A077581 -id:A077581 - cp's OEIS frontend

A077582 Sum of terms of n-th row of A077581.

1, 4, 7, 16, 16, 54, 29, 64, 61, 124, 67, 216, 92, 228, 211, 256, 154, 486, 191, 500, 385, 532, 277, 864, 391, 732, 547, 914, 436, 1688, 497, 1024, 895, 1228, 890, 1944, 704, 1524, 1232, 2000, 862, 3090, 947, 2128, 1897, 2212, 1129, 3456, 1401, 3124, 2068

Offset: 1

Amarnath Murthy, Nov 14 2002

nonn

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A069213, A077581.

a077582 = sum . a077581_row  -- Reinhard Zumkeller, Sep 26 2014

More terms from Sascha Kurz, Jan 11 2003

A247815 Number of primes in n-th row of triangle A077581.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 3, 5, 5, 7, 4, 9, 5, 9, 7, 10, 6, 14, 7, 13, 10, 13, 8, 18, 10, 14, 11, 16, 9, 26, 10, 17, 14, 18, 13, 26, 11, 20, 16, 23, 12, 31, 13, 22, 21, 22, 14, 32, 15, 28, 20, 27, 15, 35, 19, 29, 22, 28, 16, 45, 17, 29, 27, 30, 21, 44, 18, 32, 26

Offset: 1

Reinhard Zumkeller, Sep 26 2014

nonn

a(n) = n - A247892(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A010051, A069213, A077581, A247892.

a247815 = sum . map a010051' . a077581_row

a(n) = A000720(A069213(n)) - A001221(n).

A247892 Number of nonprimes in n-th row of triangle A077581.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 4, 3, 7, 3, 8, 5, 8, 6, 11, 4, 12, 7, 11, 9, 15, 6, 15, 12, 16, 12, 20, 4, 21, 15, 19, 16, 22, 10, 26, 18, 23, 17, 29, 11, 30, 22, 24, 24, 33, 16, 34, 22, 31, 25, 38, 19, 36, 27, 35, 30, 43, 15, 44, 33, 36, 34, 44, 22, 49, 36, 43, 27

Offset: 1

Reinhard Zumkeller, Sep 26 2014

nonn

a(n) = n - A247815(n)

Cf. A077581, A247815.

Haskell
```
a247892 n = n - a247815 n
```

A126577 a(n) = numerator of the sum of reciprocals of the terms in n-th row of triangle A077581.

Original entry on oeis.org

1, 4, 7, 176, 9, 133542, 103, 91072, 99527, 131023748, 7591, 300996993816, 88001, 1403843964196, 44094737, 10686452707072, 825533, 368070779365071896502, 2895701, 8653175044141052500, 81659533540907, 3080940707518158404

Offset: 1

Leroy Quet, Dec 28 2006

			Row 4 of triangle A077581 is (1,3,5,7).
So a(4) is the numerator of 1/1 +1/3 +1/5 + 1/7 = 176/105.

Cf. A077581, A126578.

row[n_] := Take[Select[Range[n^2], GCD[ #, n] == 1 &], n]; Table[Numerator[Plus @@ (1/# &) /@ row[n]], {n, 23}] (* Ray Chandler, Dec 29 2006 *)

Extended by Ray Chandler, Dec 29 2006

A126578 a(n) = denominator of the sum of reciprocals of the terms in n-th row of triangle A077581.

Original entry on oeis.org

1, 3, 4, 105, 4, 85085, 40, 45045, 40040, 66927861, 2520, 167133741775, 27720, 644658718275, 16997552, 4512611027925, 240240, 190103424450275260925, 816816, 3873805630307495883, 28269478608800, 1257729100749186975, 15519504

Offset: 1

Leroy Quet, Dec 28 2006

			Row 4 of triangle A077581 is (1,3,5,7).
So a(4) is the denominator of 1/1 +1/3 +1/5 + 1/7 = 176/105.

Cf. A077581, A126577.

row[n_] := Take[Select[Range[n^2], GCD[ #, n] == 1 &], n]; Table[Denominator[Plus @@ (1/# &) /@ row[n]], {n, 23}] (* Ray Chandler, Dec 29 2006 *)

Extended by Ray Chandler, Dec 29 2006

A069213 a(n) = n-th positive integer relatively prime to n.

