A016726 -id:A016726 - cp's OEIS frontend

A192419 Smallest k such that 1^3, 2^3, 3^3,... n^3 are distinct modulo k.

1, 2, 3, 5, 5, 6, 10, 10, 10, 10, 11, 15, 15, 15, 15, 17, 17, 22, 22, 22, 22, 22, 23, 29, 29, 29, 29, 29, 29, 30, 33, 33, 33, 34, 41, 41, 41, 41, 41, 41, 41, 46, 46, 46, 46, 46, 47, 51, 51, 51, 51, 53, 53, 55, 55, 58, 58, 58, 59, 66, 66, 66, 66, 66, 66, 66, 69, 69, 69, 71, 71, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82

Offset: 1

R. J. Mathar, Jun 30 2011

nonn

The discriminator D(3,n).

It appears that a(n) ~ n. Is there an explicit formula as for A016726? - M. F. Hasler, May 04 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000
P. Moree, H. Roskam, On an arithmetical function related to Euler's totient and the discriminator, Fib. Quart. 33 (4) (1995) 332-340

Cf. A016726, A192420.

dis := proc(j,n) local k,s,i; for k from 1 do s := {} ; for i from 1 to n do s := s union { (i^j) mod k} ;
end do: if nops(s) = n then return k; end if; end do: end proc:
A192419 := proc(n) dis(3,n) ; end proc:

dmk[n_]:=Module[{k=1,res},While[res=Table[PowerMod[i,3,k],{i,n}]; Length[ res]!= Length[Union[res]],k++];k]; Array[dmk,90] (* Harvey P. Dale, Jan 28 2013 *)

A192419(nMax)={my(S=[],a=1);vector(nMax, n, S=concat(S,n^3); while(#Set(S%a)M. F. Hasler, May 04 2016

A192420 Smallest k such that 1^4, 2^4, 3^4,... ,n^4 are distinct modulo k.

Original entry on oeis.org

1, 2, 6, 9, 11, 14, 14, 18, 19, 22, 22, 31, 31, 31, 31, 38, 38, 38, 38, 43, 43, 46, 46, 59, 59, 59, 59, 59, 59, 62, 62, 67, 67, 71, 71, 79, 79, 79, 79, 83, 83, 86, 86, 94, 94, 94, 94, 103, 103, 103, 103, 107, 107, 118, 118, 118, 118, 118, 118, 127, 127, 127, 127, 131, 131, 134, 134, 139, 139

Offset: 1

R. J. Mathar, Jun 30 2011

nonn

The discriminator D(4,n).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
P. Moree, H. Roskam, On an arithmetical function related to Euler's totient and the discriminator, Fib. Quart. 33 (4) (1995), 332-340

Cf. A016726, A192419.

dis := proc(j,n) local k,s,i; for k from 1 do s := {} ; for i from 1 to n do s := s union { (i^j) mod k} ; end do: if nops(s) = n then return k; end if; end do: end proc:
A192420 := proc(n) dis(4,n) ; end proc:

a[n_] := For[k = 1, True, k++, If[Unequal @@ PowerMod[Range[n], 4, k], Return[k]]]; Array[a, 100] (* Jean-François Alcover, May 18 2018 *)

A192420(nMax,f=n->n^4)={my(S=[],a=1); vector(nMax,n,S=concat(S,f(n));while(#Set(S%a)M. F. Hasler, May 04 2016

A272633 Discriminator of the primes: Least m > 0 such that (prime(1),...,prime(n)) are all different mod m; a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 7, 13, 13, 13, 19, 19, 19, 23, 23, 23, 31, 31, 31, 33, 37, 37, 43, 43, 47, 49, 53, 53, 53, 55, 61, 63, 67, 73, 73, 75, 75, 79, 83, 83, 89, 89, 91, 91, 97, 103, 103, 109, 113, 113, 115, 117, 119, 121, 121, 121, 121, 121, 139, 139, 141, 141, 151, 153, 157, 157, 159, 167, 169, 169, 175, 181, 181, 183, 187

Offset: 0

M. F. Hasler, May 04 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5001 terms from M. F. Hasler)
Arnold, L. K.; Benkoski, S. J.; and McCabe, B. J.; The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.

