A057859 -id:A057859 - cp's OEIS frontend

A057860 Number of residue classes modulo n which contain only composite numbers.

0, 0, 0, 1, 0, 2, 0, 3, 2, 4, 0, 6, 0, 6, 5, 7, 0, 10, 0, 10, 7, 10, 0, 14, 4, 12, 8, 14, 0, 19, 0, 15, 11, 16, 9, 22, 0, 18, 13, 22, 0, 27, 0, 22, 19, 22, 0, 30, 6, 28, 17, 26, 0, 34, 13, 30, 19, 28, 0, 41, 0, 30, 25, 31, 15, 43, 0, 34, 23

Offset: 1

Henry Bottomley, Sep 08 2000

			a(30) = 19 since 30k+m is always composite if m = 0, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27 or 28

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A057858.

Table[n - EulerPhi[n] - PrimeNu[n], {n, 1, 100}] (* G. C. Greubel, May 13 2017 *)

for(n=1,100, print1(n - eulerphi(n) - omega(n), ", ")) \\ G. C. Greubel, May 13 2017

a(n) = n - A057859(n) = A051953(n) - A001221(n).

A179179 a(n) = phi(n) - omega(n) = A000010(n) - A001221(n).

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 3, 5, 2, 9, 2, 11, 4, 6, 7, 15, 4, 17, 6, 10, 8, 21, 6, 19, 10, 17, 10, 27, 5, 29, 15, 18, 14, 22, 10, 35, 16, 22, 14, 39, 9, 41, 18, 22, 20, 45, 14, 41, 18, 30, 22, 51, 16, 38, 22, 34, 26, 57, 13, 59, 28, 34, 31, 46, 17, 65, 30, 42, 21, 69, 22, 71, 34, 38, 34, 58

Offset: 1

Peter Luschny, Jun 30 2010

nonn

a(n) is the number of positive integers which are coprime to n minus the number of distinct primes dividing n.

			a(7) = phi(7) - omega(7) = card({1,2,3,4,5,6}) - card({7}) = 6 - 1 = 5

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Peter Luschny, Blog on OEIS, Euler's totient function

Cf. A000010, A001221, A057859.

with(numtheory): a := n -> phi(n) - nops(factorset(n));

Table[EulerPhi[n] - PrimeNu[n], {n,1,50}] (* G. C. Greubel, Apr 23 2017 *)

a(n) = phi(n) - omega(n), by definition.

A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 5, 7, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 1, 2, 3, 5, 7, 9, 11, 13, 15

Offset: 2

Roger L. Bagula and Gary W. Adamson, Aug 07 2008

			Table begins:0, 1;
0, 1, 2;
1, 2, 3;
0, 1, 2, 3, 4;
1, 2, 3, 5;
0, 1, 2, 3, 4, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 5, 7, 11;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 5, 7, 9, 11, 13;
1, 2, 3, 4, 5, 7, 8, 11, 13, 14;
1, 2, 3, 5, 7, 9, 11, 13, 15;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
1, 2, 3, 5, 7, 11, 13, 17;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18;
1, 2, 3, 5, 7, 9, 11, 13, 17, 19;

Michael De Vlieger, Table of n, a(n) for n = 2..10209 (rows n = 2..180, flattened)

Cf. A057859 (row lengths), A039701 (row n=3), A039704 (row n=6), A027748, A038566.

Table[Union[FactorInteger[n][[All, 1]] /. n -> 0, Select[Range[n - 1], CoprimeQ[n, #] &]], {n, 2, 15}] (* Michael De Vlieger, Apr 18 2022 *)

Row n = A027748(n) U A038566(n), writing n as 0 iff n is prime. - Michael De Vlieger, Apr 18 2022

A364933 a(n) = Sum_{k=1..n} A191898(n,k)*[A191904(n,k) = A191898(n,k)].

Original entry on oeis.org

0, -1, -1, 0, -1, -2, -1, 2, 3, -2, -1, 0, -1, -2, 1, 6, -1, 2, -1, 2, 3, -2, -1, 4, 15, -2, 15, 4, -1, 0, -1, 14, 7, -2, 13, 8, -1, -2, 9, 10, -1, 2, -1, 8, 17, -2, -1, 12, 35, 14, 13, 10, -1, 14, 25, 16, 15, -2, -1, 8, -1, -2, 27, 30, 31, 6, -1, 14, 19, 12

Offset: 1

Mats Granvik, Aug 13 2023

sign

Cf. A191898, A191904, A057859, A008472.

f[n_] := DivisorSum[n, MoebiusMu[#] # &]; Table[Sum[If[If[Mod[n, k] == 0, 1 - k, 1] == f[GCD[n, k]], f[GCD[n, k]], 0], {k, 1, n}], {n, 1, 70}]

a(n) = Sum_{k=1..n} A191898(n,k)*[A191904(n,k) = A191898(n,k)].

Conjecture: a(n) = A057859(n) - A008472(n) - 1.

cp's OEIS Frontend

A057860 Number of residue classes modulo n which contain only composite numbers.

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A179179 a(n) = phi(n) - omega(n) = A000010(n) - A001221(n).

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A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n.

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A364933 a(n) = Sum_{k=1..n} A191898(n,k)*[A191904(n,k) = A191898(n,k)].

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