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.

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A372742 Numbers k such that there is a unique number m for which the sum of the aliquot coreful divisors of m (A336563) is k.

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 24, 29, 31, 36, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 80, 83, 84, 89, 96, 97, 98, 101, 103, 107, 109, 112, 113, 127, 131, 135, 137, 139, 140, 149, 150, 151, 156, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198
Offset: 1

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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) = 1.
The corresponding values of m are in A372743.
Includes all prime numbers.

Links

Crossrefs

Cf. A336563, A372739, A372743.
A000040 is a subsequence.
Similar sequences: A057709, A357324, A361419.

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[1 <= i <= max, v[[i]]++], {k, 1, max^2}]; Position[v, 1] // 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] == 1, print1(k, ", ")));}

Formula

a(n) = A336563(A372743(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, ", ")));}

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.

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