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A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

2, 4, 3, 8, 4, 2, 4, 16, 12, 9, 6, 14, 6, 12, 8, 32, 10, 26, 8, 21, 14, 18, 12, 20, 30, 16, 18, 2, 14, 24, 10, 64, 16, 24, 22, 88, 14, 30, 26, 36, 18, 32, 14, 42, 26, 28, 24, 54, 56, 80, 20, 32, 26, 40, 36, 60, 38, 42, 30, 56, 18, 42, 48, 128, 42, 48, 22, 50, 28, 72, 26, 122, 26, 54, 58, 46, 48, 56, 26, 86, 120, 60, 42, 96, 54

Offset: 1

Antti Karttunen and David A. Corneth, May 26 2019, with better name from Charlie Neder, Jun 02 2019

nonn

Number of ways to form the sum sigma(n) = x+y so that n-x and n-y are coprime, with x and y in range 0..sigma(n).
From Antti Karttunen, May 28 - Jun 08 2019: (Start)
Empirically, it seems that a(n) >= A034444(n) and also that a(n) >= A034444(A000203(n)) unless n is in A000396.
Specifically, if it could be proved that a(n) >= A034444(n)/2 for n >= 2, which in turn would imply that a(n) >= A001221(n) for all n, then we would know that no odd perfect numbers could exist. Note that a(n) must be 2 on all perfect numbers, whether even or odd. See also A325819.
(End)

			For n=1, sigma(1) = 1, both gcd(1-0, 1-(1-0)) = gcd(1,0) = 1 and gcd(1-1, 1-(1-1)) = gcd(0,1) = 1, thus a(1) = 2.
--
For n=3, sigma(3) = 4, we have 5 cases to consider:
  gcd(3-0, 3-(4-0)) = 1 = gcd(3-4, 3-(4-4)),
  gcd(3-1, 3-(4-1)) = 2 = gcd(3-3, 3-(4-3)),
  gcd(3-2, 3-(4-2)) = 1,
of which three cases give 1 as a result, thus a(3) = 3.
--
For n=6, sigma(6) = 12, we have 13 cases to consider:
  gcd(6-0, 6-(12-0)) = 6 = gcd(6-12, 6-(12-12)),
  gcd(6-1, 6-(12-1)) = 5 = gcd(6-11, 6-(12-11)),
  gcd(6-2, 6-(12-2)) = 4 = gcd(6-10, 6-(12-10)),
  gcd(6-3, 6-(12-3)) = 3 = gcd(6-9, 6-(12-9)),
  gcd(6-4, 6-(12-4)) = 2 = gcd(6-8, 6-(12-8))
  gcd(6-5, 6-(12-5)) = 1 = gcd(6-7, 6-(12-7)),
  gcd(6-6, 6-(12-6)) = 0,
of which only two give 1 as a result, thus a(6) = 2.
--
For n=10, sigma(10) = 18, we have 19 cases to consider:
  gcd(10-0, 10-(18-0)) = 2 = gcd(10-18, 10-(18-18)),
  gcd(10-1, 10-(18-1)) = 1 = gcd(10-17, 10-(18-17)),
  gcd(10-2, 10-(18-2)) = 2 = gcd(10-16, 10-(18-16)),
  gcd(10-3, 10-(18-3)) = 1 = gcd(10-15, 10-(18-15)),
  gcd(10-4, 10-(18-4)) = 2 = gcd(10-14, 10-(18-14)),
  gcd(10-5, 10-(18-5)) = 1 = gcd(10-13, 10-(18-13)),
  gcd(10-6, 10-(18-6)) = 2 = gcd(10-12, 10-(18-12)),
  gcd(10-7, 10-(18-7)) = 1 = gcd(10-11, 10-(18-11)),
  gcd(10-8, 10-(18-8)) = 2 = gcd(10-10, 10-(18-10)),
  gcd(10-9, 10-(18-9)) = 1,
of which 9 cases give 1 as a result, thus a(10) = 9.
--
For n=15, sigma(15) = 24, we have 25 cases to consider:
  gcd(15-0, 15-(24-0)) = 3 = gcd(15-24, 15-(24-24)),
  gcd(15-1, 15-(24-1)) = 2 = gcd(15-23, 15-(24-23)),
  gcd(15-2, 15-(24-2)) = 1 = gcd(15-22, 15-(24-22)),
  gcd(15-3, 15-(24-3)) = 6 = gcd(15-21, 15-(24-21)),
  gcd(15-4, 15-(24-4)) = 1 = gcd(15-20, 15-(24-20)),
  gcd(15-5, 15-(24-5)) = 2 = gcd(15-19, 15-(24-19)),
  gcd(15-6, 15-(24-6)) = 3 = gcd(15-18, 15-(24-18)),
  gcd(15-7, 15-(24-7)) = 2 = gcd(15-17, 15-(24-17)),
  gcd(15-8, 15-(24-8)) = 1 = gcd(15-16, 15-(24-16)),
  gcd(15-9, 15-(24-9)) = 6 = gcd(15-15, 15-(24-15)),
  gcd(15-10, 15-(24-10)) = 1 = gcd(15-14, 15-(24-14)),
  gcd(15-11, 15-(24-11)) = 2 = gcd(15-13, 15-(24-13)),
  gcd(15-12, 15-(24-12)) = 3,
of which 2*4 = 8 cases give 1 as a result, thus a(15) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000010, A000203, A000396, A001221, A014567, A034444, A058062, A062401, A228058, A325807, A325815, A325816, A325817, A325818, A325819, A325960, A325961, A325962, A325965, A325966, A325967, A325968, A325970, A325971, A325972.

Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* Michael De Vlieger, Jun 09 2019 *)

A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };

a(n) = Sum_{i=0..sigma(n)} [1 == gcd(n-i,n-(sigma(n)-i))], where [ ] is the Iverson bracket and sigma(n) is A000203(n).
a(A000396(n)) = 2.
a(n) = A325815(n) + A034444(n).
a(n) = 1+A000203(n) - A325816(n).
a(A228058(n)) = A325819(n).

A157989 The numbers of the jingle for a popular Ontario pizza chain's phone number.

Original entry on oeis.org

9, 6, 7, 11, 11

Offset: 1

Dave [Last name?] (dave(AT)endicott.net), Mar 10 2009

Sung as: "nine, six, seven, eleven, eleven, .... ".

Jingle first introduced in 1978 and has since become a pop culture item for Canadians.

Pizza-Pizza, History of the Enterprise.
Wikipedia, Pizza Pizza.
Index entries for sequences related to songs

Cf. A104175.

A166067 Fibonacci sequence beginning 1, 69.

Original entry on oeis.org

1, 69, 70, 139, 209, 348, 557, 905, 1462, 2367, 3829, 6196, 10025, 16221, 26246, 42467, 68713, 111180, 179893, 291073, 470966, 762039, 1233005, 1995044, 3228049, 5223093, 8451142, 13674235, 22125377, 35799612, 57924989, 93724601, 151649590

Offset: 0

Submitted Oct 06 2009 by someone who later asked to have his name removed from the Author line

nonn

G. C. Greubel, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (1, 1).

CoefficientList[Series[(-68x-1)/(x^2+x-1),{x,0,40}],x]   (* Harvey P. Dale, Mar 14 2011 *)

a(n) = a(n-1) + a(n-2).
G.f.: (1+68*x)/(1-x-x^2). - Philippe Deléham, Oct 06 2009
a(n) = 68*A000045(n) + A000045(n+1). - R. J. Mathar, Oct 08 2009

A165134 Number of directed Hamiltonian paths in the n X n knight graph.

Original entry on oeis.org

1, 0, 0, 0, 1728, 6637920, 165575218320, 19591828170979904

Offset: 1

[No name given] (c.candide(AT)free.fr), Sep 04 2009

Previous name was: Number of knight's paths visiting each square of an n X n chessboard exactly once.

