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.

A351113 Sum of the balanced numbers dividing n.

Original entry on oeis.org

1, 3, 4, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 17, 19, 3, 1, 12, 1, 3, 4, 3, 1, 24, 1, 3, 4, 17, 1, 57, 1, 3, 4, 3, 36, 24, 1, 3, 4, 3, 1, 68, 1, 3, 19, 3, 1, 24, 1, 3, 4, 3, 1, 12, 1, 73, 4, 3, 1, 69, 1, 3, 4, 3, 1, 12, 1, 3, 4, 122, 1, 24, 1, 3, 19, 3, 1, 90, 1, 3, 4, 3, 1, 80
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 31 2022

Keywords

Comments

A balanced number k is a number such that phi(k) | sigma(k).

Examples

			a(4) = 3; the balanced divisors of 4 are 1 and 2 and 1+2 = 3.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 12; the balanced divisors of 6 are 1,2,3,6 and 1+2+3+6 = 12.

Links

Crossrefs

Cf. A351112 (number of balanced divisors of n).
Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A023897, A351114.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)
  • PARI
    a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), d)); \\ Michel Marcus, Feb 01 2022

Formula

a(n) = Sum_{d|n, phi(d)|sigma(d)} d.
a(n) = Sum_{d|n} d * A351114(d).
a(n) = sigma(n) - Sum_{d|n} d * sign(sigma(d) mod phi(d)).

A386591 Number of divisors of n that are not balanced numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 4, 2, 2, 2, 4, 1, 2, 2, 6, 1, 2, 1, 4, 3, 2, 1, 5, 2, 4, 2, 4, 1, 4, 3, 4, 2, 2, 1, 5, 1, 2, 4, 5, 3, 4, 1, 4, 2, 3, 1, 7, 1, 2, 3, 4, 3, 3, 1, 8, 3, 2, 1, 5, 3, 2, 2, 6, 1, 6, 3, 4, 2, 2, 3, 7, 1, 3, 4, 7
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 26 2025

Keywords

Comments

Number of divisors d of n such that phi(d) does not divide sigma(d).
Inverse Möbius transform of 1 - c(n), where c = A351114.

Links

Crossrefs

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A020492 (balanced numbers), A351112, A351114.

Programs

  • Maple
    g:= proc(n) option remember; numtheory:-sigma(n) mod numtheory:-phi(n) <> 0 end proc:
f:= n -> nops(select(g,numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Aug 26 2025
  • Mathematica
    Table[Sum[Ceiling[DivisorSigma[1, d]/EulerPhi[d]] - Floor[DivisorSigma[1, d]/EulerPhi[d]], {d, Divisors[n]}], {n, 100}]
  • PARI
    a(n) = sumdiv(n, d, sigma(d)%eulerphi(d) != 0); \\ Michel Marcus, Aug 26 2025

Formula

a(n) = Sum_{d|n} (1 - c(d)), where c = A351114.
a(n) = A000005(n) - A351112(n).

A367734 Numbers that have no balanced divisors except for 1.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203
Offset: 1

Views

Author

Robert Israel, Nov 28 2023

Keywords

Comments

Numbers k such that A351112(k) = 1.
Includes all primes except for 2 and 3, and all powers of those primes.
If k is a term, then so are all divisors of k.
For i < 271, a(i+68) = a(i) + 210, and this equation seems to be true for most i.

Examples

			a(9) = 25 is a term because of its divisors 1, 5, 25, only 1 is balanced.

Links

Crossrefs

Cf. A020492, A351112.

Programs

  • Maple
    filter:= proc(n) uses numtheory;
  andmap(t -> sigma(t) mod phi(t) <> 0, divisors(n) minus {1})
end proc:
select(filter, [$1..1000]);
  • Mathematica
    Select[Range[200], DivisorSum[#, 1 &, Divisible[DivisorSigma[1, #1], EulerPhi[#1]] &] == 1 &] (* Amiram Eldar, Nov 28 2023 *)

Formula

A351112(a(n)) = 1.

