A351894 -id:A351894 - cp's OEIS frontend

A351893 Numbers that contain only even digits in their factorial-base representation.

0, 4, 12, 16, 48, 52, 60, 64, 96, 100, 108, 112, 240, 244, 252, 256, 288, 292, 300, 304, 336, 340, 348, 352, 480, 484, 492, 496, 528, 532, 540, 544, 576, 580, 588, 592, 1440, 1444, 1452, 1456, 1488, 1492, 1500, 1504, 1536, 1540, 1548, 1552, 1680, 1684, 1692, 1696

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms are multiples of 4 (A008586).

			4 is a term since its factorial-base presentation, 20, has only even digits.
16 is a term since its factorial-base presentation, 220, has only even digits.

Amiram Eldar, Table of n, a(n) for n = 1..14400 (terms below 10!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A008586, A351894.
Subsequence: A052849 \ {2}.
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[0, max!, 2], AllTrue[fctBaseDigits[#], EvenQ] &]

A351895 Numbers with an equal number of odd and even digits in their factorial-base representation.

Original entry on oeis.org

2, 5, 25, 26, 29, 30, 34, 37, 38, 41, 42, 46, 51, 55, 56, 59, 63, 67, 68, 71, 73, 74, 77, 78, 82, 85, 86, 89, 90, 94, 99, 103, 104, 107, 111, 115, 116, 119, 723, 727, 728, 731, 735, 739, 740, 743, 745, 746, 749, 750, 754, 757, 758, 761, 762, 766, 771, 775, 776

Offset: 1

Amiram Eldar, Feb 24 2022

			5 is a term since its factorial-base representation, 21, has one odd digit, 1, and one even digit, 2.

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

Cf. A007623, A351893, A351894.
A138524 is a subsequence.
Similar sequences: A031443 (binary), A227870 (decimal).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]

A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

Original entry on oeis.org

71, 743, 791, 839, 862, 910, 983, 1031, 1079, 1102, 1150, 1223, 1271, 1319, 1342, 1390, 1583, 1631, 1823, 1871, 2063, 2111, 2183, 2231, 2279, 2302, 2350, 2423, 2471, 2519, 2542, 2590, 2663, 2711, 2759, 2782, 2830, 3023, 3071, 3263, 3311, 3503, 3551, 3623, 3671, 3719

Offset: 1

Amiram Eldar, Feb 24 2022

			71 is a term since the factorial-base representations of 71 and 73 are 2321 and 3001, respectively, and both have 2 odd digits and 2 even digits.

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

Subsequence of A351895.
Cf. A007623, A351893, A351894.
Similar sequence: A337238.

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; s = Select[Range[1, max!], EvenQ[Length[(d = fctBaseDigits[#])]] && Count[d, _?EvenQ] == Length[d]/2 &]; ind = Position[Differences[s], 2] // Flatten; s[[ind]]

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);}

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A351893 Numbers that contain only even digits in their factorial-base representation.

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A351895 Numbers with an equal number of odd and even digits in their factorial-base representation.

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A351896 Numbers k such that k and k+2 both have an equal number of odd and even digits in their factorial-base representations.

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

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