A338171 -id:A338171 - cp's OEIS frontend

A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 2, 3, 3, 2, 6, 2, 1, 4, 2, 4, 3, 2, 3, 4, 3, 2, 7, 2, 4, 5, 3, 2, 3, 3, 2, 4, 2, 2, 5, 4, 4, 4, 2, 2, 8, 2, 3, 4, 1, 4, 7, 2, 3, 4, 6, 2, 3, 2, 2, 4, 3, 4, 6, 2, 3, 3, 2, 2, 7, 4, 3, 4, 4, 2, 7, 4, 4, 4, 3, 4, 4, 2, 4, 5, 3, 2, 6, 2, 2, 8

Offset: 1

Jaroslav Krizek, Oct 14 2020

nonn

a(n) is the number of arithmetic divisors d of n.
a(n) = tau(n) = A000005(n) for numbers n from A334420.
See A338171 and A338172 for sum and product such divisors.
a(n) = 1 iff n = 2^k (A000079). - Bernard Schott, Dec 06 2020

			a(6) = 3 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.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Inverse Möbius transform of A245656.
Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334420, A334421.
Cf. A337326 (smallest numbers m with n such divisors).

[#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

a(n) = Sum_{d|n} c(d), where c(n) is the arithmetic characteristic of n (A245656).

a(p) = 2 for odd primes p (A065091).

Data section extended up to 105 terms by Antti Karttunen, Dec 12 2021

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

Jaroslav Krizek, Oct 14 2020

nonn

a(n) is the product of arithmetic divisors d of n.

a(n) = pod(n) = A007955(n) for numbers n from A334420.

			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.

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

[&*[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

a[n_] := Times @@ Select[Divisors[n],  Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

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

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

Jaroslav Krizek, Oct 20 2020

nonn

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.

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).

[Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(&+Divisors(d) / #Divisors(d))] eq n]): n in [1..30]];

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 *)

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

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.

cp's OEIS Frontend

A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

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

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

Formula