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 1-3 of 3 results.

A331972 Bi-unitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper bi-unitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 29, 31, 55, 79, 91, 115, 121, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091, 6511, 6931
Offset: 1

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The corresponding record values are 0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 14, 15, ...
The bi-unitary version of A238895.

Examples

			a(1) = 2 since it is the first number which is not the sum of proper bi-unitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper bi-unitary divisors of one number: 6 = A331970(6).
a(3) = 8 since it is the least number which is the sum of proper bi-unitary divisors of 2 numbers: 8 = A331970(10) = A331970(12).

Links

Crossrefs

Cf. A188999, A238895, A324276, A325177, A331970, A331974.

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}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

Original entry on oeis.org

0, 6, 7, 9, 11, 18, 32, 44, 56, 62, 72, 82, 94, 96, 98, 102, 104, 110, 116, 122, 132, 136, 138, 146, 150, 152, 178, 180, 182, 210, 222, 226, 230, 236, 238, 242, 248, 252, 264, 272, 284, 292, 296, 304, 322, 332, 338, 342, 350, 356, 360, 374, 382, 390, 392, 404
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

Numbers k such that A331973(k) = 1.

Links

Crossrefs

Cf. A126168, A324277, A331974, A331973, A361420.
Similar sequences: A057709, A357324.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; is[1] = 0; is[n_] := Times @@ f @@@ FactorInteger[n] - n;
seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = is[k] + 1; If[i <= max, v[[i]]++], {k, 1, max^2}]; -1 + Position[v, 1] // Flatten];
seq[500]
  • PARI
    s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(v = vector(nmax+1)); for(k=1, nmax^2, i = s(k) + 1; if(i <= nmax+1, v[i] += 1)); for(i = 1, nmax+1, if(v[i] == 1, print1(i-1, ", "))); }

Formula

a(n) = A126168(A361420(n)).

A372741 Coreful highly touchable numbers: numbers m > 0 such that a record number of numbers k have m as the sum of the aliquot coreful divisors (A336563) of k.

Original entry on oeis.org

1, 2, 6, 30, 210, 930, 2310, 2730, 30030, 71610, 84630
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).
Indices of records of A372739.
The corresponding record values are 0, 1, 3, 6, 8, 9, 11, 12, 15, 16, 17, ... .
a(12) > 2*10^5.

Examples

			a(1) = 1 since it is the least number that is not the sum of aliquot coreful divisors of any number.
a(2) = 2 since it is the least number that is the sum of aliquot coreful divisors of one number: 2 = A336563(4).
a(3) = 6 since it is the least number that is the sum of aliquot coreful divisors of 3 numbers: 6 = A336563(8) = A336563(12) = A336563(18), and there is no number between 2 and 6 that is the sum of aliquot coreful divisors of exactly 2 numbers.

Crossrefs

Cf. A307958, A336563, A372739, A372740.
Similar sequences: A238895, A325177, A331972, A331974.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[m_] := Module[{v = Table[0, {m}], vm = -1, w = {}, i}, Do[i = s[k]; If[1 <= i <= m, v[[i]]++], {k, 1, m^2}]; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[w, k]], {k, 1, m}]; w]; seq[1000]
  • 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), vmax = -1, i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); for(k = 1, nmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", ")));}
Showing 1-3 of 3 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.