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 1-5 of 5 results.

A319296 a(n) = (Sum_{d|n} sigma(d)) mod sigma(n).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 11, 5, 10, 1, 27, 1, 12, 11, 26, 1, 33, 1, 35, 13, 16, 1, 10, 7, 18, 18, 43, 1, 68, 1, 57, 17, 22, 15, 16, 1, 24, 19, 2, 1, 84, 1, 59, 48, 28, 1, 37, 9, 59, 23, 67, 1, 112, 19, 114, 25, 34, 1, 49, 1, 36, 58, 120, 21, 116, 1, 83, 29, 108
Offset: 1

Views

Author

Jaroslav Krizek, Sep 16 2018

Keywords

Examples

			For n = 4; a(4) = (sigma(1) + sigma(2) + sigma(4)) mod sigma(4) = (1+3+7) mod 7 = 11 mod 7 = 4.

Links

Crossrefs

Cf. A000040, A000203, A007429, A221219.

Programs

  • Magma
    [&+[SumOfDivisors(d): d in Divisors(n)] mod  SumOfDivisors(n): n in [1..1000]];
  • Mathematica
    Table[Mod[Sum[DivisorSigma[1, d], {d, Divisors[n]}], DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Sep 26 2018 *)
  • PARI
    A319296(n) = (sumdiv(n,d,sigma(d))%sigma(n)); \\ Antti Karttunen, Sep 16 2018

Formula

a(n) = A007429(n) mod A000203(n).
a(A221219(n)) = 0.
a(A000040(n)) = 1; the only composite number < 2*10^6 with a(n) = 1 is 636.
a(n) = n only for numbers 4, 10 and 96 < 3000000.

A322655 Numerator of (Sum_{d|n} sigma(d)) / sigma(n).

Original entry on oeis.org

1, 4, 5, 11, 7, 5, 9, 26, 18, 14, 13, 55, 15, 3, 35, 57, 19, 24, 21, 11, 45, 13, 25, 13, 38, 10, 29, 99, 31, 35, 33, 40, 65, 38, 21, 198, 39, 7, 75, 91, 43, 15, 45, 143, 21, 25, 49, 285, 22, 152, 95, 165, 55, 29, 91, 39, 21, 62, 61, 55, 63, 11, 81, 247, 5, 65
Offset: 1

Views

Author

Jaroslav Krizek, Dec 22 2018

Keywords

Comments

Numerator of A007429(n) / A000203(n).
Also numerator of Sum_{d|n} (sigma(d) / sigma(n)).

Examples

			For n = 4; a(4) = numerator((Sum_{d|4} sigma(d)) / sigma(4)) = numerator((1 + 3 + 7) / (1 + 2 + 4)) = numerator(11/7) = 11.

Links

Crossrefs

Cf. A000203, A007429, A319296, A221219, A322656 (denominator).

Programs

  • Magma
    [Numerator(&+[SumOfDivisors(d): d in Divisors(n)] /  SumOfDivisors(n)): n in [1..1000]];
  • Mathematica
    Table[Numerator[Sum[DivisorSigma[1, d], {d, Divisors[n]}] / DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 22 2018 *)
  • PARI
    a(n) = numerator(sumdiv(n, d, sigma(d))/sigma(n)); \\ Michel Marcus, Dec 22 2018

Formula

a(n) = A007429(n) / gcd(A000203(n), A007429(n)). - Antti Karttunen, Nov 15 2021

A322656 Denominator of (Sum_{d|n} sigma(d)) / sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 15, 13, 9, 12, 28, 14, 2, 24, 31, 18, 13, 20, 6, 32, 9, 24, 6, 31, 7, 20, 56, 30, 18, 32, 21, 48, 27, 16, 91, 38, 5, 56, 45, 42, 8, 44, 84, 13, 18, 48, 124, 19, 93, 72, 98, 54, 15, 72, 20, 16, 45, 60, 24, 62, 8, 52, 127, 4, 36, 68, 126, 96
Offset: 1

Views

Author

Jaroslav Krizek, Dec 22 2018

Keywords

Comments

Denominator of A007429(n) / A000203(n).
Also denominator of Sum_{d|n} (sigma(d) / sigma(n)).