Original entry on oeis.org

1, 3, 4, 7, 6, 17, 8, 15, 13, 23, 12, 35, 14, 31, 28, 31, 18, 53, 20, 49, 37, 47, 24, 71, 31, 55, 40, 65, 30, 109, 32, 63, 53, 71, 51, 107, 38, 79, 62, 99, 42, 145, 44, 95, 83, 95, 48, 143, 57, 123, 80, 111, 54, 161, 74, 129, 89, 119, 60, 223, 62, 127, 109, 127, 87, 217

Offset: 1

Leroy Quet, Apr 11 2002

nonn

Smallest k such there are exactly n integers among (1,2,3,4,...,k) relatively prime to n. - Benoit Cloitre, Jun 09 2002

			6 is relatively prime to 1, 5, 7, 11, 13, 17,..., the 6th term of this sequence being 17, so a(6) = 17.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Final term of n-th row of A077581.
Cf. A077582.
Cf. A247798, A247815.

a069213 = last . a077581_row  -- Reinhard Zumkeller, Sep 26 2014

f[n_] := Block[{c = 0, k = 1}, While[c < n, If[CoprimeQ[k, n], c++ ]; k++ ]; k - 1]; Array[f, 66] (* Robert G. Wilson v, Sep 10 2008 *)
Table[Position[CoprimeQ[Range[300],n],True,1,n][[-1]],{n,70}]//Flatten (* Harvey P. Dale, Aug 14 2020 *)

for(n=1,100,s=1; while(sum(i=1,s,if(gcd(n,i)-1,0,1))

a(p) = p+1, p is a prime, a(2^n)= 2^(n+1) - 1. What are a(pq), a(pqr), a(n) where n the product of first k primes? - Amarnath Murthy, Nov 14 2002

Let the remainder when n is divided by phi(n) be r and the quotient be k. I.e., n = k*phi(n) + r. Then k*n + r < a(n) < (k+1)*n. If the phi(n) numbers be arranged in increasing order and if the r-th number is m then a(n) = k*n + m. - Amarnath Murthy, Jul 07 2002

A126572 Array read by antidiagonals: a(n,m) = the m-th integer from among those positive integers coprime to n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 2, 5, 4, 1, 3, 4, 7, 5, 1, 2, 5, 5, 9, 6, 1, 5, 3, 7, 7, 11, 7, 1, 2, 7, 4, 9, 8, 13, 8, 1, 3, 3, 11, 6, 11, 10, 15, 9, 1, 2, 5, 4, 13, 7, 13, 11, 17, 10, 1, 3, 4, 7, 5, 17, 8, 15, 13, 19, 11, 1, 2, 7, 5, 9, 6, 19, 9, 17, 14, 21, 12, 1, 5, 3, 9, 7, 11, 8, 23, 11, 19, 16, 23, 13

Offset: 1

Leroy Quet, Dec 28 2006

From Rémy Sigrist, May 21 2017: (Start)
The n-th row only depends on the radical of n: a(n, m) = a(rad(n), m), where rad(n) = A007947(n).
The n-th row is linear: a(n, m + phi(rad(n))) = a(n, m) + rad(n), where phi(n) = A000010(n) and rad(n) = A007947(n).
(End)

			Array begins:
1,2,3,4,5,6,7,...
1,3,5,7,9,11,13,...
1,2,4,5,7,8,10,...
1,3,5,7,9,11,13,...
1,2,3,4,6,7,8,...
1,5,7,11,13,17,19,...
1,2,3,4,5,6,8,...
...

Cf. A000010, A007947, A126571, A077581.

f[m_, n_] := Block[{k = 0, c = n},While[c > 0,k++;While[GCD[k, m] > 1, k++ ];c--;];k];Flatten@Table[f[d - m + 1, m], {d, 13}, {m, d}] (* Ray Chandler, Dec 29 2006 *)

Extended by Ray Chandler, Dec 29 2006

A077664 Triangle in which the n-th row contains n smallest numbers greater than n and coprime to n.

Original entry on oeis.org

2, 3, 5, 4, 5, 7, 5, 7, 9, 11, 6, 7, 8, 9, 11, 7, 11, 13, 17, 19, 23, 8, 9, 10, 11, 12, 13, 15, 9, 11, 13, 15, 17, 19, 21, 23, 10, 11, 13, 14, 16, 17, 19, 20, 22, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47

Offset: 1

Amarnath Murthy, Nov 14 2002

A260910 gives the triangle of Frobenius numbers of n, T(n,k). - Reinhard Zumkeller, Aug 04 2015

			Triangle begins:
  2;
  3,  5;
  4,  5,  7;
  5,  7,  9, 11;
  6,  7,  8,  9, 11;
  7, 11, 13, 17, 19, 23;
  8,  9, 10, 11, 12, 13, 15;
  ...