			From _Gheorghe Coserea_, Oct 08 2016: (Start)
For n=5 the numbers in the table below give the number of knight's paths starting at the respective position on the 5 X 5 chessboard.  In total there are a(5) = 304*4 + 56*8 + 64 = 1728 solutions.
    [1]  [2]  [3]  [4]  [5]
[1] 304    0   56    0  304
[2]   0   56    0   56    0
[3]  56    0   64    0   56
[4]   0   56    0   56    0
[5] 304    0   56    0  304
(End)

Stefan Behnel, The Knight's Paths
A. Chernov, Open knight's tours
Gheorghe Coserea, Solutions for 5x5 chessboard
P. Hingston, G. Kendall, Enumerating knight's tours using an ant colony algorithm, The 2005 IEEE Congress on Evolutionary Computation, 2 (2006), 1003-1010
G. Stertenbrink, Number of Knight's Tours
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

Cf. Undirected Hamiltonian paths: A169696 (3 X n), A079137 (4 X n), A083386 (5 X n), A306281 (6 X n), A306283 (7 X n), A308131 (n X n).

Cf. A001230, A118067, A306282.

a(7) from Guenter Stertenbrink, added by Alex Chernov, Sep 01 2013
a(1)=1, a(2)=0 prepended by Max Alekseyev, Sep 22 2013
a(8) from Alex Chernov, May 10 2014
Name made more precise by Eric W. Weisstein, Apr 14 2019

A166050 a(n) = Sum_{i=0..(2n+1)} J(i,12n+7), where J(i,k) is the Jacobi symbol.

Original entry on oeis.org

1, -1, 3, -1, 4, -1, 5, -2, 5, -2, 5, -3, 7, -1, 2, -2, 9, -3, 7, -2, 6, -4, 11, -3, 8, -3, 10, -3, 1, -4, 9, -3, 14, -2, 10, -2, 15, -6, 7, 0, 7, -3, 14, -5, 14, -3, 16, -5, 8, -4, 13, -5, 13, -3, 12, -4, 18, -5, 14, -4, 13, -5, 15, -4, 0, -5, 16, -7, 9, -6, 0, -7, 22, -3, 16, -3

Offset: 0

Antti Karttunen, Oct 13 2009. Erroneous name corrected Oct 20 2009

sign

The height at the 1/6 point of "Jacobi-bridge/path", computed for each odd integer of the form 12n+7.

A. Karttunen, Table of n, a(n) for n = 0..131071
Wikipedia, Jacobi symbol

Bisections: A166268, A166269 (see conjectures there). Cf. A017605. Scheme-code for jacobi-symbol is given at A165601.

A166100 Sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Original entry on oeis.org

1, 1, 5, 7, 27, 22, 39, 15, 68, 76, 63, 92, 250, 117, 203, 186, 165, 175, 333, 156, 410, 430, 270, 423, 1029, 357, 689, 440, 513, 767, 915, 504, 780, 1072, 759, 994, 1314, 725, 1155, 1343, 2187, 1577, 1360, 957, 1958, 1547, 1395, 1330, 2328, 1485, 2525

Offset: 0

Antti Karttunen, Oct 13 2009. Erroneous name corrected Oct 20 2009

nonn

Note that this sequence is not equal to the sum of the quadratic residues of 2n+1 in range [1,2n+1], and thus NOT a bisection of A165898.

			For n=5, we get odd number 11 (2*5+1), and J(i,11) = 1,-1,1,1,1,-1,-1,-1,1,-1,0 when i ranges from 1 to 11, J(i,11) getting value 1 when i=1, 3, 4, 5 and 9, thus a(5)=22.

A. Karttunen, Table of n, a(n) for n = 0..131071
Wikipedia, Jacobi symbol

A076409(n)=a(A102781(n)). Cf. A166101-A166104, A166405, A166406.

Scheme-code for jacobi-symbol is given at A165601.

Table[Total[Flatten[Position[JacobiSymbol[Range[2n+1],2n+1],1]]],{n,0,50}] (* Harvey P. Dale, Jun 19 2013 *)

from sympy import jacobi_symbol as J
def a(n): return sum([i for i in range(1, 2*n + 2) if J(i, 2*n + 1)==1]) # Indranil Ghosh, Jun 12 2017

A155154 Number of right triangles with nonnegative integer coordinates less than or equal to n and one corner at the origin.

