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.

A147612 If n is a Jacobsthal number then 1 else 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 08 2008

Keywords

Comments

a(A001045(n)) = 1; a(A147613(n)) = 0.

Links

Crossrefs

Cf. A001045, A105348, A265746, A293431, A293432, A293433, A293434.

Programs

Formula

a(n) = 0^(j(n,1)*j(n,-1)) with j(n,i) = if n mod 2 = 0 then n else j((n+i)/2,-i).
a(n) = A105348(n), for n <> 1. - R. J. Mathar, Nov 19 2008
For n > 0, a(n) = A000035(A281228(A265746(n))), where A000035(A281228(n)) is the characteristic function of powers of 3 (A000244). - Antti Karttunen, Oct 09 2017

A293433 a(n) is the number of the proper divisors of n that are Jacobsthal numbers (A001045).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2017

Keywords

Examples

			For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 2.

Links

Crossrefs

Cf. A001045, A147612, A293431, A293434.
Cf. also A293435.

Programs

  • Mathematica
    With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)
  • PARI
    A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293433(n) = sumdiv(n,d,(dA147612(d));
  • Python
    from sympy import divisors
def A293433(n): return sum(1 for d in divisors(n,generator=True) if d(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

Formula

a(n) = Sum_{d|n, dA147612(d).
a(n) = A293431(n) - A147612(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A001045(n) = 1.718591611927... . - Amiram Eldar, Jul 05 2025

A293432 Sum of Jacobsthal numbers that divide n.

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 12, 4, 1, 1, 9, 1, 1, 4, 1, 6, 25, 12, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 15, 1, 6, 4, 1, 1, 4, 6, 1, 25, 44, 12, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 17, 1, 4, 1, 1, 9, 1, 1, 25, 1, 6, 15, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 12, 4, 1, 6, 4, 1, 1, 25, 91, 44, 4, 12, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 15, 6, 1, 4, 1, 1, 30
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2017

Keywords

Comments

a(n) is the sum of the divisors of n that are Jacobsthal numbers (A001045).

Examples

			For n = 15, whose divisors are [1, 3, 5, 15], the first three, 1, 3 and 5 are all in A001045, thus a(15) = 1 + 3 + 5 = 9.
For n = 105, whose divisors are [1, 3, 5, 7, 15, 21, 35, 105], only the divisors 1, 3, 5 and 21 are in A001045, thus a(105) = 1 + 3 + 5 + 21 = 30.
For n = 21845, whose divisors are [1, 5, 17, 85, 257, 1285, 4369, 21845], the divisors 1, 5, 85 and 21845 are in A001045, thus a(21845) = 1 + 5 + 85 + 21845 = 21936.

Links

Crossrefs

Cf. A000203, A001045, A147612, A293431, A293434.
Cf. also A005092.

Programs

  • Mathematica
    With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, # &, MemberQ[s, #] &] &, 105]] (* Michael De Vlieger, Oct 09 2017 *)
  • PARI
    A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293432(n) = sumdiv(n,d,A147612(d)*d);
  • Python
    from sympy import divisors
def A293432(n): return sum(d for d in divisors(n,generator=True) if (m:=3*d+1).bit_length()>(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

Formula

a(n) = Sum_{d|n} A147612(d)*d.
a(n) = A293434(n) + (A147612(n)*n).

A293436 a(n) is the sum of the proper divisors of n that are Fibonacci numbers (A000045).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 3, 4, 8, 1, 6, 1, 3, 9, 11, 1, 6, 1, 8, 4, 3, 1, 14, 6, 16, 4, 3, 1, 11, 1, 11, 4, 3, 6, 6, 1, 3, 17, 16, 1, 27, 1, 3, 9, 3, 1, 14, 1, 8, 4, 16, 1, 6, 6, 11, 4, 3, 1, 11, 1, 3, 25, 11, 19, 6, 1, 37, 4, 8, 1, 14, 1, 3, 9, 3, 1, 19, 1, 16, 4, 3, 1, 27, 6, 3, 4, 11, 1, 11, 14, 3, 4, 3, 6, 14, 1, 3, 4, 8, 1, 40, 1, 24, 30
Offset: 1

Views

Author

Antti Karttunen, Oct 09 2017

Keywords

Examples

			For n = 55, its proper divisors are [1, 5, 11], of which only 1 and 5 are in A000045, thus a(55) = 1 + 5 = 6.
For n = 10946, its proper divisors are [1, 2, 13, 26, 421, 842, 5473], and only 1, 2 and 13 are Fibonacci numbers, thus a(10946) = 1 + 2 + 13 = 16.

Links

Crossrefs

Cf. A000045, A005092, A010056, A293435.
Cf. also A293434.

Programs

  • Mathematica
    With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, # &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)
  • PARI
    A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ This function from Charles R Greathouse IV, Jul 30 2012
A293436(n) = sumdiv(n,d,(dA010056(d)*d);

Formula

a(n) = Sum_{d|n, dA010056(d)*d.
a(n) = A005092(n) - (A010056(n)*n).
G.f.: Sum_{k>=2} Fibonacci(k) * x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021
Showing 1-4 of 4 results.

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