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

Cf. A077665, A077666.

Cf. A077581, A260895 (number of primes per row), A260910.

a077664 n k = a077664_tabl !! (n-1) !! (k-1)
a077664_row n = a077664_tabl !! (n-1)
a077664_tabl = map (\x -> take x $ filter ((== 1). gcd x) [x + 1 ..]) [1..]
-- Reinhard Zumkeller, Aug 03 2015

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

from math import gcd
def arow(n):
    rown, k = [], n + 1
    while len(rown) < n:
        if gcd(k, n) == 1: rown.append(k)
        k += 1
    return rown
def agen(rows):
    for n in range(1, rows+1): yield from arow(n)
print([an for an in agen(12)]) # Michael S. Branicky, Sep 21 2021

More terms from Sascha Kurz, Jan 03 2003

A126571 Triangle where the m-th term in row n is the n-th integer from among those positive integers coprime to m.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 4, 7, 5, 7, 5, 9, 7, 9, 6, 6, 11, 8, 11, 7, 17, 7, 13, 10, 13, 8, 19, 8, 8, 15, 11, 15, 9, 23, 9, 15, 9, 17, 13, 17, 11, 25, 10, 17, 13, 10, 19, 14, 19, 12, 29, 11, 19, 14, 23, 11, 21, 16, 21, 13, 31, 12, 21, 16, 27, 12, 12, 23, 17, 23, 14, 35, 13, 23, 17, 29, 13, 35

Offset: 1

Leroy Quet, Dec 28 2006

			The fifth positive integer coprime to 1 is 5. The fifth positive integer coprime to 2 is 9. The fifth positive integer coprime to 3 is 7. The fifth positive integer coprime to 4 is 9. And the fifth positive integer coprime to 5 is 6. So row 5 of the triangle is (5,9,7,9,6).
From _Michael De Vlieger_, Aug 21 2017: (Start)
Triangle begins:
   1
   2    3
   3    5    4
   4    7    5    7
   5    9    7    9    6
   6   11    8   11    7   17
   7   13   10   13    8   19    8
   8   15   11   15    9   23    9   15
   9   17   13   17   11   25   10   17   13
  10   19   14   19   12   29   11   19   14   23
  11   21   16   21   13   31   12   21   16   27   12
  12   23   17   23   14   35   13   23   17   29   13   35
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).

Cf. A126572, A077581.

f[m_, n_] := Block[{k = 0, c = n},While[c > 0,k++;While[GCD[k, m] > 1, k++ ];c--;];k];Flatten@Table[f[m, n], {n, 12}, {m, n}] (* Ray Chandler, Dec 29 2006 *)

Extended by Ray Chandler, Dec 29 2006

A247798 n-th positive integer relatively prime to 2*n - 1.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 7, 14, 9, 10, 19, 12, 16, 20, 15, 16, 28, 26, 19, 32, 21, 22, 43, 24, 29, 41, 27, 38, 46, 30, 31, 55, 44, 34, 55, 36, 37, 71, 50, 40, 61, 42, 57, 68, 45, 58, 73, 63, 49, 82, 51, 52, 116, 54, 55, 86, 57, 76, 95, 74, 67, 95, 78, 64, 100, 66

Offset: 1

Reinhard Zumkeller, Sep 26 2014

nonn

Central terms of triangle A077581.

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

Cf. A077581, A069213.

Haskell
```
a247798 n = a077581 (2 * n - 1) n
```

cp's OEIS Frontend

A077582 Sum of terms of n-th row of A077581.

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A247815 Number of primes in n-th row of triangle A077581.

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A247892 Number of nonprimes in n-th row of triangle A077581.

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A126577 a(n) = numerator of the sum of reciprocals of the terms in n-th row of triangle A077581.

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A126578 a(n) = denominator of the sum of reciprocals of the terms in n-th row of triangle A077581.

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A069213 a(n) = n-th positive integer relatively prime to n.

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A126572 Array read by antidiagonals: a(n,m) = the m-th integer from among those positive integers coprime to n.

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A077664 Triangle in which the n-th row contains n smallest numbers greater than n and coprime to n.

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