A293433 -id:A293433 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Nov 08 2008

nonn

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

Antti Karttunen, Table of n, a(n) for n = 0..43692
Index entries for characteristic functions

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

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[If[FreeQ[s, #], 0, 1] &, 105, 0]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1)); \\ Antti Karttunen, Oct 09 2017

def A147612(n): return int((m:=3*n+1).bit_length()>(m-3).bit_length()) if n else 1 # Chai Wah Wu, Apr 18 2025

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

A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 4, 1, 3, 2, 2, 2, 3, 1, 2, 3, 4, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 4, 1, 4, 4

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 55, its proper divisors are [1, 5, 11], of which only two, namely 1 and 5 are in A000045, thus a(55) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10946

Cf. A000045, A005086, A010056, A079586, A293436.

Cf. also A293433.

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

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
A293435(n) = sumdiv(n,d,(dA010056(d));

a(n) = Sum_{d|n, dA010056(d).
a(n) = A005086(n) - A010056(n).
G.f.: Sum_{k>=2} x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - Amiram Eldar, Jul 05 2025

A293431 a(n) is the number of Jacobsthal numbers dividing n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 2, 1, 3, 2, 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, 3, 2, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 15, whose divisors are [1, 3, 5, 15], the first three, 1, 3 and 5 are all in A001045, thus a(15) = 3.
For n = 21, whose divisors are [1, 3, 7, 21], 1, 3 and 21 are in A001045, thus a(21) = 3.
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) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..21845

Cf. A000005, A001045, A147612, A293432, A293433.

Cf. also A005086.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, 1 &, MemberQ[s, #] &] &, 105]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293431(n) = sumdiv(n,d,A147612(d));

from sympy import divisors
def A293431(n): return sum(1 for d in divisors(n,generator=True) if (m:=3*d+1).bit_length()>(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

a(n) = Sum_{d|n} A147612(d).
a(n) = A293433(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, Jan 01 2024

A293434 a(n) is the sum of the proper divisors of n that are Jacobsthal numbers (A001045).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 12, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 15, 1, 6, 4, 1, 1, 4, 6, 1, 25, 1, 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, 6, 44, 4, 12, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 15, 6, 1, 4, 1, 1, 30

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 15, whose proper divisors are [1, 3, 5], all of them are in A001045, thus a(15) = 1 + 3 + 5 = 9.
For n = 21, whose proper divisors are [1, 3, 7], both 1 and 3 are in A001045, thus a(21) = 1 + 3 = 4.
For n = 21845, whose proper divisors are [1, 5, 17, 85, 257, 1285, 4369], only 1, 5, 85 are in A001045, thus a(21845) = 1 + 5 + 85 = 91.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to sums of divisors

Cf. A001045, A147612, A293432, A293433.

Cf. also A293436.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, # &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

A147612aux(n,i) = if(!(n%2),n,A147612aux((n+i)/2,-i));
A147612(n) = 0^(A147612aux(n,1)*A147612aux(n,-1));
A293434(n) = sumdiv(n,d,(dA147612(d)*d);

from sympy import divisors
def A293434(n): return sum(d for d in divisors(n,generator=True) if d(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

a(n) = Sum_{d|n, dA147612(d)*d.

a(n) = A293432(n) - (A147612(n)*n).

A294879 Number of proper divisors of n that are in Perrin sequence, A001608.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 3, 2, 1, 0, 3, 1, 1, 1, 2, 0, 4, 0, 1, 1, 2, 2, 3, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 3, 1, 3, 2, 1, 0, 2, 1, 2, 1, 2, 0, 5, 0, 1, 2, 1, 1, 3, 0, 2, 1, 4, 0, 3, 0, 1, 2, 1, 1, 3, 0, 3, 1, 1, 0, 4, 2, 1, 2, 2, 0, 4, 1, 1, 1, 1, 1, 3, 0, 2, 1, 3, 0, 4, 0, 1, 3, 1, 0, 3, 0, 4, 1, 2, 0, 2, 1, 2, 2, 1, 2, 5, 0

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n = 22, with proper divisors [1, 2, 11], only 2 is in A001608, thus a(22) = 1.
For n = 121, with proper divisors [1, 11], neither of them is in A001608, thus a(121) = 0. Note that this is the first zero not in A008578.
For n = 644, with proper divisors [1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644], both 2 and 7 are in A001608, thus a(644) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..24914

Cf. A001608, A014981, A074788, A294878, A294880.

Cf. also A293433, A293435.

A001608(n) = if(n<0, 0, polsym(x^3-x-1, n)[n+1]);
A294878(n) = { my(k=1,v); while((v=A001608(k))A294879(n) = sumdiv(n,d,(dA294878(d));

a(n) = Sum_{d|n, dA294878(d).

a(n) = A294880(n) - A294878(n).

cp's OEIS Frontend

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

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A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

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A293431 a(n) is the number of Jacobsthal numbers dividing n.

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A293434 a(n) is the sum of the proper divisors of n that are Jacobsthal numbers (A001045).

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A294879 Number of proper divisors of n that are in Perrin sequence, A001608.

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