A073454 -id:A073454 - cp's OEIS frontend

A274320 Least inverse of A073454: Smallest m such that m divided by the primes up to m have exactly n repeated residues.

6, 15, 35, 95, 187, 259, 671, 903, 905, 1273, 1967, 2938, 3161, 4382, 6004, 6005, 9718, 11049, 12371, 14194, 16181, 17285, 20842, 27242, 27257, 31937, 35758, 35767, 50407, 54071, 56345, 59917, 59923, 75898, 86833, 86839, 106999, 116651, 116653, 134027, 134034, 134041, 156138, 171613, 173499, 188170, 194554, 194555, 228122, 253291, 253327, 260374, 302371, 302395, 302396, 346837, 368983, 376262, 376267, 376268, 376270

Offset: 1

Charles R Greathouse IV, Jun 17 2016

nonn

Trivially a(n) >= prime(n+1). I would like to see a better lower bound.

			The primes up to 15 are (2, 3, 5, 7, 11, 13) and 15 mod each of these primes leaves residues of (1, 0, 0, 1, 4, 2). There are two duplicates (1 appears twice and so does 0) and no smaller number has this property, so a(2) = 15.

Charles R Greathouse IV, Table of n, a(n) for n = 1..134

Cf. A073453, A073454.

a(n)=my(P=List(),m=1); while(#P-#Set(apply(p->m%p, P)) != n, if(isprime(m++), listput(P,m))); m

A073453 Number of distinct remainders arising when n is divided by all primes up to n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 6, 6, 7, 7, 6, 6, 7, 8, 8, 8, 8, 8, 9, 8, 9, 10, 10, 9, 8, 9, 10, 10, 9, 10, 11, 11, 12, 12, 12, 12, 13, 14, 14, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 16, 16, 16, 17, 16, 15, 16, 16, 16, 16, 17, 18, 19, 19, 19, 19, 18, 18, 19, 19, 20

Offset: 1

Labos Elemer, Aug 02 2002

nonn

			n=25: Primes are (2,3,5,7,11,13,17,19,23), remainders are (1,1,0,4,3,12,8,6,2), distinct remainders are {0,1,2,3,4,6,8,12} which has 8 members, so a(25) = 8.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A072530, A072531, A073454, A000720.

Table[Length[Union[Table[Mod[w, Prime[j]], {j, 1, PrimePi[w]}]]], {w, 1, 256}]
Table[Length[Union[Mod[n,Prime[Range[PrimePi[n]]]]]],{n,100}] (* Harvey P. Dale, Jul 04 2021 *)

a(n) = #Set(vector(primepi(n), k, n % prime(k))); \\ Michel Marcus, May 28 2016

a(n) = #Set(apply(p->n%p, primes([2,n]))) \\ Charles R Greathouse IV, Jun 17 2016

See program below.

a(n) = n + 1 - Sum_{k=1..n-1} ( floor((k-1)!^(n-1)/(n-k+1))-floor(((k-1)!^(n-1)-1)/(n-k+1)) ). - Anthony Browne, May 27 2016

A073464 a(n) = phi(n) mod PrimePi(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 4, 0, 0, 2, 2, 2, 6, 2, 0, 4, 2, 4, 8, 2, 3, 0, 3, 8, 8, 8, 5, 9, 5, 2, 1, 0, 6, 0, 4, 1, 12, 0, 6, 10, 8, 1, 1, 12, 5, 2, 9, 4, 2, 8, 8, 4, 12, 7, 16, 6, 12, 0, 14, 12, 2, 9, 13, 6, 5, 10, 4, 9, 15, 19, 15, 18, 3, 12, 10, 10, 18, 13, 1, 18, 19, 10, 17, 16, 0, 0, 20, 12

Offset: 2

Labos Elemer, Aug 02 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..10000

Cf. A000010, A000720.
Cf. A037228, A037171.
Cf. A073453, A073454, A073455, A073456, A073457, A072530, A072531, A073461, A073461.

[EulerPhi(n) mod #PrimesUpTo(n): n in [2..100]]; // Vincenzo Librandi, Dec 11 2018

a(n)=Table[Mod[EulerPhi[w], PrimePi[w]], {w, 2, 1000}]

a(n) = eulerphi(n) % primepi(n); \\ Michel Marcus, Dec 11 2018

a(n) = A000010(n) mod A000720(n).

cp's OEIS Frontend

A274320 Least inverse of A073454: Smallest m such that m divided by the primes up to m have exactly n repeated residues.

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A073453 Number of distinct remainders arising when n is divided by all primes up to n.

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A073464 a(n) = phi(n) mod PrimePi(n).

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