Cf. A016726, A192419, A192420.

a:= proc(n) option remember; local m;
      for m from `if`(n=1, 1, a(n-1)) while
        n<>nops({seq(ithprime(i) mod m, i=1..n)})
      do od; m
    end: a(0):=0:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2016

a[0]=0; a[n_]:=Block[{m=1}, While[Length@ DeleteDuplicates@ Mod[Prime@ Range@ n, m] != n, m++]; m]; a /@ Range[0, 74] (* Giovanni Resta, May 04 2016 *)

A272633(nMax)={my(S=[],a=1);vector(nMax, n, S=concat(S,prime(n)); while(#Set(S%a)

A270151 Discriminator of the Fibonacci numbers; least positive integer r such that F(2), F(3), ..., F(n+1) are all incongruent modulo r.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 14, 14, 15, 15, 15, 30, 30, 30, 30, 30, 35, 35, 35, 35, 59, 59, 59, 59, 79, 79, 83, 83, 83, 83, 83, 83, 120, 120, 120, 157, 157, 157, 157, 173, 173, 173, 173, 173, 193, 193, 193, 193, 193, 193, 193, 193, 193, 193, 193, 311, 311, 311, 311, 337, 337, 337, 337, 337, 409, 409, 409, 409, 431

Offset: 1

Jeffrey Shallit, Mar 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000
Arnold, L. K.; Benkoski, S. J.; and McCabe, B. J.; The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.
A. de Clercq, F. Luca, L. Martirosyan, M. Matthis, P. Moree, M.A. Stoumen and M. Weiß, Binary recurrences for which powers of two are discriminating moduli, arXiv:2003.01559 [math.NT], 2020. See Table 1 p. 7.

Cf. A016726.

A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 3, 4, 2, 4, 3, 4, 1, 3, 2, 5, 3, 3, 4, 5, 2, 5, 4, 5, 3, 5, 4, 5, 1, 2, 3, 3, 2, 4, 5, 6, 3, 4, 3, 6, 4, 4, 5, 6, 2, 3, 5, 6, 4, 6, 5, 6, 3, 6, 5, 6, 4, 6, 5, 6, 1, 4, 2, 4, 3, 5, 3, 7, 2, 4, 4, 4, 5, 5, 6, 7, 3, 5, 4, 7, 3, 5, 6

Offset: 0

Rémy Sigrist, Oct 08 2017

The set of exponents in expression for n as a sum of distinct powers of 2 corresponds to the n-th row of A133457.
The sum of digits of n in base 2^a(n), say s, can be computed without carry in base 2; the Hamming weight of s equals the Hamming weight of n.
a(n) >= A000120(n) for any n > 0.
Apparently, a(n) = A000120(n) iff n = 0 or n belongs to A100290.
a(n) <= A070939(n) for any n >= 0.
For any sequence s of distinct nonnegative integers (s(n) being defined for n >= 0):
- let D_s be defined for any n > 0 by D_s(n) = a(Sum_{k=0..n-1} 2^s(k)),
- then D_s is the discriminator of s as introduced by Arnold, Benkoski, and McCabe in 1985,
- D_s(1) = 1,
- D_s(n) >= n for any n >= 1,
- D_s(n+1) >= D_s(n) for any n >= 1.

			For n=42:
- 42 = 2^5 + 2^3 + 2^1,
- 5 mod 1 = 3 mod 1,
- 5 mod 2 = 3 mod 2,
- 5 mod 3, 3 mod 3 and 1 mod 3 are all distinct,
- hence a(42) = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.

Cf. A000041, A000045, A000058, A000069, A000108, A000110, A000120, A000215, A001147, A001566, A001969, A002808, A003095, A005823, A016726, A062383, A070939, A076793, A100290, A133457, A173195, A270097, A270151, A270176, A272633, A272881, A272882, A273037, A273041, A273043, A273044, A273056, A273062, A273064, A273068, A273237, A273377

f:= proc(n) local L,D,k;
  L:= convert(n,base,2);
  L:= select(t -> L[t+1]=1, [$0..nops(L)-1]);
  if nops(L) = 1 then return 1 fi;
  D:= {seq(seq(L[j]-L[i],i=1..j-1),j=2..nops(L))};
  D:= `union`(seq(numtheory:-divisors(i),i=D));
  min({$2..max(D)+1} minus D)
end proc:
0, seq(f(i),i=1..100); # Robert Israel, Oct 08 2017

