A046807 -id:A046807 - cp's OEIS frontend

A265350 Numbers whose factorial base representation (A007623) contains at least one of the nonzero digits occurs more than once (although not necessarily in adjacent positions).

3, 7, 8, 9, 11, 15, 16, 17, 21, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 69, 70, 71, 75, 79, 80, 81, 83, 87, 88, 89, 90, 91, 92, 93, 94, 95, 99, 103, 104, 105, 107, 111, 112, 113, 117, 121, 122, 123, 125, 126, 127, 128, 129, 130

Offset: 1

Antti Karttunen, Dec 22 2015

Positions of terms larger than ones in A264990.

			For n=7 the factorial base representation (A007623) is "101" as 7 = 3!+1! = 6+1. Digit "1" occurs twice in it, thus 7 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Indranil Ghosh, Python program for computing this sequence.
Index entries for sequences related to factorial base representation.

Cf. A007623, A264990.
Cf. A265349 (complement).
Cf. A007489, A046807 (subsequences from 3 onward).

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Max[Tally[Select[s, # > 0 &]][[;;,2]]] > 1]; Select[Range[130], q] (* Amiram Eldar, Jan 24 2024 *)

A342725 Numbers that are palindromic in base i-1.

Original entry on oeis.org

0, 1, 13, 17, 189, 205, 257, 273, 3005, 3069, 3277, 3341, 4033, 4097, 4305, 4369, 48061, 48317, 49149, 49405, 52173, 52429, 53261, 53517, 64449, 64705, 65537, 65793, 68561, 68817, 69649, 69905, 768957, 769981, 773309, 774333, 785405, 786429, 789757, 790781, 834509

Offset: 1

Amiram Eldar, Mar 19 2021

Amiram Eldar, Table of n, a(n) for n = 1..512
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.

Cf. A066321, A066323, A271472, A342726, A342727, A342728, A342729.

Similar sequences: A002113 (decimal), A006995 (binary), A014190 (base 3), A014192 (base 4), A029952 (base 5), A029953 (base 6), A029954 (base 7), A029803 (base 8), A029955 (base 9), A046807 (factorial base), A094202 (Zeckendorf), A331191 (dual Zeckendorf), A331891 (negabinary), A333423 (primorial base).

v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := PalindromeQ @ FromDigits[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[0, 10^4], q]

13 is a term since its base-(i-1) presentation is 100010001 which is palindromic.

A333421 Primes that are palindromic in factorial base.

Original entry on oeis.org

3, 7, 11, 41, 127, 139, 173, 179, 191, 751, 811, 5113, 5167, 5419, 5443, 6581, 6659, 6737, 6761, 6833, 6863, 6911, 6959, 40609, 40897, 41047, 41479, 42061, 42349, 42499, 42643, 42787, 50549, 51131, 51419, 51563, 52289, 52433, 52583, 52727, 363361, 363481, 365473

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

			3 is a term since it is a prime number and its factorial base representation is 11 which is a palindrome.

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

Intersection of A000040 and A046807.
Cf. A007623, A108731.
Cf. A002385, A016041, A029971, A029972, A029973, A029974, A029975, A029976, A029977, A029978, A029979, A029980, A029981, A029982.

max = 9; Select[Range[0, max! - 1], PrimeQ[#] && PalindromeQ @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]] &]

A333423 Numbers that are palindromes in primorial base.

Original entry on oeis.org

0, 1, 3, 7, 9, 11, 31, 39, 47, 211, 217, 223, 229, 235, 243, 249, 255, 261, 267, 275, 281, 287, 293, 299, 2311, 2347, 2383, 2419, 2455, 2523, 2559, 2595, 2631, 2667, 2735, 2771, 2807, 2843, 2879, 30031, 30061, 30091, 30121, 30151, 30181, 30211, 30247, 30277, 30307

Offset: 1

Amiram Eldar, Mar 20 2020

			3 is a term since its representation in primorial base is 11 (1 * 2# + 1) which is a palindrome.
7 is a term since its representation in primorial base is 101 (1 * 3# + 0 * 2# + 1 = 6 + 1) which is a palindrome.

Amiram Eldar, Table of n, a(n) for n = 1..2500
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A049345, A235168.

Cf. A002113, A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A046807.

max = 6; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Select[Range[0, nmax], PalindromeQ @ IntegerDigits[#, MixedRadix[bases]] &]

A333422 Factorial base emirps: prime numbers whose factorial base reversal is a different prime.

Original entry on oeis.org

29, 37, 137, 181, 733, 743, 769, 977, 1013, 1031, 1033, 1049, 5107, 5119, 5171, 5179, 5233, 5273, 5297, 5323, 5387, 5393, 5399, 5407, 5437, 5441, 5449, 5471, 5477, 5483, 6571, 6607, 6689, 6691, 6709, 6719, 6733, 6763, 6803, 6823, 6829, 6907, 6947, 6949, 40343

Offset: 1

Amiram Eldar, Mar 20 2020

			29 is a term since it is a prime number and its representation in factorial base is 1021, whose reversal, 1201, is the factorial base representation of another prime number, 37.

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, A046807, A108731, A333421.

Cf. A006567, A080790.

max = 8; bases = Range[max, 2, -1]; nmax = max! - 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]

A373085 Numbers k such that the factorial base representation of 1/k without the leading zeros is palindromic.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 12, 20, 24, 30, 40, 60, 120, 126, 144, 160, 180, 189, 210, 240, 315, 360, 384, 630, 720, 840, 896, 1008, 1056, 1120, 1260, 1680, 2240, 2520, 4480, 5040, 5184, 5760, 6048, 6300, 6720, 6912, 8064, 9072, 9450, 10080, 12096, 13440, 14400, 18144

Offset: 1

Amiram Eldar, May 23 2024

All the factorials (A000142) are terms, since the factorial base representation of 1/k! is k-1 0's followed by 1.

If k > 4 is composite then (k-1)!/k is a term.

			The first 10 terms are:
   n  a(n)   1/a(n) in factorial base
  --  ----   ------------------------
   1    1    1.
   2    2    0.1
   3    3    0.02
   4    6    0.01
   5    8    0.003
   6    9    0.00232
   7   10    0.0022
   8   12    0.002
   9   20    0.0011
  10   24    0.001

Cf. A000142, A007623, A046807, A294168.

q[n_] := Module[{d = NumberDecompose[1/n, 1/Range[n]!], i}, i = Position[d, _?(# > 0&)] // Flatten; PalindromeQ[d[[First[i];;Last[i]]]]]; q[1] = True; Select[Range[1000], q]

cp's OEIS Frontend

A265350 Numbers whose factorial base representation (A007623) contains at least one of the nonzero digits occurs more than once (although not necessarily in adjacent positions).

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A342725 Numbers that are palindromic in base i-1.

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A333421 Primes that are palindromic in factorial base.

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A333423 Numbers that are palindromes in primorial base.

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A333422 Factorial base emirps: prime numbers whose factorial base reversal is a different prime.

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A373085 Numbers k such that the factorial base representation of 1/k without the leading zeros is palindromic.

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