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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* Michael De Vlieger, Jun 09 2019 *)
  • PARI
    A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };

Formula

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).

A325970 a(n) is the largest k <= A066503(n) such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 0, -1, 2, -1, -1, -1, 6, 6, -1, -1, 5, -1, -1, -1, 14, -1, 11, -1, 9, -1, -1, -1, 17, 20, -1, 23, 1, -1, -1, -1, 30, -1, -1, -1, 30, -1, -1, -1, 29, -1, -1, -1, 21, 29, -1, -1, 41, 42, 40, -1, 25, -1, 47, -1, 41, -1, -1, -1, 29, -1, -1, 41, 62, -1, -1, -1, 33, -1, -1, -1, 65, -1, -1, 59, 37, -1, -1, -1, 69, 78
Offset: 1

Views

Author

Antti Karttunen, May 31 2019

Keywords

Comments

a(n) = n-k for the least k >= A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Links

Crossrefs

Cf. A000203, A005117, A007947, A033879, A066503, A324213, A325826, A325960, A325967, A325971, A325972.

Programs

  • PARI
    A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325970(n) = { my(s=sigma(n)); forstep(k=n-A007947(n), -oo, -1, if(1==gcd(k, n+n-sigma(n)), return(k))); };
\\ Or alternatively:
A325970(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(n-i))); };

Formula

a(n) = n - A325971(n).
For n >= 3, a(A005117(n)) = -1.

A325972 a(n) is the largest k <= sigma(n)-A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 5, 0, 13, 10, 7, 0, 21, 0, 9, 8, 29, 0, 32, 0, 31, 10, 13, 0, 53, 26, 15, 36, 29, 0, 41, 0, 61, 14, 19, 12, 85, 0, 21, 16, 79, 0, 53, 0, 61, 62, 25, 0, 117, 50, 83, 20, 71, 0, 113, 16, 105, 22, 31, 0, 137, 0, 33, 82, 125, 18, 77, 0, 91, 26, 73, 0, 188, 0, 39, 108, 101, 18, 89, 0, 175, 118, 43, 0, 181, 22
Offset: 1

Views

Author

Antti Karttunen, May 31 2019

Keywords

Links

Crossrefs

Cf. A000203, A007947, A324213, A325818, A325962, A325966, A325970, A325971.

Programs

  • PARI
    A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325972(n) = { my(s=sigma(n)); forstep(i=s-A007947(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

Formula

a(n) = A000203(n) - A325971(n).
Showing 1-3 of 3 results.

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