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 1-6 of 6 results.

A334420 Numbers m such that sigma(d)/tau(d) is an integer for all divisors d of m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145
Offset: 1

Views

Author

Jaroslav Krizek, Apr 29 2020

Keywords

Comments

Sequences of numbers m from this sequence with k such divisors for 1 < k < 6:
k = 2: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... (A065091 - odd primes).
k = 3: 49, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 9409, ...
k = 4: 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, ...
k = 5: 923521, 13845841, 519885601, 1073283121, 1982119441, ...
See A334421 for sequence of the smallest numbers m with k such divisors.
All divisors of a member of the sequence are members of the sequence. - Robert Israel, May 01 2020
Numbers for which all divisors are in A003601. - Michel Marcus, May 02 2020

Examples

			Number 15 with divisors 1, 3, 5 and 15 is a term because sigma(1)/tau(1) = 1/1 = 1, sigma(3)/tau(3) = 4/2 = 2, sigma(5)/tau(5) = 6/2 = 3, sigma(15)/tau(15) = 24/4 = 6.

Links

Crossrefs

Subsequence of A306639.
Cf. A000005 (tau), A000203 (sigma), A003601, A324499, A324500, A334421.
Includes A056911.

Programs

  • Magma
    [m: m in [1..10^6] | &+[SumOfDivisors(d) mod NumberOfDivisors(d): d in Divisors(m)] eq 0];
  • Maple
    filter:= n -> andmap(d -> numtheory:-sigma(d) mod numtheory:-tau(d)=0, numtheory:-divisors(n)):
select(filter, [$1..200]); # Robert Israel, May 01 2020
  • Mathematica
    divQ[n_] := Divisible[DivisorSigma[1, n], DivisorSigma[0, n]]; Select[Range[150], AllTrue[Divisors[#], divQ] &] (* Amiram Eldar, Apr 29 2020 *)
  • PARI
    isok(m) = fordiv(m, d, if (sigma(d) % numdiv(d), return (0))); return(1); \\ Michel Marcus, Apr 29 2020

Formula

A324500(a(n)) = 1.

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

Original entry on oeis.org

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

Views

Author

Jaroslav Krizek, Oct 14 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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).

Programs

  • Magma
    [#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];
  • Mathematica
    a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)
  • PARI
    a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

Formula

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).

Extensions

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

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.

A359261 a(n) is the least term of A359260 whose number of divisors is n.

Original entry on oeis.org

1, 3, 49, 15, 923521, 1519, 88245939632761, 3913, 1117249, 3131659711, 4345096786921664259621718196367601, 238483, 9024585590445680759701490904755712009585829774768244676951841, 2772760313554466311, 198528059518891985825881, 32748812641
Offset: 1

Views

Author

Amiram Eldar, Dec 23 2022

Keywords

Comments

a(n) is the least number m whose number of divisors is A000005(m) = n such that the arithmetic mean of the first k divisors of m is an integer for all k in 1..n.
a(17) = 4084081^16 = 5.991...*10^105 is too large to include in the data section.
a(n) exists for all n >= 1. For n > 1, consider a prime p of the form m*lcm(1,2,...n-1) + 1, with m >= 1. Such a prime exists by Dirichlet's theorem on arithmetic progressions. Then, p^(n-1) has n divisors, and p^k == 1 (mod lcm(1..n-1)) for k = 0..(n-1). Therefore, Sum_{k=0..n-1} p^k == k (mod lcm(1,2,...n-1)), or equivalently, Sum_{k=0..n-1} p^k is divisible by k for k = 0..(n-1). Thus, p^(n-1) is in A359260.

Examples

			a(3) = 49 since 49 is the least number with 3 divisors in A359260. Its divisors are {1, 7, 49}, 1/1 = 1, (1+7)/2 = 4, and (1+7+49)/3 = 19 are all integers.

Crossrefs

Cf. A000005, A003418, A359260, A359262.
Similar sequence: A334421.
Showing 1-6 of 6 results.

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