A235168 -id:A235168 - cp's OEIS frontend

A333424 Primes that are palindromes in primorial base.

3, 7, 11, 31, 47, 211, 223, 229, 281, 293, 2311, 2347, 2383, 2843, 2879, 30091, 30181, 30211, 30307, 30367, 30427, 30493, 30553, 30643, 30829, 30859, 34871, 34961, 35051, 35117, 35267, 35363, 35393, 35423, 510751, 511711, 513067, 513307, 515143, 517459, 518179

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			3 is a term since it is a prime number and its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..300
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Intersection of A000040 and A333423.
Cf. A049345, A235168.
Cf. A002385, A016041, A029971, A029972, A029973, A029974, A029975, A029976, A029977, A029978, A029979, A029980, A029981, A029982, A333421.

max = 8; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[nmax], PrimeQ[#] && PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

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]

A342117 Lexicographically latest sequence of distinct nonnegative integers such that the multisets of frequencies of digits in the primorial base representations of n and of a(n) are the same.

Original entry on oeis.org

0, 1, 5, 3, 4, 2, 27, 24, 21, 9, 29, 18, 17, 28, 26, 16, 15, 12, 11, 25, 23, 8, 22, 20, 7, 19, 14, 6, 13, 10, 189, 147, 144, 180, 207, 204, 111, 159, 150, 39, 201, 129, 198, 197, 196, 120, 195, 108, 192, 191, 188, 99, 209, 187, 186, 184, 183, 90, 208, 182, 77

Offset: 0

Rémy Sigrist, Feb 28 2021

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, in decimal and in primorial base, alongside the corresponding multisets of frequencies (ignoring 0's), are:
  n   a(n)  prim(n)  prim(a(n))  freq(n)
  --  ----  -------  ----------  ------------
   0     0        0           0  ()
   1     1        1           1  (1)
   2     5       10          21  (1, 1)
   3     3       11          11  (2)
   4     4       20          20  (1, 1)
   5     2       21          10  (1, 1)
   6    27      100         411  (1, 2)
   7    24      101         400  (1, 2)
   8    21      110         311  (1, 2)
   9     9      111         111  (3)
  10    29      120         421  (1, 1, 1)
  11    18      121         300  (1, 2)
  12    17      200         221  (1, 2)

Rémy Sigrist, Table of n, a(n) for n = 0..2309
Rémy Sigrist, Colored scatterplot of the first 2*3*5*7*11*13 terms (where the color is function of the multiset of frequencies of n)
Rémy Sigrist, PARI program for A342117
Index entries for sequences related to primorial base
Index entries for sequences that are permutations of the natural numbers

See A342102 for similar sequences.

Cf. A002110, A235168.

