A094387 -id:A094387 - cp's OEIS frontend

A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 28, 29, 31, 32, 33, 37, 39, 41, 43, 44, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 88, 89, 92, 93, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A034968(k)) = 1.
The factorial numbers (A000142) are terms. These are also the only factorial base Niven numbers (A118363) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 601, 6064, 60729, 607567, 6083420, 60827602, 607643918, 6079478119, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A034968(3) = 2, and gcd(3, 2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034968, A059956, A118363.
Subsequences: A000040, A000142.
Similar sequences: A094387, A339076, A358975, A358977, A358978.

q[n_] := Module[{k = 2, s = 0, m = n, r}, While[m > 0, r=Mod[m,k]; s+=r; m=(m-r)/k; k++]; CoprimeQ[n, s]]; Select[Range[120], q]

is(n)={my(k=2, s=0, m=n); while(m>0, s+=m%k; m\=k; k++); gcd(s,n)==1;}

A358978 Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 15, 17, 19, 20, 21, 23, 25, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 47, 49, 50, 51, 52, 53, 55, 57, 59, 61, 62, 63, 64, 65, 67, 70, 71, 73, 75, 77, 79, 83, 85, 87, 88, 89, 91, 95, 97, 98, 100, 101, 103, 104, 107, 109

Offset: 1

Amiram Eldar, Dec 07 2022

First differs from A063743 at n = 22.
Numbers k such that gcd(k, A007895(k)) = 1.
The Fibonacci numbers (A000045) are terms. These are also the only Zeckendorf-Niven numbers (A328208) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 61, 614, 6028, 61226, 606367, 6041106, 61235023, 612542436, 6034626175, 60093287082, 609082612171, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A007895(3) = 1, and gcd(3, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007895, A059956, A063743, A328208.
Subsequences: A000040, A000045.
Similar sequences: A094387, A339076, A358975, A358976, A358977.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; Select[Range[120], CoprimeQ[#, z[#]] &] (* after Alonso del Arte at A007895 *)

is(n) = if(n<4, 1, my(k=2, m=n, s, t); while(fibonacci(k++)<=m, ); while(k && m, t=fibonacci(k); if(t<=m, m-=t; s++); k--); gcd(n, s)==1); \\ after Charles R Greathouse IV at A007895

A348416 For n >= 1; a(n) = gcd(n,w(n)) where w(n) is the binary weight of n, A000120(n).

Original entry on oeis.org

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

Offset: 1

Ctibor O. Zizka, Oct 19 2021

a(n) is even for n in A125592, a(n) = 1 for n in A094387.

			n = 6; gcd(6, A000120(6)) = 2, thus a(6) = 2.

Cf. A000120, A001969, A049445, A094387, A125592.

Array[GCD[#, Total@ IntegerDigits[#, 2]] &, 105] (* Michael De Vlieger, Oct 19 2021 *)

a(n) = gcd(n, hammingweight(n)); \\ Michel Marcus, Oct 19 2021

a(n) = gcd(n, A000120(n)).

a(n) = A000120(n) if and only if n is in A049445. - Amiram Eldar, Oct 19 2021

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A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

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A358978 Numbers that are coprime to the number of terms in their Zeckendorf representation (A007895).

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A348416 For n >= 1; a(n) = gcd(n,w(n)) where w(n) is the binary weight of n, A000120(n).

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