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 1-10 of 10 results.

A336040 Characteristic function of refactorable numbers (A033950).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 07 2020

Keywords

Examples

			a(1) = 1 since d(1) = 1 and 1 divides 1.
a(2) = 1 since d(2) = 2 and 2 divides 2.
a(3) = 0 since d(3) = 2, but 2 does not divide 3.

Links

Crossrefs

Cf. A000005, A033950, A054008, A336041 (inverse Möbius transform), A335182, A335665, A349322.

Programs

  • Mathematica
    a[n_] := Boole @ Divisible[n, DivisorSigma[0, n]]; Array[a, 100] (* Amiram Eldar, Jul 08 2020 *)
  • PARI
    a(n) = n%numdiv(n) == 0; \\ Michel Marcus, Jul 07 2020

Formula

a(n) = 1 - ceiling(n/d(n)) + floor(n/d(n)), where d(n) is the number of divisors of n (A000005).
a(n) = [A054008(n) == 0], where [ ] is the Iverson bracket. - Antti Karttunen, Nov 24 2021
a(p) = 0 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A335182 Sum of the refactorable divisors of n.

Original entry on oeis.org

1, 3, 1, 3, 1, 3, 1, 11, 10, 3, 1, 15, 1, 3, 1, 11, 1, 30, 1, 3, 1, 3, 1, 47, 1, 3, 10, 3, 1, 3, 1, 11, 1, 3, 1, 78, 1, 3, 1, 51, 1, 3, 1, 3, 10, 3, 1, 47, 1, 3, 1, 3, 1, 30, 1, 67, 1, 3, 1, 75, 1, 3, 10, 11, 1, 3, 1, 3, 1, 3, 1, 182, 1, 3, 1, 3, 1, 3, 1, 131, 10, 3, 1, 99
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 17 2020

Keywords

Comments

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of refactorable numbers (A336040). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(6) = 3; The divisors of 6 are {1,2,3,6}. 1 and 2 are refactorable since d(1) = 1|1 and d(2) = 2|2, so a(6) = 1 + 2 = 3.
a(7) = 1; The divisors of 7 are {1,7} and 1 is the only refactorable divisor of 7. So a(7) = 1.
a(8) = 11; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8, so a(8) = 1 + 2 + 8 = 11.
a(9) = 10; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9, so a(9) = 1 + 9 = 10.

Links

Crossrefs

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335665 (their product).
Difference of A349322 and A349658.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, Divisible[#, DivisorSigma[0, #]] &]; Array[a, 80] (* Amiram Eldar, Nov 24 2021 *)
  • PARI
    isr(n) = n%numdiv(n)==0; \\ A033950
a(n) = sumdiv(n, d, if (isr(d), d)); \\ Michel Marcus, Jul 20 2020

Formula

a(n) = Sum_{d|n} d * c(d), where c = A336040.
a(n) = Sum_{d|n} d * (1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
a(n) = A349322(n) - A349658(n). - Antti Karttunen, Nov 24 2021
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A335665 Product of the refactorable divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 16, 9, 2, 1, 24, 1, 2, 1, 16, 1, 324, 1, 2, 1, 2, 1, 4608, 1, 2, 9, 2, 1, 2, 1, 16, 1, 2, 1, 139968, 1, 2, 1, 640, 1, 2, 1, 2, 9, 2, 1, 4608, 1, 2, 1, 2, 1, 324, 1, 896, 1, 2, 1, 1440, 1, 2, 9, 16, 1, 2, 1, 2, 1, 2, 1, 1934917632, 1, 2, 1, 2, 1, 2, 1, 51200
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 17 2020

Keywords

Examples

			a(6) = 2; The divisors of 6 are {1,2,3,6}. 1 and 2 are refactorable since d(1) = 1|1 and d(2) = 2|2, so a(6) = 1 * 2 = 2.
a(7) = 1; The divisors of 7 are {1,7} and 1 is the only refactorable divisor of 7. So a(7) = 1.
a(8) = 16; The divisors of 8 are {1,2,4,8}. 1, 2 and 8 are refactorable since d(1) = 1|1, d(2) = 2|2 and d(8) = 4|8, so a(8) = 1 * 2 * 8 = 16.
a(9) = 9; The divisors of 9 are {1,3,9}. 1 and 9 are refactorable since d(1) = 1|1 and d(9) = 3|9, so a(9) = 1 * 9 = 9.

Links

Crossrefs

Cf. A000005 (tau), A033950 (refactorable numbers), A336040 (refactorable characteristic), A336041 (number of refactorable divisors), A335182 (their sum).
Cf. also A349322 (similar formula, but with sum instead of product).

