A321682 -id:A321682 - cp's OEIS frontend

A321683 Numbers with distinct digits in primorial base.

0, 1, 2, 4, 5, 10, 13, 14, 19, 20, 22, 23, 25, 26, 28, 29, 52, 58, 79, 80, 85, 86, 95, 100, 103, 104, 115, 116, 118, 119, 125, 130, 133, 134, 139, 140, 142, 143, 155, 160, 163, 164, 169, 170, 172, 173, 175, 176, 178, 179, 185, 190, 193, 194, 199, 200, 202, 203

Offset: 1

Rémy Sigrist, Nov 17 2018

This sequence is a variant of A010784 (numbers with distinct digits in decimal). The final term of that sequence is 9876543210. This sequence, by contrast, has infinitely many terms (for example, all the terms of A057588 belong to this sequence).

			13 in primorial base is 201, which has no repeated digits, hence 13 is in the sequence.
14 in primorial base is 210, which has no repeated digits, hence 14 is also in the sequence.
15 in primorial base is 211, so 15 is not in the sequence on account of the digit 1 appearing twice in its primorial base representation.

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

See A321682 for the factorial base variant.

Cf. A049345, A057588, A276150, A321682.

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]]; UnsameQ @@ s]; Select[Range[0, 210], q] (* Amiram Eldar, Mar 13 2024 *)

is(n) = my (s=0); forprime (p=2, oo, if (n==0, return (1)); my (d=n%p); if (bittest(s,d), return (0), s+=2^d; n\=p))

A322845 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has distinct digits in factorial base.

Original entry on oeis.org

1, 3, 2, 8, 5, 9, 4, 6, 7, 12, 10, 13, 33, 34, 43, 24, 22, 45, 23, 44, 38, 29, 17, 50, 18, 28, 39, 46, 21, 25, 42, 26, 20, 47, 30, 16, 51, 31, 15, 52, 49, 19, 27, 40, 37, 48, 53, 14, 32, 35, 11, 56, 54, 55, 60, 41, 36, 65, 173, 182, 174, 64, 291, 170, 68, 287

Offset: 1

Rémy Sigrist, Dec 29 2018

In other words, for any n > 0, a(n) + a(n+1) belongs to A321682.
Apparently, all the positive integers appear in the sequence.
This sequence has interesting graphical features (see scatterplots in Links section).
This sequence is to A321682 what A228730 is to A002113.

			The first terms, alongside the factorial representation of a(n)+a(n+1), are:
  n   a(n)  fact(a(n)+a(n+1))
  --  ----  -----------------
   1     1              (2,0)
   2     3              (2,1)
   3     2            (1,2,0)
   4     8            (2,0,1)
   5     5            (2,1,0)
   6     9            (2,0,1)
   7     4            (1,2,0)
   8     6            (2,0,1)
   9     7            (3,0,1)
  10    12            (3,2,0)
  11    10            (3,2,1)
  12    13          (1,3,2,0)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 181425 terms
Rémy Sigrist, Scatterplot of the first 19958408 terms
Rémy Sigrist, C program for A322845
Index entries for sequences related to factorial base representation

Cf. A002113, A228730, A321682.

C
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// See Links section.
```

A307710 a(n) is the determinant of the Vandermonde matrix of the digits of n in factorial base.

Original entry on oeis.org

1, 1, -1, 0, -2, -1, 0, 0, 0, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 6, -6, 0, -6, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, -12, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0

Offset: 0

Rémy Sigrist, Apr 23 2019

This sequence is a variant of A307651, and has infinitely many nonzero terms.

			                                    | 3^0 3^1 2^2 |
a(22) = a(3*3! + 2*2! + 0*1!) = det | 2^0 2^1 2^2 | = -6.
                                    | 0^0 0^1 0^2 |

See A307651 for the decimal variant.

Cf. A108731, A321682.

a(n) = my (d=[]); for (r=2, oo, if (n, d=concat(n%r,d); n\=r, return (matdet(matrix(#d, #d, r, c, d[r]^(c-1))))))

a(n) != 0 iff n belongs to A321682.

cp's OEIS Frontend

A321683 Numbers with distinct digits in primorial base.

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A322845 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has distinct digits in factorial base.

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C

A307710 a(n) is the determinant of the Vandermonde matrix of the digits of n in factorial base.

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