Examples

			For n = 4; a(4) = denominator((Sum_{d|4} sigma(d)) / sigma(4)) = denominator((1 + 3 + 7) / (1 + 2 + 4)) = denominator(11/7) = 7.

Links

Crossrefs

Cf. A000203, A007429, A319296, A221219, A322655 (numerator).

Programs

  • Magma
    [Denominator(&+[SumOfDivisors(d): d in Divisors(n)] /  SumOfDivisors(n)): n in [1..1000]];
  • Mathematica
    Table[Denominator[Sum[DivisorSigma[1, d], {d, Divisors[n]}] / DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Dec 22 2018 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, sigma(d))/sigma(n)); \\ Michel Marcus, Dec 22 2018

Formula

a(n) = 1 for numbers in A221219.
a(n) = A000203(n) / gcd(A000203(n), A007429(n)). - Antti Karttunen, Nov 15 2021

A224488 Numbers m such that k*m = Sum_{j|m, j < m} sigma(j), where k >= 1 is an integer.

Original entry on oeis.org

4, 10, 42, 90, 2448, 4680, 27144, 117000, 154498509, 222970077, 11049088050, 63554826816
Offset: 1

Views

Author

Giovanni Resta, Apr 08 2013

Keywords

Comments

a(13) > 10^11.

Examples

			The divisors of 10 smaller than 10 are 1, 2 and 5. Since sigma(1) + sigma(2) + sigma(5) = 10, 10 is in the sequence.

Crossrefs

Cf. A211779, A221219.

Programs

  • Mathematica
    Select[Range[2, 120000], Mod[Total@DivisorSigma[1, Most@Divisors@#], #] == 0 &]
  • PARI
    isok(m) = sumdiv(m, d, if (d!=m, sigma(d))) % m == 0; \\ Michel Marcus, Jul 13 2021

Extensions

a(9)-a(12) and bound on a(13) from Donovan Johnson.

A243373 Numbers m such that k*phi(n) = Sum_{j|n} sigma(j), where k >= 1 is an integer.

Original entry on oeis.org

1, 2, 6, 9, 10, 14, 18, 26, 42, 66, 90, 126, 150, 186, 234, 266, 342, 490, 666, 1426, 1634, 2394, 4410, 12834, 14706, 16758, 18846, 209754, 308602, 350154, 385434, 1122786, 2777418, 12130734, 15616986, 29682342, 223843466, 270397974, 300398714, 559894482
Offset: 1

Views

Author

Paolo P. Lava, Jun 04 2014

Keywords

Comments

a(49) > 10^11. - Hiroaki Yamanouchi, Aug 24 2014

Examples

			The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 and sigma(1) + sigma(2) + sigma(3) + sigma(5) + sigma(6) + sigma(9) + sigma(10) + sigma(15) + sigma(18) + sigma(30) + sigma(45) + sigma(90) = 1 + 3 + 4 + 6 + 12 + 13 + 18 + 24 + 39 + 72 + 78 + 234 = 504 and phi(n) = 24. Finally 504 / 24 = 21.

Links

Crossrefs

Cf. A000010, A000203, A221219.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=divisors(n); b:=0;
b:=add(sigma(a[k]), k=1..nops(a)); if type(b/phi(n),integer)
then print(n); fi; od; end: P(10^10);
  • PARI
    isok(n) = (sumdiv(n, d, sigma(d)) % eulerphi(n)) == 0; \\ Michel Marcus, Jun 04 2014

Extensions

a(37)-a(40) from Hiroaki Yamanouchi, Aug 24 2014
Showing 1-5 of 5 results.

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