{0}~Join~Table[Function[r, SelectFirst[Range@ 10, Length@ Union@ Mod[r, #] == Length@ r &]][Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[n, 2]], {n, 86}] (* Michael De Vlieger, Oct 08 2017 *)

a(n) = if (n, my (d=Vecrev(binary(n)), x = []); for (i=1, #d, if (d[i], x = concat(x, i-1))); for (m=1, oo, if (#Set(vector(#x, i, x[i]%m))==#x, return (m))), return (0))

a(2*n) = a(n) for any n >= 0.
a(2^k-1) = k for any k >= 0.
a(n) = 1 iff n = 2^k for some k >= 0.
a(n) = 2 iff n belongs to A173195.
a(Sum_{k=1..n} 2^(k^2)) = A016726(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000069(k)) = A062383(n) for any n >= 1.
a(Sum_{k=0..n} 2^(2^k)) = A270097(n) for any n >= 0.
a(Sum_{k=1..n} 2^A000045(k+1)) = A270151(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000041(k)) = A270176(n) for any n >= 1.
a(A076793(n)) = A272633(n) for any n >= 0.
a(Sum_{k=1..n} 2^A001969(k)) = A272881(n) for any n >= 1.
a(Sum_{k=1..n} 2^A005823(k)) = A272882(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000215(k-1)) = A273037(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000108(k)) = A273041(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001566(k)) = A273043(n) for any n >= 1.
a(Sum_{k=1..n} 2^A003095(k)) = A273044(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000058(k-1)) = A273056(n) for any n >= 1.
a(Sum_{k=1..n} 2^A002808(k)) = A273062(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k!)) = A273064(n) for any n >= 1.
a(Sum_{k=1..n} 2^(k^k)) = A273068(n) for any n >= 1.
a(Sum_{k=1..n} 2^A000110(k)) = A273237(n) for any n >= 1.
a(Sum_{k=1..n} 2^A001147(k)) = A273377(n) for any n >= 1.

A347693 Smallest k >= 2*n such that k is a prime or twice a prime.

Original entry on oeis.org

2, 2, 4, 6, 10, 10, 13, 14, 17, 19, 22, 22, 26, 26, 29, 31, 34, 34, 37, 38, 41, 43, 46, 46, 53, 53, 53, 58, 58, 58, 61, 62, 67, 67, 71, 71, 73, 74, 79, 79, 82, 82, 86, 86, 89, 94, 94, 94, 97, 101, 101, 103, 106, 106, 109, 113, 113, 118, 118, 118, 122, 122, 127, 127, 131, 131, 134

Offset: 0

N. J. A. Sloane, Oct 04 2021

nonn

Suggested by the formula for A016726.

Cf. A016726, A347694.

A347694 Largest k <= 2*n such that k is a prime or twice a prime.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 14, 14, 17, 19, 22, 23, 26, 26, 29, 31, 34, 34, 38, 38, 41, 43, 46, 47, 47, 47, 53, 53, 58, 59, 62, 62, 62, 67, 67, 71, 74, 74, 74, 79, 82, 83, 86, 86, 89, 89, 94, 94, 97, 97, 101, 103, 106, 107, 109, 109, 113, 113, 118, 118, 122, 122, 122, 127, 127, 131, 134, 134, 137, 139

Offset: 1

N. J. A. Sloane, Oct 05 2021

nonn

Suggested by the formula for A016726.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A016726, A347693.

lkp2p[n_]:=Module[{k=2n},While[NoneTrue[{k,k/2},PrimeQ],k--];k]; Array[lkp2p,200] (* Harvey P. Dale, Mar 01 2025 *)

cp's OEIS Frontend

A192419 Smallest k such that 1^3, 2^3, 3^3,... n^3 are distinct modulo k.

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A192420 Smallest k such that 1^4, 2^4, 3^4,... ,n^4 are distinct modulo k.

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A272633 Discriminator of the primes: Least m > 0 such that (prime(1),...,prime(n)) are all different mod m; a(0) = 0 by convention.

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A270151 Discriminator of the Fibonacci numbers; least positive integer r such that F(2), F(3), ..., F(n+1) are all incongruent modulo r.

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A293390 Least m such that the exponents in expression for n as a sum of distinct powers of 2 are pairwise distinct mod m; a(0) = 0 by convention.

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A347693 Smallest k >= 2*n such that k is a prime or twice a prime.

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A347694 Largest k <= 2*n such that k is a prime or twice a prime.

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