A033034 -id:A033034 - cp's OEIS frontend

A351894 Numbers that contain only odd digits in their factorial-base representation.

1, 3, 9, 21, 33, 45, 81, 93, 153, 165, 201, 213, 393, 405, 441, 453, 633, 645, 681, 693, 873, 885, 921, 933, 1113, 1125, 1161, 1173, 1353, 1365, 1401, 1413, 2313, 2325, 2361, 2373, 2553, 2565, 2601, 2613, 2793, 2805, 2841, 2853, 3753, 3765, 3801, 3813, 3993, 4005

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms above 1 are odd multiples of 3.

			3 is a term since its factorial-base presentation, 11, has only odd digits.
21 is a term since its factorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..18056 (terms below 11!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A016945, A351893.
Subsequence: A007489
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!, 2], AllTrue[fctBaseDigits[#], OddQ] &]

A363242 Numbers whose primorial-base representation contains only odd digits.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 111, 159, 171, 249, 261, 309, 321, 369, 381, 669, 681, 729, 741, 789, 801, 1089, 1101, 1149, 1161, 1209, 1221, 1509, 1521, 1569, 1581, 1629, 1641, 1929, 1941, 1989, 2001, 2049, 2061, 2559, 2571, 2619, 2631, 2679, 2691, 2979, 2991, 3039

Offset: 1

Amiram Eldar, May 23 2023

All the terms above 1 are odd multiples of 3.

The partial sums of the primorials (A143293) are terms, since the primorial-base representation of A143293(n) is n+1 1's.

			3 is a term since its primorial-base presentation, 11, has only odd digits.
21 is a term since its primorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..14620 (terms below A002110(8) = 19#)
Index entries for sequences related to primorial base.

Cf. A002110, A049345, A328849.
Subsequence: A143293.
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary), A351894 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[1, nmax, 2], AllTrue[prmBaseDigits[#], OddQ] &]]

is(n) = {my(p = 2); if(n < 1, return(0)); while(n > 0, if((n%p)%2 == 0, return(0)); n \= p; p = nextprime(p+1)); return(1);}

cp's OEIS Frontend

A351894 Numbers that contain only odd digits in their factorial-base representation.

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