A362707 a(n) = Sum_{d|n, phi(d)|sigma(d)} (n-d).

Original entry on oeis.org

0, 1, 2, 5, 4, 12, 6, 13, 14, 17, 10, 36, 12, 25, 26, 29, 16, 60, 18, 37, 38, 41, 22, 96, 24, 49, 50, 67, 28, 123, 30, 61, 62, 65, 34, 156, 36, 73, 74, 77, 40, 184, 42, 85, 116, 89, 46, 216, 48, 97, 98, 101, 52, 204, 54, 151, 110, 113, 58, 351, 60, 121, 122, 125, 64, 252, 66, 133
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 30 2023

Keywords

Comments

Total distance from n to each balanced divisor of n (see example).

Examples

			a(12) = 36; 12 has 5 balanced divisors 1,2,3,6,12 and the sum of their distances to n is (12-1)+(12-2)+(12-3)+(12-6)+(12-12) = 36.

Crossrefs

Cf. A000005 (tau), A000010 (phi), A020492 (balanced numbers), A351112, A351113, A351114.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n - #) &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100]
  • PARI
    a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), n-d)); \\ Michel Marcus, Apr 30 2023

Formula

a(n) = Sum_{d|n} (n-d) * A351114(d).
a(n) = n*A351112(n) - A351113(n).

A386589 a(n) = Sum_{d|n} d^c(d), where c = A351114.

Original entry on oeis.org

1, 3, 4, 4, 2, 12, 2, 5, 5, 5, 2, 25, 2, 18, 20, 6, 2, 14, 2, 7, 6, 5, 2, 27, 3, 5, 6, 20, 2, 59, 2, 7, 6, 5, 38, 28, 2, 5, 6, 9, 2, 70, 2, 7, 22, 5, 2, 29, 3, 7, 6, 7, 2, 16, 4, 77, 6, 5, 2, 74, 2, 5, 8, 8, 4, 16, 2, 7, 6, 125, 2, 31, 2, 5, 22, 7, 4, 93, 2, 11, 7, 5, 2, 85, 4, 5, 6, 9, 2, 63, 4, 7, 6, 5, 4, 31, 2, 20, 8, 10
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 26 2025

Keywords

Comments

Inverse Möbius transform of n^c(n), where c = A351114.
For each divisor d of n, add d if d is a balanced number (A020492), else add 1.

Crossrefs

Cf. A000005 (tau), A020492 (balanced numbers), A351112, A351113, A351114.

Programs

  • Mathematica
    Table[Sum[d^(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

Formula

a(n) = Sum_{d|n} 1 + (d - 1)*c(d), where c = A351114.
a(n) = A000005(n) + A351113(n) - A351112(n).

A386622 a(n) = Sum_{d|n} n^c(d), where c = A351114.

Original entry on oeis.org

1, 4, 6, 9, 6, 24, 8, 18, 19, 22, 12, 61, 14, 43, 46, 35, 18, 74, 20, 44, 44, 46, 24, 123, 27, 54, 56, 87, 30, 182, 32, 68, 68, 70, 72, 184, 38, 78, 80, 86, 42, 254, 44, 92, 138, 94, 48, 245, 51, 104, 104, 108, 54, 220, 58, 228, 116, 118, 60, 425, 62, 126, 130, 133, 68, 268, 68, 140, 140, 353, 72, 367, 74, 150, 228, 156, 80, 393, 80, 168, 165, 166, 84, 593
Offset: 1

Views

Author

Wesley Ivan Hurt, Jul 27 2025

Keywords

Crossrefs

Cf. A000005 (tau), A020492 (balanced numbers), A351112, A351114, A386591.

Programs

  • Mathematica
    Table[Sum[n^(1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]), {d, Divisors[n]}], {n, 100}]

Formula

a(n) = tau(n) + (n - 1) * Sum_{d|n} c(d), where c = A351114.
a(n) = n*A351112(n) + A386591(n).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.