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.

A325177 Unitary highly touchable numbers: Numbers m > 1 such that a record number of numbers k have m as the sum of the proper unitary divisors of k.

Original entry on oeis.org

2, 6, 8, 12, 22, 33, 49, 55, 67, 79, 91, 115, 121, 151, 169, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1171, 1231, 1261, 1321, 1471, 1561, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251
Offset: 1

Views

Author

Amiram Eldar, Sep 05 2019

Keywords

Comments

The unitary version of A238895.

Links

Crossrefs

Cf. A034448, A034460, A063948, A238895, A324938.

Programs

  • Mathematica
    us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  m = 300; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A331973 a(n) is the number of values of m such that the sum of proper infinitary divisors of m (A126168) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 1, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 4, 1, 4, 3, 4, 3, 5, 0, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 6, 2, 5, 2, 5, 3, 7, 1, 6, 2, 4, 2, 7, 1, 5, 4, 5, 3, 8, 0, 5, 2, 6, 1, 8, 2, 5, 4, 6, 4, 9, 0, 6, 1, 5, 3, 10, 2, 8, 2
Offset: 2

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The infinitary version of A048138.
The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A126168(k) = 1.

Examples

			a(8) = 2 since 8 is the sum of the proper infinitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Links

Crossrefs

Cf. A048138, A049417, A077609, A126168, A324938, A331971.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ (fun @@@ FactorInteger[n]); is[n_] := isigma[n] - n; m = 300; v = Table[0, {m}]; Do[i = is[k]; If[2 <= i <= m, v[[i]]++], {k, 1, m^2}]; Rest@v

A357324 Numbers k such that there is a unique m for which the sum of the aliquot unitary divisors of m (A034460) is k.

Original entry on oeis.org

6, 9, 11, 13, 128, 150, 164, 222, 224, 332, 338, 390, 404, 416, 420, 458, 510, 548, 558, 570, 576, 582, 584, 598, 660, 668, 750, 788, 800, 810, 818, 822, 836, 852, 878, 884, 926, 930, 1046, 1118, 1200, 1202, 1230, 1244, 1250, 1260, 1284, 1298, 1304, 1382, 1422, 1472, 1478
Offset: 1

Views

Author

Amiram Eldar, Sep 24 2022

Keywords

Comments

Numbers k such that A324938(k) = 1.

Links

Crossrefs

The unitary version of A057709.
Cf. A034460, A063948, A324938, A357325.

Programs

  • Mathematica
    us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; m = 1500; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; Position[v, 1] // Flatten

Formula

a(n) = A034460(A357325(n)).

A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, May 12 2024

Keywords

Comments

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).

Examples

			a(2) = 1 since there is 1 possible value of k, k = 4, such that A336563(k) = 2.
a(6) = 3 since there are 3 possible values of k, k = 8, 12 and 18, such that A336563(k) = 6.

Links

Crossrefs

Cf. A307958, A336563, A372740, A372742.
Similar sequences: A048138, A324938, A331971, A331973.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; v]; seq[100]
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n;}
lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); v;}

Formula

a(n) = 0 if and only if n is in A372740.
a(n) = 1 if and only if n is in A372742.

A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 6, 1, 4, 2, 4, 2, 5, 0, 3, 1, 4, 2, 5, 1, 4, 2, 4, 1, 6, 2, 5, 2, 5, 2, 8, 1, 6, 1, 4, 2, 7, 1, 5, 3, 5, 2, 8, 0, 5, 1, 6, 1, 8, 2, 5, 3, 6, 3, 9, 0, 6, 2, 5, 1, 9, 1, 7, 1
Offset: 2

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The bi-unitary version of A048138.
The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

Examples

			a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Links

Crossrefs

Cf. A048138, A188999, A222266, A324938, A331970, A331973.

Programs

  • Mathematica
    fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

A325202 Number of times that A325177(n) occurs in the sum of proper unitary divisors function (A034460).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 21, 24, 26, 28, 31, 33, 35, 37, 38, 45, 49, 56, 60, 63, 64, 65, 72, 73, 81, 83, 94, 100, 105, 121, 138, 145, 149, 169, 175, 176, 180, 182, 202, 210, 234, 236, 256, 285, 288, 306, 319, 343, 347, 362, 382
Offset: 1

Views

Author

Amiram Eldar, Sep 05 2019

Keywords

Comments

The unitary version of A238896.

Crossrefs

Cf. A034448, A034460, A063948, A238896, A324938, A325177.

Programs

  • Mathematica
    us[1] = 0; us[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n;  m = 300; v = Table[0, {m}]; Do[u = us[k]; If[2 <= u <= m, v[[u]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, vm]], {k, 2, m}]; s

Formula

a(n) = A324938(A325177(n)).
Showing 1-6 of 6 results.

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