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.

A324276 Bi-unitary untouchable numbers: numbers that are not the sum of aliquot bi-unitary divisors of any number.

Original entry on oeis.org

2, 3, 4, 5, 38, 68, 80, 96, 98, 128, 138, 146, 158, 164, 180, 188, 192, 206, 208, 210, 212, 222, 224, 248, 264, 278, 290, 300, 304, 308, 324, 326, 328, 338, 360, 374, 380, 390, 398, 416, 418, 420, 430, 432, 458, 476, 480, 488, 498, 516, 518, 530, 536, 542, 548
Offset: 1

Views

Author

Amiram Eldar, Feb 20 2019

Keywords

Links

Crossrefs

Cf. A188999, A005114, A063948 (unitary), A324277 (infinitary), A324278 (exponential), A331970.

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_] := bsigma[n] = Times @@ (fun @@@ FactorInteger[n]); untouchableQ[n_] := Catch[ Do[ If[n == bsigma[k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 2, 550}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

A331974 Infinitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper infinitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 21, 37, 49, 55, 67, 79, 85, 91, 121, 151, 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
Offset: 1

Views

Author

Amiram Eldar, Feb 03 2020

Keywords

Comments

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
The infinitary version of A238895.

Examples

			a(1) = 2 since it is the first number which is not the sum of proper infinitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper infinitary divisors of one number: 6 = A126168(6).
a(3) = 8 since it is the least number which is the sum of proper infinitary divisors of 2 numbers: 8 = A126168(10) = A126168(12).

Links

Crossrefs

Cf. A049417, A077609, A126168, A238895, A324277, A325177, A331972, A331973.

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}]; 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)).

A372740 Coreful untouchable numbers: numbers that are not the sum of aliquot coreful divisors (A336563) of any number.

Original entry on oeis.org

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 63, 64, 68, 72, 75, 76, 81, 88, 92, 99, 100, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208
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).
Numbers k such that A372739(k) = 0.
Numbers that are not in the range of A336563.
Except for 1, all the terms are not squarefree (A013929), because if k is squarefree (A005117), and there is a prime p such that p|k, then A336563(p*k) = k.
Includes all the squares of primes (A001248).
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are , 4, 29, 281, 2762, 27690, ... . Apparently, the asymptotic density of this sequence exists and equals 0.27... .

Links

Crossrefs

Cf. A005117, A013929, A307958, A336563, A372739.
A001248 is a subsequence.
Similar sequences: A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

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}]; Position[v, _?(# == 0 &)] // Flatten]; seq[200]
  • 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]++)); for(k = 1, nmax, if(v[k] == 0, print1(k, ", ")));}

A324278 Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.

Original entry on oeis.org

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 49, 52, 54, 63, 64, 68, 75, 76, 81, 88, 92, 96, 99, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224
Offset: 1

Views

Author

Amiram Eldar, Feb 20 2019

Keywords

Comments

The terms are conjectural and based on a search for solutions to esigma(x) - x = k for k in the range of the data section and x < 10^12 (esigma(x) - x = A051377(x) - x = A126164(x) is the sum of aliquot exponential divisors of x). - Amiram Eldar, Jan 22 2020

Crossrefs

Cf. A051377, A126164, A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := esigma[n] = Times @@ fun @@@ FactorInteger[n]; untouchableQ[n_] := Catch[ Do[ If[n == esigma[k]-k, Throw[True]], {k, 0, (n+1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 1, 130}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

Extensions

Data corrected by Amiram Eldar, Jan 22 2020
Showing 1-5 of 5 results.

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