PARI
```
See Links section.
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a(n) < A002110(k) for any n < A002110(k).

A354470 Square array A(n, k), n, k >= 0, read by antidiagonals; the primorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the primorial base expansions of n and k.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 4, 2, 4, 5, 5, 5, 5, 5, 5, 6, 4, 0, 4, 0, 4, 6, 7, 7, 1, 1, 1, 1, 7, 7, 8, 6, 8, 0, 2, 0, 8, 6, 8, 9, 9, 9, 9, 3, 3, 9, 9, 9, 9, 10, 8, 10, 8, 10, 2, 10, 8, 10, 8, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Rémy Sigrist, Jun 02 2022

The nonnegative integers together with A form an abelian group; A354469 gives inverse elements.

Each row is a permutation of the nonnegative integers.

			Square array A(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   0   3   2   5   4   7   6   9   8  11  10  13  12  15  14
    2|   2   3   4   5   0   1   8   9  10  11   6   7  14  15  16  17
    3|   3   2   5   4   1   0   9   8  11  10   7   6  15  14  17  16
    4|   4   5   0   1   2   3  10  11   6   7   8   9  16  17  12  13
    5|   5   4   1   0   3   2  11  10   7   6   9   8  17  16  13  12
    6|   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21
    7|   7   6   9   8  11  10  13  12  15  14  17  16  19  18  21  20
    8|   8   9  10  11   6   7  14  15  16  17  12  13  20  21  22  23
    9|   9   8  11  10   7   6  15  14  17  16  13  12  21  20  23  22
   10|  10  11   6   7   8   9  16  17  12  13  14  15  22  23  18  19
   11|  11  10   7   6   9   8  17  16  13  12  15  14  23  22  19  18
   12|  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27
   13|  13  12  15  14  17  16  19  18  21  20  23  22  25  24  27  26
   14|  14  15  16  17  12  13  20  21  22  23  18  19  26  27  28  29
   15|  15  14  17  16  13  12  21  20  23  22  19  18  27  26  29  28

Rémy Sigrist, Colored representation of the array A(n, k) for n, k < 2*3*5*7*11 (the hue is function of A(n, k), black pixels correspond to 0's)
Index entries for sequences related to primorial base

Cf. A004442, A235168, A354438 (factorial base analog), A354469.

A(n,k, s=i->prime(i)) = { my (v=0, f=1, r); for (i=1, oo, if (n==0 && k==0, return (v), r=s(i); v+=f*((n+k)%r); f*=r; n\=r; k\=r)) }

A(n, k) = A(k, n).
A(m, A(n, k)) = A(A(m, n), k).
A(n, 0) = n.
A(n, k) = 0 iff k = A354469(n).
A(n, 1) = A004442(n).

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}.

A333425 Primorial base emirps: prime numbers whose primorial base reversal is a different prime.

Original entry on oeis.org

227, 283, 2339, 2351, 2357, 2393, 2767, 2777, 2789, 2797, 2801, 2833, 2851, 2861, 30059, 30089, 30161, 30169, 30187, 30197, 30223, 30293, 30313, 30319, 30389, 30391, 30449, 30467, 30509, 30517, 30559, 30631, 30649, 30677, 30689, 30727, 30763, 30781, 30803, 30851

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			227 is a term since it is a prime number and its representation in primorial base is 10221, whose reversal, 12201, is the primorial base representation of another prime number, 283.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A049345, A235168, A333424.

Cf. A006567, A080790, A333422.

max = 7; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; emirpQ[n_] := PrimeQ[n] && Module[{d = IntegerDigits[n, MixedRadix[bases]]}, r = Reverse @ d; IntegerDigits[(m = FromDigits[r, MixedRadix[bases]]), MixedRadix[bases]] == r && m != n && PrimeQ[m]]; Select[Range[nmax], emirpQ]

A354469 Write n in primorial base, then replace each nonzero digit d of radix p with p-d.

Original entry on oeis.org

0, 1, 4, 5, 2, 3, 24, 25, 28, 29, 26, 27, 18, 19, 22, 23, 20, 21, 12, 13, 16, 17, 14, 15, 6, 7, 10, 11, 8, 9, 180, 181, 184, 185, 182, 183, 204, 205, 208, 209, 206, 207, 198, 199, 202, 203, 200, 201, 192, 193, 196, 197, 194, 195, 186, 187, 190, 191, 188, 189

Offset: 0

Rémy Sigrist, Jun 02 2022

This sequence is a self-inverse permutation of the nonnegative integers, similar to A225901.

There are exactly two fixed points: a(0) = 0 and a(1) = 1.

			For n = 42:
- the primorial base expansion of 42 is: (1, 2, 0, 0),
- the corresponding radixes are: (7, 5, 3, 2),
- so the primorial base expansion of a(42) is: (7-1, 5-2, 0, 0) = (6, 3, 0, 0),
- and a(42) = 198.

Cf. A225901, A235168.

a(n, s=i->prime(i)) = { my (v=0, f=1, r); for (i=1, oo, if (n==0, return (v), r=s(i); v+=f*((-n)%r); f*=r; n\=r)) }

cp's OEIS Frontend

A333424 Primes that are palindromes in primorial base.

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

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A342117 Lexicographically latest sequence of distinct nonnegative integers such that the multisets of frequencies of digits in the primorial base representations of n and of a(n) are the same.

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A354470 Square array A(n, k), n, k >= 0, read by antidiagonals; the primorial base expansion of A(n, k) is obtained by adding componentwise and reducing modulo their radix the digits of the primorial base expansions of n and k.

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

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PARI

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A333425 Primorial base emirps: prime numbers whose primorial base reversal is a different prime.

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Mathematica

A354469 Write n in primorial base, then replace each nonzero digit d of radix p with p-d.

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