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 1-5 of 5 results.

A348004 Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91
Offset: 1

Views

Author

Amiram Eldar, Sep 23 2021

Keywords

Comments

First differs from A042965 \ {0} at n=63, and from A122906 at n=53.
Since Sum_{d|k, gcd(d,k/d)=1} uphi(d) = k, these are numbers k such that the set {uphi(d) | d|k, gcd(d,k/d)=1} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345), since an odd prime power p^e has 2 unitary divisors, 1 and p^e, whose uphi values are 1 and p^e - 1. It also includes all the powers of 2, except for 2 (A151821).
If k is a term, then all the unitary divisors of k are also terms.
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 74, 741, 7386, 73798, 737570, 7374534, 73740561, 737389031, 7373830133, ... Apparently, this sequence has an asymptotic density 0.73738...

Examples

			4 is a term since it has 2 unitary divisors, 1 and 4, and uphi(1) = 1 != uphi(4) = 3.
12 is a term since the uphi values of its unitary divisors, {1, 3, 4, 12}, are distinct: {1, 2, 3, 6}.

Links

Crossrefs

The unitary version of A326835.
Cf. A034444, A042965, A047994, A061345, A151821, A122906, A348001.

Programs

  • Mathematica
    f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; Select[Range[100], q]
  • Python
    from math import prod
from sympy.ntheory.factor_ import udivisors, factorint
A348004_list = []
for n in range(1,10**3):
    pset = set()
    for d in udivisors(n,generator=True):
        u = prod(p**e-1 for p, e in factorint(d).items())
        if u in pset:
            break
        pset.add(u)
    else:
        A348004_list.append(n) # Chai Wah Wu, Sep 24 2021

Formula

Numbers k such that A348001(k) = A034444(k).

A122904 Numbers m such that in Z/mZ the number of squares is strictly less than the number of invertible elements.

Original entry on oeis.org

5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96, 97, 99, 100
Offset: 1

Views

Author

Max Alekseyev, Sep 18 2006

Keywords

Comments

Numbers m such that A000224(m) < A000010(m).

Links

Crossrefs

Cf. A000010, A000224, A122903, A122905, A122906, A122907.

Programs

  • Mathematica
    Select[Range@100, Length@Union@Mod[Range[#]^2, #] < EulerPhi[#] &] (* Ivan Neretin, Dec 14 2016 *)
f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] < 1; Select[Range[100], q] (* Amiram Eldar, Nov 11 2024 *)

A122905 Numbers m such that in Z/mZ the number of squares is strictly greater than the number of invertible elements.

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 74, 78, 82, 86, 94, 98, 102, 106, 114, 118, 122, 134, 138, 142, 146, 150, 158, 162, 166, 174, 178, 186, 194, 202, 206, 214, 218, 222, 226, 242, 246, 250, 254, 258, 262, 274, 278, 282, 294, 298, 302
Offset: 1

Views

Author

Max Alekseyev, Sep 18 2006

Keywords

Comments

Numbers m such that A000224(m) > A000010(m).

Links

Crossrefs

Complement of A122906 in positive integers.
Cf. A000010, A000224, A122903, A122904, A122907.

Programs

  • Mathematica
    Select[Range@310, Length@Union@Mod[Range[#]^2, #] > EulerPhi[#] &] (* Ivan Neretin, Dec 14 2016 *)
f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] > 1; Select[Range[300], q] (* Amiram Eldar, Nov 11 2024 *)

A122907 Numbers m such that in Z/mZ the number of squares is greater than or equal to the number of invertible elements.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 114, 118, 122, 134, 138, 142, 146, 150, 158, 162, 166, 174, 178, 186, 194, 202, 206, 210, 214, 218, 222, 226, 242, 246, 250, 254, 258, 262, 274
Offset: 1

Views

Author

Max Alekseyev, Sep 18 2006

Keywords

Comments

Numbers m such that A000224(m) >= A000010(m).

Links

Crossrefs

Union of A122903 and A122905.
Complement of A122904 in positive integers.
Cf. A000010, A000224, A122906.

Programs

  • Mathematica
    Select[Range@275, Length@Union@Mod[Range[#]^2, #] >= EulerPhi[#] &] (* Ivan Neretin, Dec 14 2016 *)
f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = True; q[k_] := Times @@ f @@@ FactorInteger[k] >= 1; Select[Range[300], q] (* Amiram Eldar, Nov 11 2024 *)

A122903 Numbers m such that in Z/mZ the number of squares is equal to the number of invertible elements.

Original entry on oeis.org

1, 3, 4, 12, 70, 90, 210
Offset: 1

Views

Author

Max Alekseyev, Sep 18 2006

Keywords

Comments

Numbers m such that A000224(m) = A000010(m).

Crossrefs

Cf. A000010, A000224, A122904, A122905, A122906, A122907.

Programs

  • Mathematica
    f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = True; q[k_] := Times @@ f @@@ FactorInteger[k] == 1; Select[Range[210], q] (* Amiram Eldar, Nov 11 2024 *)
Showing 1-5 of 5 results.

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