Programs

  • Mathematica
    a[n_] := Product[If[Divisible[d, DivisorSigma[0, d]], d, 1], {d, Divisors[n]}]; Array[a, 60] (* Amiram Eldar, Nov 24 2021 *)
  • PARI
    isr(n) = n%numdiv(n)==0; \\ A033950
a(n) = my(d=divisors(n)); prod(k=1, #d, if (isr(d[k]), d[k], 1)); \\ Michel Marcus, Jul 18 2020

Formula

a(n) = Product_{d|n} d^c(d), where c(n) is the refactorable characteristic of n (A336040).
a(n) = Product_{d|n} d^(1 - ceiling(d/tau(d)) + floor(d/tau(d))), where tau(n) is the number of divisors of n (A000005).
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A349322 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of refactorable numbers (A336040).

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 2, 12, 11, 5, 2, 18, 2, 5, 4, 13, 2, 32, 2, 7, 4, 5, 2, 50, 3, 5, 12, 7, 2, 9, 2, 14, 4, 5, 4, 81, 2, 5, 4, 55, 2, 9, 2, 7, 14, 5, 2, 52, 3, 7, 4, 7, 2, 34, 4, 71, 4, 5, 2, 83, 2, 5, 14, 15, 4, 9, 2, 7, 4, 9, 2, 185, 2, 5, 6, 7, 4, 9, 2, 136, 13, 5, 2, 107
Offset: 1

Views

Author

Wesley Ivan Hurt, Nov 14 2021

Keywords

Comments

For each divisor d of n, add d if d is refactorable (i.e., if the number of divisors of d divides d), otherwise add 1. For example, the divisors of 8 are 1,2,4,8 and the refactorable divisors of 8 are 1,2,8. The sum is then a(8) = 1 + 2 + 1 + 8 = 12.
Inverse Möbius transform of n^c(n), where c = A336040. - Wesley Ivan Hurt, Jun 29 2024

Links

Crossrefs

Cf. A033950, A208251, A335182, A335665, A336040, A336041, A349658.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, If[Divisible[#, DivisorSigma[0, #]], #, 1] &]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    isrf(n) = n%numdiv(n)==0; \\ A336040
a(n) = sumdiv(n, d, if (isrf(d), d, 1)); \\ Michel Marcus, Nov 16 2021

Formula

a(n) = A335182(n) + A349658(n). - Antti Karttunen, Nov 24 2021
a(p) = 2 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A349658 Number of nonrefactorable divisors of n.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 6, 1, 3, 3, 2, 3, 3, 1, 2, 3, 4, 1, 6, 1, 4, 4, 2, 1, 5, 2, 4, 3, 4, 1, 4, 3, 4, 3, 2, 1, 8, 1, 2, 4, 4, 3, 6, 1, 4, 3, 6, 1, 3, 1, 2, 5, 4, 3, 6, 1, 5, 3, 2, 1, 8, 3, 2, 3, 4, 1, 8, 3, 4
Offset: 1

Views

Author

Wesley Ivan Hurt, Nov 23 2021

Keywords

Links

Crossrefs

Cf. A000005, A033950, A336041, A335182, A349322.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, !Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Nov 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d%numdiv(d) != 0); \\ Michel Marcus, Nov 24 2021

Formula

a(n) = A000005(n) - A336041(n).
a(n) = A349322(n) - A335182(n). - Antti Karttunen, Nov 24 2021
a(p) = 1 for odd primes p. - Wesley Ivan Hurt, Nov 28 2021

A363085 Sum of the refactorable unitary divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 9, 10, 3, 1, 13, 1, 3, 1, 1, 1, 30, 1, 1, 1, 3, 1, 33, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 46, 1, 3, 1, 49, 1, 3, 1, 1, 10, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 65, 1, 3, 1, 73, 1, 3, 10, 1, 1, 3, 1, 1, 1, 3, 1, 90, 1, 3, 1, 1, 1, 3, 1, 81, 1, 3, 1, 97, 1, 3, 1, 97
Offset: 1

Views

Author

Wesley Ivan Hurt, May 27 2023

Keywords

Crossrefs

Cf. A034444, A336041, A363298.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, CoprimeQ[#, n/#] && Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100]

Formula

a(n) = Sum_{d|n, tau(d)|d, gcd(d,n/d)=1} d.

