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 1-10 of 11 results. Next

A336566 a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 3, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 1, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 1, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2020

Keywords

Links

Crossrefs

Cf. A057723, A308135, A336563, A336564, A336565.
Differs from A326144 at the positions given by A336555, for the first time at n=45, where a(45) = 3, while A326144(45) = 6.

Programs

Formula

a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(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.

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

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) = 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, ", ")));}

A372743 a(n) is the unique number m such that A336563(m) = A372742(n).

Original entry on oeis.org

4, 9, 25, 49, 121, 27, 169, 289, 24, 361, 529, 54, 841, 961, 36, 1369, 1681, 1849, 2209, 2809, 343, 3481, 3721, 4489, 5041, 5329, 6241, 100, 6889, 189, 7921, 72, 9409, 112, 10201, 10609, 11449, 11881, 686, 12769, 16129, 17161, 225, 18769, 19321, 196, 22201, 160
Offset: 1

Views

Author

Amiram Eldar, May 12 2024

Keywords

Comments

Includes all the squares of primes (A001248).

Links

Crossrefs

Cf. A336563, A372742.
A001248 is a subsequence.
Similar sequences: A357313, A357325, A361420.

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 = w = Table[0, {max}], i}, Do[i = s[k]; If[1 <= i <= max, v[[i]]++; w[[i]] = k], {k, 1, max^2}]; w[[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 = w = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++; w[i] = k)); for(k = 1, nmax, if(v[k] == 1, print1(w[k], ", ")));}

Formula

A336563(a(n)) = A372742(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, ", ")));}

A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 0, 6, 7, 8, 2, 10, 2, 12, 4, 6, 15, 16, 3, 18, 8, 10, 8, 22, 6, 24, 10, 26, 14, 28, -12, 30, 31, 18, 14, 22, 17, 36, 16, 22, 20, 40, -12, 42, 26, 27, 20, 46, 14, 48, 17, 30, 32, 52, 12, 38, 34, 34, 26, 58, -18, 60, 28, 43, 63, 46, -12, 66, 44, 42, -4, 70, 45, 72, 34, 41, 50, 58, -12, 78, 44, 80, 38, 82, -14, 62
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2020

Keywords

Links

Crossrefs

Cf. A000203, A033879, A057723, A308135, A336563, A336565, A336566.
Cf. A013661, A065487.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 1; a[n_] := n - Times @@ f @@@ (fct = FactorInteger[n]) + Times @@ fc @@@ fct; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)
  • PARI
    A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336564(n) = (n - A308135(n));

Formula

a(n) = n - A308135(n) = n - (sigma(n) - A057723(n)).
a(n) = A336563(n) + A033879(n). [Corrected by Georg Fischer, Dec 13 2022]
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - A013661 + 1 = 0.586357... . - Amiram Eldar, Dec 08 2023

A336565 Numbers k for which (A057723(k)-k) is equal to gcd(k-A308135(k), A057723(k)-k).

Original entry on oeis.org

6, 28, 234, 496, 588, 600, 1521, 1638, 6552, 8128, 55860, 89376, 33550336, 168836850
Offset: 1

Views

Author

Antti Karttunen, Jul 26 2020

Keywords

Comments

Numbers k for which A336563(k) = A336566(n) [= gcd(A336563(n), A336564(n))].
Numbers k such that either both A336563(k) and A336564(k) are zero (in which case k is squarefree), or A336563(k) divides A336564(k), in which case k is not squarefree.
Also numbers k for which A336647(n) = 2*n - A057723(n).
Question: Are there any other odd terms apart from 1521 = 39^2 ?

Links

Crossrefs

Cf. A057723, A308135, A336563, A336564, A336566, A336647.
Cf. A000396 (a subsequence).
Cf. also A326145.

Programs

  • PARI
    A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
isA336565(n) = { my(b=A057723(n), c=(sigma(n)-b), d=(b-n)); (gcd(d,(n-c))==d); };

A336567 Sum of proper divisors of {n divided by its largest squarefree divisor}.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 7, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 4, 1, 0, 0, 0, 15, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 4, 0, 3, 0, 0, 0, 1, 0, 0, 1, 31, 0, 0, 0, 1, 0, 0, 0, 16, 0, 0, 1, 1, 0, 0, 0, 7, 13, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 15, 0, 1, 1, 8, 0, 0, 0, 3, 0
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2020

Keywords

Links

Crossrefs

Cf. A000203, A001065, A003557, A007947, A057723, A335341, A336563.

Programs

Formula

a(n) = A001065(A003557(n)).
a(n) = A335341(n) - A003557(n) = A336563(n) / A007947(n).

A336647 a(n) = n - A336566(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 42, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 62, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 74, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86
Offset: 1

Views

Author

Antti Karttunen, Jul 30 2020

Keywords

Comments

Some terms, for example a(600) and a(6552), are negative. - Georg Fischer, Jul 31 2020

Links

Crossrefs

Cf. A000203, A007947, A057723, A308135, A336563, A336564, A336566.
Cf. A336555 (positions where differs from A336646).
Cf. A336565 (positions where a(n) = 2*n - A057723(n) = n - A336563(n)).
Cf. also A336645.

Programs

Formula

a(n) = n - A336566(n).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.