cp's OEIS Frontend

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.

Showing 1-7 of 7 results.

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).

A325817 a(n) is the least k >= 0 such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 27, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 0, 0, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 5, 0, 0, 2, 0, 0, 1, 0, 1, 2
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Comments

a(n) is the least k >= 0 such that -n + k and (n-sigma(n))+k are coprime.

Examples

			For n=15, gcd(15-0, 15-(24-0)) = 3, gcd(15-1, 15-(24-1)) = 2 and gcd(15-2, 15-(24-2)) = 1, thus a(15) = 2.

Links

Crossrefs

Cf. A000203, A000396, A001065, A009194, A014567 (positions of zeros), A324213, A325818, A325826, A325962, A325965, A325967, A325976.

Programs

  • Mathematica
    lk[n_]:=Module[{k=0},While[!CoprimeQ[n-k,n-(DivisorSigma[1,n]-k)],k++];k]; Array[lk,110] (* Harvey P. Dale, Nov 24 2024 *)
  • PARI
    A325817(n) = { my(s=sigma(n)); for(k=0, s, if(1==gcd(-n + k, (n-s)+k), return(k))); };
  • PARI
    A325817(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(i))); };

Formula

a(n) = A000203(n) - A325818(n) = A001065(n) - A325826(n) = n - A325976(n).
For all n:
a(A000396(n)) = A000396(n)-1.
a(n) <= n-1.
a(n) <= A325965(n).
a(n) <= A325967(n).

A325961 a(n) is the least k >= A061228(n)-1 such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

Original entry on oeis.org

1, 3, 0, 5, 0, 7, 0, 9, 11, 11, 0, 13, 0, 15, 20, 17, 0, 19, 0, 21, 24, 23, 0, 25, 29, 27, 30, 29, 0, 31, 0, 33, 38, 35, 40, 37, 0, 39, 42, 41, 0, 43, 0, 45, 50, 47, 0, 49, 55, 51, 58, 53, 0, 55, 60, 57, 60, 59, 0, 61, 0, 63, 66, 65, 70, 67, 0, 69, 74, 71, 0, 73, 0, 75, 78, 77, 84, 79, 0, 81, 83, 83, 0, 85, 90, 87, 92, 89, 0, 91, 100
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Comments

a(n) attains the value of A325818(n) only with n = 1, 2 and the even terms of A000396. Note that A000203(n) > ((n+A020639(n))-1) with composite n.

Links

Crossrefs

Cf. A000203, A000396, A020639, A061228, A324213, A325817, A325818, A325960, A325962, A325965, A325966.

Programs

  • PARI
    A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325961(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); (0); };

Formula

a(n) = 0 if and only if n is either an odd prime or an odd perfect number, but if n is neither, then a(n) = 2n - A325962(n).

A325966 a(n) is the largest i <= sigma(n)-A020639(n) such that n-i and n-(sigma(n)-i) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 7, 0, 13, 10, 15, 0, 25, 0, 21, 20, 29, 0, 37, 0, 39, 28, 33, 0, 55, 26, 39, 36, 29, 0, 67, 0, 61, 44, 51, 42, 89, 0, 57, 52, 87, 0, 91, 0, 81, 74, 69, 0, 121, 50, 91, 68, 95, 0, 115, 66, 117, 76, 87, 0, 163, 0, 93, 100, 125, 78, 139, 0, 121, 92, 141, 0, 193, 0, 111, 120, 137, 88, 163, 0, 183, 118, 123, 0
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Links

Crossrefs

Cf. A000203, A000396, A020639, A324213, A325818, A325961, A325962, A325965.

Programs

  • PARI
    A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325966(n) = { my(s=sigma(n)); forstep(i=s-A020639(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

Formula

a(n) = A000203(n) - A325965(n).
For all n:
a(A000396(n)) = A000396(n)+1.
a(n) <= A325818(n).

A325962 a(1) = 1; for n > 1, a(n) is the largest k <= 1+A046666(n) such that n-k and n-(sigma(n)-k) are relatively prime, or -1 if no such nonnegative k exists.

Original entry on oeis.org

1, 1, 0, 3, 0, 5, 0, 7, 7, 9, 0, 11, 0, 13, 10, 15, 0, 17, 0, 19, 18, 21, 0, 23, 21, 25, 24, 27, 0, 29, 0, 31, 28, 33, 30, 35, 0, 37, 36, 39, 0, 41, 0, 43, 40, 45, 0, 47, 43, 49, 44, 51, 0, 53, 50, 55, 54, 57, 0, 59, 0, 61, 60, 63, 60, 65, 0, 67, 64, 69, 0, 71, 0, 73, 72, 75, 70, 77, 0, 79, 79, 81, 0, 83, 80, 85, 82, 87, 0, 89, 82
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Comments

a(n) is equal to A325817(n) only with odd primes and the even terms of A000396. a(n) = -1 only on odd perfect numbers, if such numbers exist. Otherwise a(n) = 2n - A325961(n).

Links

Crossrefs

Cf. A000203, A000396, A020639, A046666, A065091, A324213, A325817, A325818, A325960, A325961, A325965, A325966.

Programs

  • PARI
    A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325962(n) = { my(s=sigma(n)); forstep(i=1+n-A020639(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); (-1); };

Formula

For all n, a(A065091(n)) = 0.

A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 0, 1, 4, 1, 3, 1, 0, 1, 0, 1, 5, 1, 5, 1, 0, 1, 3, 1, 0, 1, 0, 1, 5, 1, 0, 1, 6, 1, 7, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 1, 3, 1, 5, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 7, 1, 0, 1, 2, 1, 0, 1, 5, 1, 5, 1, 0, 1, 9, 1, 3, 1, 9, 1, 0, 1, 5, 1, 0, 1, 0, 1, 5
Offset: 1

Views

Author

Antti Karttunen, May 29 2019

Keywords

Comments

By definition, if n is neither an odd prime nor an odd perfect number, then a(n) >= (A020639(n)-1).

Links

Crossrefs

Cf. A000203, A000396, A020639, A324213, A325817, A325818, A325826, A325961, A325962, A325965, A325966, A325970.
Cf. A006005 (positions of zeros, provided no odd perfect numbers exist).

Programs

  • PARI
    A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325960(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i-n))); (0); };

Formula

a(n) = (A325961(n) - A325962(n)) / 2, assuming no odd perfect numbers exist.
a(2n) = 1.

A325971 a(n) is the least k >= A007947(n) such that -n + k and (n-sigma(n))+k are relatively prime.

Original entry on oeis.org

1, 2, 4, 2, 6, 7, 8, 2, 3, 11, 12, 7, 14, 15, 16, 2, 18, 7, 20, 11, 22, 23, 24, 7, 5, 27, 4, 27, 30, 31, 32, 2, 34, 35, 36, 6, 38, 39, 40, 11, 42, 43, 44, 23, 16, 47, 48, 7, 7, 10, 52, 27, 54, 7, 56, 15, 58, 59, 60, 31, 62, 63, 22, 2, 66, 67, 68, 35, 70, 71, 72, 7, 74, 75, 16, 39, 78, 79, 80, 11, 3, 83, 84, 43, 86, 87, 88, 23, 90, 31
Offset: 1

Views

Author

Antti Karttunen, May 31 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A000203, A007947, A324213, A325817, A325961, A325965, A325970, A325972.

Programs

  • PARI
    A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325971(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); (0); };
A325971(n) = { my(s=sigma(n)); for(k=A007947(n), s, if(1==gcd(-n + k, (n-sigma(n))+k), return(k))); };

Formula

a(n) = A000203(n) - A325972(n).
a(n) = n - A325970(n).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.