A329027 -id:A329027 - cp's OEIS frontend

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

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

Antti Karttunen, Oct 20 2019

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

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

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.

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 *)

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));

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));

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

Antti Karttunen, Nov 03 2019

			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.

Antti Karttunen, Table of n, a(n) for n = 0..32786
Index entries for sequences related to primorial base.

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

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 *)

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))); };

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

a(A276086(n)) = A329030(n).

A328840 Numbers with no digit 1 in their primorial base expansion (A049345).

Original entry on oeis.org

0, 4, 12, 16, 18, 22, 24, 28, 60, 64, 72, 76, 78, 82, 84, 88, 90, 94, 102, 106, 108, 112, 114, 118, 120, 124, 132, 136, 138, 142, 144, 148, 150, 154, 162, 166, 168, 172, 174, 178, 180, 184, 192, 196, 198, 202, 204, 208, 420, 424, 432, 436, 438, 442, 444, 448, 480, 484, 492, 496, 498, 502, 504, 508, 510, 514, 522, 526

Offset: 1

Antti Karttunen, Nov 07 2019

Numbers for which the least missing nonzero digit (A329028) in their primorial base expansion is 1.

Antti Karttunen, Table of n, a(n) for n = 1..92160
Index entries for sequences related to primorial base.

Cf. A049345.
Positions of ones in A329028.
Cf. also A328574 and A329027.
Cf. A255411 for an analogous sequence.

q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; FreeQ[s, 1]]; Select[Range[0, 600], q] (* Amiram Eldar, Mar 06 2024 *)

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))); };
isA328840(n) = (1 == A329028(n));

cp's OEIS Frontend

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

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

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A328840 Numbers with no digit 1 in their primorial base expansion (A049345).

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