A363298 Number of refactorable unitary divisors of n.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4
Offset: 1

Views

Author

Wesley Ivan Hurt, May 26 2023

Keywords

Crossrefs

Cf. A034444, A336041, A363085.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, CoprimeQ[#, n/#] && Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, May 26 2023 *)

Formula

a(n) = Sum_{d|n, tau(d)|d, gcd(d,n/d)=1} 1.

A345211 Numbers with the same number of odd / even, refactorable divisors.

Original entry on oeis.org

2, 4, 6, 10, 14, 18, 20, 22, 26, 28, 30, 34, 38, 42, 44, 46, 50, 52, 54, 58, 62, 66, 68, 70, 74, 76, 78, 82, 86, 90, 92, 94, 98, 100, 102, 106, 110, 114, 116, 118, 122, 124, 126, 130, 134, 138, 140, 142, 146, 148, 150, 154, 158, 162, 164, 166, 170, 172, 174, 178, 182, 186, 188
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 10 2021

Keywords

Comments

All the terms are even. Conjecture: The sequence a(n)/2 is the complement of A157932 (verified for a(n) < 4*10^7). - Amiram Eldar, Oct 11 2023

Examples

			18 is in the sequence since it has 2 odd and 2 even, refactorable divisors: 1, 2, 9, and 18.

Links

Crossrefs

Cf. A033950, A157932, A336041.

Programs

  • Mathematica
    q[n_] := DivisorSum[n, (-1)^# &, Divisible[#, DivisorSigma[0, #]] &] == 0; Select[Range[2, 200, 2], q] (* Amiram Eldar, Oct 11 2023 *)
  • PARI
    is(n) = (n%2)! && sumdiv(n, d, (-1)^d * !(d % numdiv(d))) == 0; \\ Amiram Eldar, Oct 11 2023

A338140 a(n) is the smallest number with n refactorable divisors.

Original entry on oeis.org

1, 2, 8, 18, 24, 36, 108, 180, 72, 216, 288, 1944, 360, 1080, 1920, 720, 1800, 2160, 5400, 1440, 6720, 3600, 12600, 4320, 16200, 5760, 12960, 38016, 13440, 45360, 35280, 10080, 21600, 28800, 67200, 51840, 215040, 20160, 30240, 97200, 50400, 64800, 144000
Offset: 1

Views

Author

Jaroslav Krizek, Oct 24 2020

Keywords

Comments

a(n) is the greedy inverse of A336041: the smallest number with exactly n divisors d such that d / tau(d) is also an integer.
Numbers 1 and 2 are only numbers m such that d / tau(d) is an integer for all divisors d of m.

Examples

			a(3) = 8 because 8 with divisors 1, 2, 4 and 8 is the smallest number with 3 refactorable divisors: 1 / tau(1) = 1, 2 / tau(2) = 1, 8 / tau(8) = 2.

Crossrefs

Cf. A336041, A033950 (refactorable numbers).
Cf. A335182, A335665.

Programs

  • Magma
    [Min([m: m in[1..10^5] | #[d: d in Divisors(m) | IsIntegral(d / #Divisors(d))] eq n]): n in [1..12]]
  • Mathematica
    f[n_] := DivisorSum[n, 1 &, Divisible[#, DivisorSigma[0, #]] &]; m = 43; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = f[n]; If[i <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* Amiram Eldar, Oct 24 2020 *)

Formula

a(n) = min{ k: A336041(k)=n}. - R. J. Mathar, Nov 24 2020

A363091 Sum of the divisor complements of the refactorable unitary divisors of n.

Original entry on oeis.org

1, 3, 3, 4, 5, 9, 7, 9, 10, 15, 11, 13, 13, 21, 15, 16, 17, 30, 19, 20, 21, 33, 23, 28, 25, 39, 27, 28, 29, 45, 31, 32, 33, 51, 35, 41, 37, 57, 39, 46, 41, 63, 43, 44, 50, 69, 47, 48, 49, 75, 51, 52, 53, 81, 55, 64, 57, 87, 59, 66, 61, 93, 70, 64, 65, 99, 67, 68, 69, 105, 71
Offset: 1

Views

Author

Wesley Ivan Hurt, May 27 2023

Keywords

Crossrefs

Cf. A034444, A336041, A363085, A363298.

Programs

  • Mathematica
    a[n_] := n*DivisorSum[n, 1/# &, CoprimeQ[#, n/#] && Divisible[#, DivisorSigma[0, #]] &]; Array[a, 100]

Formula

a(n) = n * Sum_{d|n, tau(d)|d, gcd(d,n/d)=1} 1 / d.
Showing 1-10 of 10 results.

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