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 1-3 of 3 results.

A338171 a(n) is the sum of those divisors d of n such that tau(d) divides sigma(d).

Original entry on oeis.org

1, 1, 4, 1, 6, 10, 8, 1, 4, 6, 12, 10, 14, 22, 24, 1, 18, 10, 20, 26, 32, 34, 24, 10, 6, 14, 31, 22, 30, 60, 32, 1, 48, 18, 48, 10, 38, 58, 56, 26, 42, 94, 44, 78, 69, 70, 48, 10, 57, 6, 72, 14, 54, 91, 72, 78, 80, 30, 60, 140, 62, 94, 32, 1, 84, 142, 68, 86
Offset: 1

Views

Author

Jaroslav Krizek, Oct 14 2020

Keywords

Comments

a(n) is the sum of arithmetic divisors d of n.
a(n) = sigma(n) = A000203(n) for numbers n from A334420.
See A338170 and A338172 for number and product such divisors.

Examples

			a(6) = 10 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Sum of this divisors is 10.

Links

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334420, A334421.

Programs

  • Magma
    [&+[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];
  • Maple
    f:= proc(n) uses numtheory;
convert(select(t -> sigma(t) mod tau(t) = 0, divisors(n)),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Oct 27 2020
  • Mathematica
    a[n_] := DivisorSum[n, # &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)
  • PARI
    a(n) = sumdiv(n, d, d*!(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

Formula

a(p) = p + 1 for odd primes p (A065091).

A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).

Original entry on oeis.org

1, 1, 3, 1, 5, 18, 7, 1, 3, 5, 11, 18, 13, 98, 225, 1, 17, 18, 19, 100, 441, 242, 23, 18, 5, 13, 81, 98, 29, 40500, 31, 1, 1089, 17, 1225, 18, 37, 722, 1521, 100, 41, 1555848, 43, 10648, 10125, 1058, 47, 18, 343, 5, 2601, 13, 53, 26244, 3025, 5488, 3249, 29
Offset: 1

Views

Author

Jaroslav Krizek, Oct 14 2020

Keywords

Comments

a(n) is the product of arithmetic divisors d of n.
a(n) = pod(n) = A007955(n) for numbers n from A334420.

Examples

			a(6) = 18 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Product of this divisors is 18.

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334420, A334421.
See A338170 and A338171 for number and sum of such divisors.

Programs

  • Magma
    [&*[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];
  • Mathematica
    a[n_] := Times @@ Select[Divisors[n],  Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, if (sigma(d[k]) % numdiv(d[k]), 1, d[k])); \\ Michel Marcus, Oct 15 2020

Formula

a(p) = p for odd primes p (A065091).

A337326 a(n) is the smallest number with n divisors d such that sigma(d) / tau(d) is an integer.

Original entry on oeis.org

1, 3, 6, 15, 45, 30, 42, 60, 132, 264, 270, 378, 594, 210, 462, 780, 1050, 420, 924, 660, 2100, 840, 3060, 1848, 3300, 1890, 2970, 2520, 9702, 2310, 5544, 3780, 11592, 8316, 18216, 5460, 5940, 7980, 16830, 7140, 11550, 4620, 21252, 10920, 23760, 22440, 49500
Offset: 1

Views

Author

Jaroslav Krizek, Oct 20 2020

Keywords

Comments

a(n) is the smallest number m with n arithmetic divisors d (terms of A003601).
See A338170, A338171 and A338172 for number, sum and product of such divisors for n>=1.

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334421 (smallest number with n divisors d such that sigma(d)/tau(d) is an integer for all divisors).

Programs

  • Magma
    [Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(&+Divisors(d) / #Divisors(d))] eq n]): n in [1..30]];
  • Mathematica
    f[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; m = 50; s = Table[0, {m}]; c = 0; n = 1; While[c < m, If[(i = f[n]) <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 21 2020 *)
  • PARI
    isok(m, n) = sumdiv(m, d, !(sigma(d) % numdiv(d))) == n;
a(n) = my(m=1); while(!isok(m,n), m++); m; \\ Michel Marcus, Oct 21 2020

Formula

a(3) = 6 because number 6 is the smallest number with 3 such divisors (1, 3 and 6): sigma(1) / tau(1) = 1 / 1 = 1; sigma(3) / tau(3) = 4 / 2 = 2; sigma(6) / tau(6) = 12 / 4 = 3.
Showing 1-3 of 3 results.

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

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