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 1-8 of 8 results.

A353901 Numbers k such that A353900(k) is divisible by k.

Original entry on oeis.org

1, 6, 28, 56, 1104, 2208, 2178540, 4357080, 6499584, 12999168
Offset: 1

Views

Author

Amiram Eldar, May 10 2022

Keywords

Comments

a(11) > 8*10^10, if it exists.
The corresponding ratios A353900(k)/k are 1, 2, 2, 1, 2, 1, 4, 2, 2, 1, ...

Examples

			6 is a term since A353900(6) = 12 is divisible by 6.
56 is a term since A353900(56) = 56 is divisible by 56.

Crossrefs

Cf. A353900.
Similar sequences: A007691, A189000, A327158, A348601.

Programs

  • Mathematica
    f[p_, e_] := 1 + Sum[p^(2^k), {k, 0, Floor[Log2[e]]}]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[6.5*10^6], Divisible[s[#], #] &]

A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 1, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 3, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96
Offset: 1

Views

Author

Amiram Eldar, Jun 16 2025

Keywords

Comments

The number of these divisors is A385042(n), and the largest of them is A367168(n).

Links

Crossrefs

The unitary analog of A353900.
Cf. A005117, A034448, A138302, A209229, A385042.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), this sequence (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

Programs

  • Mathematica
    f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e + 1, 1]; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1<

Formula

Multiplicative with a(p^e) = p^(A209229(e)) + 1.
a(n) <= A034448(n), with equality if and only if n is in A138302.
a(n) <= A353900(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1/(p*(p+1)) + Sum_{k>=2} (1/p^(2^k)-1/p^(2^k-1))) = 1.21427559551509410114... .

A366904 The sum of exponentially evil divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 105, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

The number of these divisors is A366902(n) and the largest of them is A366906(n).

Links

Crossrefs

Cf. A004709, A366902, A366906.
Similar sequences: A353900, A365682, A366903.

Programs

  • Mathematica
    f[p_, e_] := 1 + Total[p^Select[Range[e], EvenQ[DigitCount[#, 2, 1]] &]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i, 2], !(hammingweight(k)%2) * f[i, 1]^k));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{k = 1..e, k is evil} p^k.
a(n) >= 1, with equality if and only if n is a cubefree number (A004709).

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Jun 15 2025

Keywords

Comments

First differs from A365682 and A366992 at n = 32.
The number of these divisors is A252505(n), and the largest of them is A058035(n).

Links

Crossrefs

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.
The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

Programs

  • Mathematica
    f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Formula

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 29, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 58, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 116, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 116, 84, 144
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2025

Keywords

Links

Crossrefs

Cf. A004709, A138302, A353900.
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).

Programs

  • Mathematica
    f[p_, e_] := p^e + Sum[p^(e - 2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}

Formula

Multiplicative with a(p^e) = p^e + Sum_{k=0..floor(log_2(e))} p^(e-2^k).
a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .

A366903 The sum of exponentially odious divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 23, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 23, 84, 144, 68
Offset: 1

Views

Author

Amiram Eldar, Oct 27 2023

Keywords

Comments

First differs from A353900 at n = 128.
The number of these divisors is A366901(n) and the largest of them is A366905(n).

Links

Crossrefs

Cf. A004709, A000203, A366901, A366905.
Similar sequences: A353900, A365682, A366904.

Programs

  • Mathematica
    f[p_, e_] := 1 + Total[p^Select[Range[e], OddQ[DigitCount[#, 2, 1]] &]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i, 2], (hammingweight(k)%2) * f[i, 1]^k));}

Formula

Multiplicative with a(p^e) = 1 + Sum_{k = 1..e, k is odious} p^k.
a(n) <= A000203(n), with equality if and only if n is a cubefree number (A004709).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1-1/p)*(1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 0.721190607... .

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Jun 15 2025

Keywords

Comments

The sum of the terms in A036966 that divide n.
The number of these divisors is A190867(n), and the largest of them is A360540(n).

Links

Crossrefs

Cf. A036966, A190867, A183097, A360540.
The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Formula

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A382788 The sum of divisors of n that are numbers whose number of divisors is a power of 2.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 11, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 11, 84, 144, 68, 54
Offset: 1

Views

Author

Amiram Eldar, Apr 05 2025

Keywords

Comments

First differs from A033634 at n = 32.
The sum of the terms of A036537 that divide n.
The number of these divisors is A372380(n) and the largest of them is A372379(n).

Links

Crossrefs

Cf. A000523, A033634, A036537, A372379, A353900, A372380.

Programs

  • Mathematica
    f[p_, e_] := Sum[p^(2^k-1), {k, 0, Floor[Log2[e + 1]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, sum(k = 0, exponent(e[i]+1), p[i]^(2^k-1)));

Formula

Multiplicative with a(p^e) = Sum_{k = 0..floor(log_2(e+1))} p^(2^k-1).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=1} a(p^k)/p^(2*k)) = 1.13143029377358401678... .
Showing 1-8 of 8 results.

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

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