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 1-4 of 4 results.

A308135 Sum of non-coreful divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 10, 1, 10, 9, 1, 1, 15, 1, 12, 11, 14, 1, 18, 1, 16, 1, 14, 1, 42, 1, 1, 15, 20, 13, 19, 1, 22, 17, 20, 1, 54, 1, 18, 18, 26, 1, 34, 1, 33, 21, 20, 1, 42, 17, 22, 23, 32, 1, 78, 1, 34, 20, 1, 19, 78, 1, 24, 27, 74, 1, 27, 1, 40
Offset: 1

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Author

Amiram Eldar and Paolo P. Lava, May 14 2019

Keywords

Comments

Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Examples

			a(15) = 9. Prime factors of 15 are 3, 5 and its divisors are 1, 3, 5, 15. The non-coreful divisors are 1, 3, 5 and their sum is 9.

Links

Crossrefs

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308127.
Cf. A013661, A065487.

Programs

  • Maple
    with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
sigma(k)-a*sigma(k/a); end: seq(P(i),i=1..74);
  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 0; a[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Array[a, 100]

Formula

a(n) = A000203(n) - A057723(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065487 = 0.413642... . - Amiram Eldar, Dec 08 2023

A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.

Original entry on oeis.org

6, 1638, 55860, 168836850, 12854283750
Offset: 1

Views

Author

Paolo P. Lava, May 10 2019

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Sequence is a subset of A083207.
Tested up to 10^12. - Giovanni Resta, May 10 2019

Examples

			Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.

Links

  • G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

Crossrefs

Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
then print(n); fi; od; end: P(10^7);
  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* Amiram Eldar, May 11 2019 *)
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
isok(n) = 2*s(n) == sigma(n); \\ Michel Marcus, May 11 2019

Formula

Solutions of A000203(k) = 2*A057723(k).

Extensions

a(4)-a(5) from Giovanni Resta, May 10 2019

A308127 Non-coreful abundant numbers: numbers k such that ncsigma(k) > k, where ncsigma(k) is the sum of the non-coreful divisors of k (A308135).

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 186, 198, 210, 222, 240, 246, 258, 270, 282, 294, 300, 318, 330, 336, 354, 366, 378, 390, 402, 420, 426, 438, 450, 462, 474, 480, 498, 510, 534, 546, 570, 582, 606, 618, 630
Offset: 1

Views

Author

Amiram Eldar and Paolo P. Lava, May 14 2019

Keywords

Comments

Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Examples

			60 is in the sequence since its non-coreful divisors are 1, 2, 3, 4, 5, 6, 10, 12, 15, and 20 whose sum is 78 > 60.

Links

Crossrefs

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308135.

Programs

  • Maple
    with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
if sigma(k)-a*sigma(k/a)>k then k; fi;  end: seq(P(i),i=1..630);
  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncAbQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]) > n; Select[Range[2, 1000], ncAbQ]
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(r=rad(n)); sumdiv(n, d, if (rad(d)!=r, d));
isok(n) = s(n) > n; \\ Michel Marcus, May 14 2019

A307986 Amicable pairs {x, y} such that y is the sum of the divisors of x that are not divided by every prime factor of x and vice versa.

Original entry on oeis.org

42, 54, 198, 204, 582, 594, 142310, 168730, 1077890, 1099390, 1156870, 1292570, 1511930, 1598470, 1669910, 2062570, 2236570, 2429030, 2728726, 3077354, 4246130, 4488910, 4532710, 5123090, 5385310, 5504110, 5812130, 6135962, 6993610, 7158710, 7288930, 8221598
Offset: 1

Views

Author

Paolo P. Lava, May 09 2019

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Here, only the non-coreful divisors of k are considered.
The non-coreful perfect numbers listed in A307888 are not considered here.
The first time a pair ordered by its first element is not adjacent is for x = 4532710 and y = 6135962, which correspond to a(23) and a(28), respectively.

Examples

			Divisors of x = 42 are 1, 2, 3, 6, 7, 14, 21, 42 and prime factors are 2, 3, 7. Among the divisors, 42 is the only one that is divisible by every prime factor, so we have 1 + 2 + 3 + 6 + 7 + 14 + 21 = 54 = y.
Divisors of y = 54 are 1, 2, 3, 6, 9, 18, 27, 54 and prime factors are 2, 3. Among the divisors, 6, 18, 54 are the only ones that are divisible by every prime factor, so we have 1 + 2 + 3 + 9 + 27 = 42 = x.

Links

Crossrefs

Cf. A000203, A007947, A057723, A307888, A307958, A307962, A307963, A308029.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do
a:=mul(k,k=factorset(n)); b:=sigma(n)-a*sigma(n/a);
a:=mul(k,k=factorset(b)); c:=sigma(b)-a*sigma(b/a);
if c=n and b<>c then print(n); fi; od; end: P(10^8);
  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncs[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); seq = {}; Do[m = ncs[n]; If[m > 1 && m != n && n == ncs[m], AppendTo[seq, n]], {n, 2, 10^6}]; seq (* Amiram Eldar, May 11 2019 *)
Showing 1-4 of 4 results.

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