, 26, 18, 24, 13, 44, 12, 24, 21, 25, 18, 43, 13, 28, 21, 39, 14

Offset: 1

Reinhard Zumkeller, Aug 03 2015

nonn

a(n) = sum(A010051(A077664(n,k)): k = 1..n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A077664, A010051.

a260895 = sum . map a010051' . a077664_row

A077581 Triangle in which row n contains the n smallest numbers starting from 1 and coprime to n.

Original entry on oeis.org

1, 1, 3, 1, 2, 4, 1, 3, 5, 7, 1, 2, 3, 4, 6, 1, 5, 7, 11, 13, 17, 1, 2, 3, 4, 5, 6, 8, 1, 3, 5, 7, 9, 11, 13, 15, 1, 2, 4, 5, 7, 8, 10, 11, 13, 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Offset: 1

Amarnath Murthy, Nov 14 2002

A247815 and A247892 give number of primes and nonprimes per row. - Reinhard Zumkeller, Sep 26 2014

			1;
1,  3;
1,  2,  4;
1,  3,  5,  7;
1,  2,  3,  4,  6;
1,  5,  7, 11, 13, 17;
1,  2,  3,  4,  5,  6,  8;
1,  3,  5,  7,  9, ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A069213, A077582.
Cf. A247798 (central terms), A247815, A247892.
Cf. A077664.

a077581 n k = a077581_tabl !! (n-1) !! (k-1)
a077581_row n = a077581_tabl !! (n-1)
a077581_tabl = map (\x -> take x [z | z <- [1..], gcd x z == 1]) [1..]
-- Reinhard Zumkeller, Sep 26 2014

row[n_] := Take[Select[Range[n^2], GCD[ #, n]==1&], n]; Join@@row/@Range[13]

More terms from Sascha Kurz, Jan 11 2003

cp's OEIS Frontend

A077665 Final term of n-th row of A077664.

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A077666 Sum of terms of n-th row of A077664.

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A260910 Triangle read by rows: Fresenius numbers of n and A077664(n,k), k = 1..n.

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A260895 Number of primes in n-th row of triangle A077664.

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Haskell

A077581 Triangle in which row n contains the n smallest numbers starting from 1 and coprime to n.

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Author

Keywords

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Examples

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Programs

Haskell

Mathematica

Extensions