A328574 -id:A328574 - cp's OEIS frontend

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

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));

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

Antti Karttunen, Nov 03 2019

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

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

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to primorial base.

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).

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 *)

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))); };

A328573 a(n) = A328475(n) / A328572(n).

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 5, 5, 15, 1, 15, 1, 5, 5, 15, 1, 15, 1, 5, 5, 15, 1, 15, 1, 5, 5, 15, 1, 15, 1, 7, 7, 21, 7, 21, 7, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 7, 7, 21, 7, 21, 7, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 35, 35, 105, 1, 105, 1, 7, 7, 21, 7, 21

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A002110, A276086, A328475, A328572, A328580.

Cf. A328574 (positions of 1's).

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); };
A328573(n) = (A328475(n) / A328572(n));

a(n) = A328475(n) / A328572(n).

A341433 Numbers that are divisible by the product of their digits in primorial base representation.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 249, 261, 309, 669, 729, 2559, 2571, 2619, 2979, 3051, 4239, 7179, 7191, 32589, 32601, 32649, 32661, 33009, 33021, 37209, 37269, 37629, 51489, 92649, 92709, 93069, 97281, 272889, 274509, 543099, 543111, 543159, 543519, 543591, 544779

Offset: 1

Amiram Eldar, Feb 11 2021

The primorial base repunits (A143293) are all terms since their product of digits in primorial base is 1.

All the terms are zeroless in primorial base, and therefore they are terms of A328574. In particular, since the last digit of even numbers in primorial base is 0, all the terms are odd numbers.

			9 is a term since 9 in primorial base is 111 (9 = 3! + 2! + 1!) and 9 is divisible by 1*1*1 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..150
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

A143293 is a subsequence.
Subsequence of A328574.
Cf. A007602, A049345, A235168, A286590, A333426.

max = 12; bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; q[n_] := FreeQ[(d = IntegerDigits[n, MixedRadix[bases]]), 0] && Divisible[n, Times @@ d]; Select[Range[1, 10^5, 2], q]

A355037 a(n) is the product of the digits of n in primorial base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 12, 0, 0, 0

Offset: 0

Rémy Sigrist, Jun 16 2022

			The first terms, alongside the digits of n in primorial base, are:
  n   a(n)  pr(n)
  --  ----  -----
   0     0      0
   1     1      1
   2     0    1_0
   3     1    1_1
   4     0    2_0
   5     2    2_1
   6     0  1_0_0
   7     0  1_0_1
   8     0  1_1_0
   9     1  1_1_1
  10     0  1_2_0
  11     2  1_2_1
  12     0  2_0_0
  13     0  2_0_1
  14     0  2_1_0
  15     2  2_1_1

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to primorial base

Cf. A143293, A208575, A235168, A328574, A328581.

a(n) = { my (p=1); forprime (r=2, oo, p*=n%r; n\=r; if (p==0 || n==0, return (p))) }

a(n) = 1 iff n belongs to A143293.

a(n) > 0 iff n belongs to A328574 \ {0}.

cp's OEIS Frontend

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

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

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A328573 a(n) = A328475(n) / A328572(n).

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A341433 Numbers that are divisible by the product of their digits in primorial base representation.

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A355037 a(n) is the product of the digits of n in primorial base.

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