cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A328574 a(1) = 0, and, for n >= 2, numbers n whose primorial base expansion doesn't contain any nonleading zeros.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 39, 41, 45, 47, 51, 53, 57, 59, 69, 71, 75, 77, 81, 83, 87, 89, 99, 101, 105, 107, 111, 113, 117, 119, 129, 131, 135, 137, 141, 143, 147, 149, 159, 161, 165, 167, 171, 173, 177, 179, 189, 191, 195, 197, 201, 203, 207, 209, 249, 251, 255, 257, 261, 263, 267, 269, 279, 281, 285
Offset: 1

Views

Author

Antti Karttunen, Oct 20 2019

Keywords

Comments

After the initial zero, numbers n for which A276086(n) produces an even number with no gaps in its prime factorization.
Numbers n such that A276086(n) is in A055932; numbers for which A328475(n) is equal to A328572(n) = A003557(A276086(n)).
The number of positive terms below prime(m)# = A002110(m) is Sum_{k=1..m} A005867(k). - Amiram Eldar, Feb 16 2021

Links

Crossrefs

Cf. A002110, A003557, A005867, A055932, A276086, A328475, A328572.
Positions of 1's in A328573, positions of 0's in A329027, cf. also A328840.
Cf. A227157 for analogous sequence.

Programs

  • Mathematica
    max = 4; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Join[{0}, Select[Range[nmax], FreeQ[IntegerDigits[#, MixedRadix[bases]], 0] &]] (* Amiram Eldar, Feb 16 2021 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA055932(n) = { my(f=factor(n)[, 1]~); f==primes(#f); }; \\ From A055932
isA328574(n) = isA055932(A276086(n));
  • PARI
    A328475(n) = { my(m=1, p=2, y=1); while(n, if(n%p, m *= p^((n%p)-y), y=0); n = n\p; p = nextprime(1+p)); (m); };
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
isA328574(n) = (A328475(n) == A328572(n));

Extensions

Primary definition changed, the old definition moved to comment section by Antti Karttunen, Nov 03 2019

A329028 The least missing nonzero digit in the primorial base expansion of n.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 1, 4, 4, 4, 1, 4, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 4, 1, 4, 4, 4
Offset: 0

Views

Author

Antti Karttunen, Nov 03 2019

Keywords

Examples

			19 in primorial base (A049345) is written as "301". The least missing nonzero digit is 2, thus a(19) = 2.
809 in primorial base is written as "35421". The least missing nonzero digit is 6, thus a(809) = 6, and this is also the first position where 6 appears in this sequence.

Links

Crossrefs

Cf. A049345, A134193, A257993, A276086, A328835, A329027, A329030.
Cf. A328840 (the positions of ones in this sequence).
Cf. A257079 for analogous sequence.

Programs

  • Mathematica
    a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Min[Complement[Range[Max[s] + 1], s]]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)
  • PARI
    A329028(n) = { my(m=Map(), p=2); while(n, mapput(m,(n%p),1); n = n\p; p = nextprime(1+p)); for(k=1,oo,if(!mapisdefined(m,k),return(k))); };

Formula

a(n) = A134193(A276086(n)) = A257993(A328835(n)).
a(A276086(n)) = A329030(n).

A329027 The least missing digit in the primorial base expansion of n. Only significant digits are considered, as the leading zeros are ignored.

Original entry on oeis.org

0, 2, 0, 1, 0, 2, 2, 2, 0, 3, 0, 1, 3, 3, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 2, 2, 2, 2, 3, 3, 2, 2, 2, 0, 3, 0, 3, 3, 3, 0, 3, 0, 2, 2, 2, 0, 4, 0, 2, 2, 2, 0, 3, 0, 1, 3, 3, 3, 1, 3, 3, 3, 3, 0, 3, 0, 1, 3, 3, 0, 1, 0, 1, 4, 4, 0, 1, 0, 1, 3, 3, 0, 1, 0, 1, 2, 2, 2, 1, 4, 2, 2, 2, 0, 4, 0, 1, 4, 4, 0
Offset: 1

Views

Author

Antti Karttunen, Nov 03 2019

Keywords

Comments

For n = 0 the value is ambiguous, thus the sequence starts from n=1.

Examples

			19 in primorial base (A049345) is written as "301". The least missing digit is 2, thus a(19) = 2.

Links

Crossrefs

Cf. A049345, A329028.
Cf. A328574 (after its initial term, gives the positions of zeros in this sequence), A328840 (after its initial term, gives the positions of ones in this sequence).

Programs

  • Mathematica
    a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Min[Complement[Range[0, Max[s]+1], s]]]; Array[a, 100] (* Amiram Eldar, Mar 13 2024 *)
  • PARI
    A329027(n) = { my(m=Map(), p=2); while(n, mapput(m,(n%p),1); n = n\p; p = nextprime(1+p)); for(k=0,oo,if(!mapisdefined(m,k),return(k))); };

A353512 n multiplied by the least nonzero digit missing from its primorial base representation.

Original entry on oeis.org

0, 2, 4, 6, 4, 15, 12, 14, 16, 18, 30, 33, 12, 39, 42, 45, 16, 51, 18, 38, 40, 42, 22, 92, 24, 50, 52, 54, 28, 87, 60, 62, 64, 66, 102, 105, 72, 74, 76, 78, 120, 123, 126, 129, 132, 135, 138, 141, 96, 98, 100, 102, 208, 212, 108, 110, 112, 114, 174, 177, 60, 183, 186, 189, 64, 195, 198, 201, 204, 207, 210, 213, 72
Offset: 0

Views

Author

Antti Karttunen, Apr 27 2022

Keywords

Examples

			19 in primorial base (A049345) is written as "301". The least missing nonzero digit is 2, thus A329028(19) = 2 and a(19) = 2*19 = 38.

Links

Crossrefs

Cf. A049345, A328840 (the fixed points, positions where a(n) = n), A329028.
Cf. also A257080 for analogous sequence.

Programs

  • Mathematica
    a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; n * Min[Complement[Range[Max[s]+1], s]]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)
  • PARI
    A329028(n) = { my(m=Map(), p=2); while(n, mapput(m,(n%p),1); n = n\p; p = nextprime(1+p)); for(k=1,oo,if(!mapisdefined(m,k),return(k))); };
A353512(n) = (n * A329028(n));

Formula

a(n) = n * A329028(n).
Showing 1-4 of 4 results.

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