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.

A069822 Numbers k for which there is at least one x < k such that sigma(x) = sigma(k).

Original entry on oeis.org

11, 15, 17, 23, 25, 26, 31, 35, 38, 39, 41, 46, 47, 51, 53, 55, 56, 58, 59, 62, 65, 69, 70, 71, 75, 77, 78, 79, 82, 83, 87, 89, 92, 94, 95, 97, 103, 107, 110, 113, 115, 116, 118, 119, 122, 123, 124, 125, 127, 130, 131, 135, 139, 140, 141, 142, 143, 145, 151, 153
Offset: 1

Views

Author

Benoit Cloitre, Apr 28 2002

Keywords

Comments

Numbers k for which A263025(k) > 1. - Antti Karttunen, Dec 08 2017

Links

Crossrefs

Cf. A263025, A296214.
Cf. A296087 (a subsequence).

Programs

  • PARI
    for(s=1,200,if(prod(i=1,s-1, sigma(i)-sigma(s))==0,print1(s,",")))
  • PARI
    for(n=1,16384,y=0;s=sigma(n);for(k=1,(n-1),if(sigma(k)==s,y=1;break)); if(y,i++;write("b069822.txt", i," ", n))); \\ Antti Karttunen, Dec 08 2017
  • PARI
    is(k) = invsigmaMin(sigma(k)) < k; \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

A380653 Number of positive integers less than or equal to n that have the same sum of prime factors (with repetition) as n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 1, 4, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 5, 2, 3, 1, 2, 1, 6, 2, 3, 1, 2, 1, 4, 1, 1, 4, 5, 1, 3, 1, 2, 1, 3, 1, 6, 1, 1, 5, 2, 2, 3, 1, 6, 7, 2, 1, 4, 2, 1, 1, 3, 1, 7, 2, 1, 1, 1, 1, 8, 1, 4, 4, 5
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 29 2025

Keywords

Comments

Ordinal transform of A001414.

Links

Crossrefs

Cf. A001414, A058933, A263025, A380654.

Programs

  • Maple
    b:= n-> add(i[1]*i[2], i=ifactors(n)[2]):
p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= b(n); p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 30 2025
  • Mathematica
    sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger@ n; Table[Length[Select[Range[n], sopfr[#] == sopfr[n] &]], {n, 1, 100}]
  • Python
    from sympy import factorint
from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    sopfrcount = Counter()
    for n in count(1):
        key = sum(p*e for p, e in factorint(n).items())
        sopfrcount[key] += 1
        yield sopfrcount[key]
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 30 2025

Formula

a(n) = |{j <= n : sopfr(j) = sopfr(n)}|.

A380654 Number of positive integers less than or equal to n that have the same sum of distinct prime factors as n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 4, 1, 3, 1, 2, 1, 5, 6, 1, 3, 2, 1, 2, 1, 5, 1, 2, 1, 7, 1, 1, 1, 4, 1, 2, 1, 3, 2, 1, 1, 8, 5, 6, 1, 2, 1, 9, 2, 3, 1, 2, 1, 3, 1, 1, 4, 6, 1, 3, 1, 3, 1, 2, 1, 10, 1, 1, 3, 2, 2, 3, 1, 7, 4, 2, 1, 3, 2, 1, 1, 4, 1, 5, 2, 2, 1, 1, 1, 11, 1, 4, 3, 8
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 29 2025

Keywords

Comments

Ordinal transform of A008472.

Links

Crossrefs

Cf. A008472, A067003, A263025, A380653.

Programs

  • Maple
    b:= n-> add(i[1], i=ifactors(n)[2]):
p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= b(n); p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 30 2025
  • Mathematica
    sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; Table[Length[Select[Range[n], sopf[#] == sopf[n] &]], {n, 1, 100}]
  • Python
    from sympy import factorint
from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    sopfcount = Counter()
    for n in count(1):
        key = sum(p for p in factorint(n))
        sopfcount[key] += 1
        yield sopfcount[key]
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 30 2025

Formula

a(n) = |{j <= n : sopf(j) = sopf(n)}|.

A338504 Number of integers less than n with the same sum of proper divisors as n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 4, 0, 5, 0, 0, 0, 6, 0, 7, 0, 0, 0, 8, 0, 1, 1, 0, 0, 9, 0, 10, 0, 1, 0, 1, 0, 11, 1, 0, 0, 12, 0, 13, 0, 0, 0, 14, 0, 1, 0, 1, 0, 15, 0, 1, 0, 0, 0, 16, 0, 17, 0, 0, 0, 0, 0, 18, 0, 0, 0, 19, 0, 20, 1, 0, 1, 1, 0, 21, 0, 2, 0, 22, 0, 1, 1, 1, 0, 23, 0, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 31 2020

Keywords

Examples

			a(5) = 2 because A001065(5) = 1 and also A001065(2) = A001065(3) = 1.

Crossrefs

Cf. A001065, A047983, A263025.

Programs

  • Mathematica
    Table[Length[Select[Range[n - 1], DivisorSigma[1, #] - # == DivisorSigma[1, n] - n &]], {n, 91}]
  • PARI
    a(n)={my(t=sigma(n)-n); sum(k=1, n-1, sigma(k)-k==t)} \\ Andrew Howroyd, Oct 31 2020

Formula

a(n) = |{j < n : sigma(j) - j = sigma(n) - n}|.

A380584 Number of positive integers <= n that have the same sum of odd divisors as n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 3, 2, 6, 1, 2, 2, 3, 1, 2, 1, 4, 1, 3, 1, 3, 1, 4, 3, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 5, 1, 4, 1, 7, 1, 4, 1, 3, 1, 5, 3, 4, 1, 2, 1, 3, 2, 2, 2, 5, 1, 2, 2, 5, 1, 2, 1, 4, 1, 2, 1, 6, 1, 6, 2, 6, 1, 2, 1, 3
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2025

Keywords

Comments

Ordinal transform of A000593.

Links

Crossrefs

Cf. A000593, A263025, A337557, A380653, A380654.

Programs

  • Mathematica
    Table[Length[Select[Range[n], Sum[Mod[d, 2] d, {d, Divisors[#]}] == Sum[Mod[d, 2] d, {d, Divisors[n]}] &]], {n, 1, 100}]

Formula

a(n) = |{j <= n : A000593(j) = A000593(n)}|.
Showing 1-5 of 5 results.

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