A293435 -id:A293435 - cp's OEIS frontend

A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

			For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000045, A007895, A014417, A072649, A300834, A300837.

Cf. also A292257, A293435.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A300836(n) = sumdiv(n,d,(dA007895(d));

a(n) = Sum_{d|n, dA007895(d).
a(n) = A300837(n) - A007895(n).
a(n) = A001222(A300834(n)).
For all n >=1, a(n) >= A293435(n).

A304102 a(n) = Product_{d|n, dA304101(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 120, 2, 12, 8, 24, 2, 200, 2, 120, 12, 12, 2, 1680, 4, 8, 20, 180, 2, 2000, 2, 120, 12, 44, 12, 12600, 2, 44, 8, 1680, 2, 1200, 2, 180, 200, 20, 2, 42000, 6, 440, 44, 120, 2, 7800, 12, 3960, 44, 12, 2, 3234000, 2, 44, 120, 840, 8, 10200, 2, 264, 20, 3000, 2, 630000, 2, 20, 440, 1452, 18, 2000, 2, 109200, 260, 44, 2, 1386000

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304103 (restricted growth sequence transform of this sequence), A304104.

\\ Needs also code from A304101:
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };

a(n) = Product_{d|n, dA000040(A304101(d)-1).
a(n) = 2*A304104(n) / A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A293435(n).
A007949(a(n)) = A304095(n).

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

Antti Karttunen, Oct 09 2017

nonn

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

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

Cf. A001045, A147612, A293431, A293434.

Cf. also A293435.

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Table[DivisorSum[n, 1 &, 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));
A293433(n) = sumdiv(n,d,(dA147612(d));

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

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

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 5, 4, 7, 2, 8, 2, 6, 5, 5, 2, 9, 3, 4, 10, 11, 2, 12, 2, 6, 5, 13, 5, 14, 2, 13, 4, 9, 2, 15, 2, 11, 8, 10, 2, 16, 17, 18, 13, 6, 2, 19, 5, 20, 13, 5, 2, 21, 2, 13, 6, 22, 4, 23, 2, 24, 10, 25, 2, 26, 2, 10, 18, 27, 28, 12, 2, 29, 30, 13, 2, 31, 13, 32, 5, 33, 2, 34, 5, 35, 13, 5, 13, 21, 2, 36, 37, 38, 2, 39, 2, 9, 15

Offset: 1

Antti Karttunen, May 13 2018

nonn

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A293435, A304095, A300836, A304102.
Cf. also A300835, A304105, A305800.
Cf. A305793 (analogous filter for base 2).

\\ Needs also code from A304101.
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304102(n))),"b304103.txt");

A304091 a(n) is the number of the proper divisors of n that are Lucas numbers (A000032, with 2 included).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000032, A093540, A102460, A304092, A304093.

Cf. also A293435, A300836.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304091(n) = sumdiv(n,d,(dA102460(d));

a(n) = Sum_{d|n, dA102460(d).
a(n) = A304092(n) - A102460(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2 + A093540 = 2.462858... . - Amiram Eldar, Jul 05 2025

A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000204, A093540, A304091, A304094, A304095.

Cf. also A293435, A300836.

isA000204(n) = { my(u1=1,u2=3,old_u1); if(n<=2,(n%2),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304093(n) = sumdiv(n,d,(dA000204(d));

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540. - Amiram Eldar, Jul 05 2025

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

Antti Karttunen, Oct 09 2017

nonn

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10946
Index entries for sequences related to sums of divisors

Cf. A000045, A005092, A010056, A293435.

Cf. also A293434.

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, # &, 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
A293436(n) = sumdiv(n,d,(dA010056(d)*d);

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

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

A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.

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A304102 a(n) = Product_{d|n, dA304101(d)-1).

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

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A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

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A304091 a(n) is the number of the proper divisors of n that are Lucas numbers (A000032, with 2 included).

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A304093 a(n) is the number of the proper divisors of n that are Lucas numbers (A000204, with 2 excluded).

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

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

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