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A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n)

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).

EXAMPLE

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.

MATHEMATICA

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

PROG

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

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.

Sequence in context: A124256 A108503 A331595 * A052131 A329486 A051145

Adjacent sequences:  A324210 A324211 A324212 * A324214 A324215 A324216

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified August 1 05:51 EDT 2021. Contains 346384 sequences. (Running on oeis4.)