Original entry on oeis.org

0, 3, 14, 33, 62, 101, 148, 207, 276, 353, 448, 547, 662, 785, 920, 1067, 1230, 1397, 1586, 1773, 1986, 2205, 2436, 2671, 2940, 3207, 3494, 3787, 4096, 4399, 4764, 5095, 5458, 5825, 6208, 6603, 7050, 7453, 7880, 8331, 8826, 9277, 9800, 10279, 10808

Offset: 0

Emlyn Corrin (emlyn(AT)corrin.name), Jan 21 2009

nonn

Project Euler, Problem 91: Right triangles with integer coordinates.

a(n)=3*n^2+sum(a=1, n, sum(b = 1, n, 2*min(b*gcd(a, b)\a, (n - a)*gcd(a, b)\b) ) ) \\ Yurii Ivanov, Jun 25 2021

A121049 Let p_n be the polynomial of degree n-1 that interpolates the first n primes (i.e., p_n(i) = prime(i) for 1 <= i <= n.) Then a(n) = p_n(n+1)/2.

Original entry on oeis.org

1, 2, 4, 4, 11, -3, 36, -46, 133, -213, 419, -586, 716, -199, -1807, 7570, -20637, 47563, -97849, 185438, -326192, 531721, -785058, 980926, -780084, -700944, 5511613, -18000159, 46704269, -107137804, 225187101, -439627178, 799622938, -1347732434, 2069035230

Offset: 1

Joseph Van Name (prism720(AT)yahoo.com), Aug 08 2006

As n approaches infinity, |a(n)|^(1/n) converges to 2, but a(n+1)/a(n) does not appear to converge.

			p_3(x) = (x^2-x+4)/2. p_3(1) = 2, p_3(2) = 3, p_3(3) = 5, so
a(3) = p_3(4)/2 = 4.

Table[ Sum[(-1)^(j + r)Prime[j] Binomial[r, j - 1]/2, {j, r}], {r, 50}]

a(n) = Sum_{j=1..n} (-1)^(j+n)*prime(j)*binomial(n,j-1)/2.

Edited and extended by David Wasserman, Aug 16 2006

Corrected by N. J. A. Sloane, Oct 29 2006

A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 24, 25, 26, 27, 35, 36, 38, 39, 60, 62, 63, 65, 84, 85, 86, 87, 91, 95, 132, 133, 134, 135, 143, 161, 205, 215, 355, 377, 445, 451, 455, 493, 1651, 1673, 1885, 1891, 1895, 8797

Offset: 1

Walter Kehowski, Sep 14 2005; name corrected Sep 18 2005

nonn

Maple worksheet available upon request. Here is the sequence of minimal composites in base 12, where X is 10 and E is 11. 4, 6, 8, 9, X, 10, 12, 13, 20, 21, 22, 23, 2E, 30, 32, 33, 50, 52, 53, 55, 70, 71, 72, 73, 77, 7E, E0, E1, E2, E3, EE, 115, 151, 15E, 257, 275, 311, 317, 31E, 351, E57, E75, 1111, 1117, 111E, 5111.

			a(13)=35=2E since no earlier composite is of the form "*2*E*". The list of minimal composites can be constructed using a sieve-like process: subsequently eliminate from the list of composites all composites of the form "*2*E*". Assuming all previous terms have been similarly determined, then the next remaining composite should be 30.

Cf. A071062, A071070.

A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185

Offset: 0

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

nonn

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).

On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials
Index entries for sequences that are permutations of the natural numbers

Inverse: A091205.
Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.
Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).
As a composition of related permutations:
a(n) = A245703(A245822(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]
For all n >= 1, the following holds:
A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]

cp's OEIS Frontend

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A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

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A157989 The numbers of the jingle for a popular Ontario pizza chain's phone number.

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A166067 Fibonacci sequence beginning 1, 69.

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A165134 Number of directed Hamiltonian paths in the n X n knight graph.

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A166050 a(n) = Sum_{i=0..(2n+1)} J(i,12n+7), where J(i,k) is the Jacobi symbol.

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A166100 Sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

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A155154 Number of right triangles with nonnegative integer coordinates less than or equal to n and one corner at the origin.

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A121049 Let p_n be the polynomial of degree n-1 that interpolates the first n primes (i.e., p_n(i) = prime(i) for 1 <= i <= n.) Then a(n) = p_n(n+1)/2.

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A110615 Minimal set of composite-strings in base 12 in the sense of A071070.

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