A257079 -id:A257079 - cp's OEIS frontend

A257080 n multiplied by the least nonzero digit missing from its factorial base representation: a(n) = n * A257079(n).

0, 2, 4, 6, 4, 15, 12, 14, 16, 18, 30, 33, 12, 39, 42, 45, 16, 51, 18, 38, 40, 42, 22, 92, 48, 50, 52, 54, 84, 87, 60, 62, 64, 66, 102, 105, 108, 111, 114, 117, 120, 123, 84, 86, 88, 90, 184, 188, 48, 147, 150, 153, 52, 159, 162, 165, 168, 171, 174, 177, 60, 183, 186, 189, 64, 195, 66, 268, 272, 276, 70, 284, 72

Offset: 0

Antti Karttunen, Apr 15 2015

Antti Karttunen, Table of n, a(n) for n = 0..10080
Eric Angelini, et al, "Multiply by the fantom digit", Discussion on Seqfan-list

A255411 gives the fixed points, positions where a(n) = n.

Cf. A257079, A257081 (number of iterations needed before a fixed point is reached).

Scheme
```
(define (A257080 n) (* n (A257079 n)))
```

a(n) = n * A257079(n).

A257262 Numbers such that the least missing nonzero digit (A257079) in their factorial base representation is 2.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 33, 42, 43, 44, 45, 73, 74, 75, 78, 79, 80, 81, 91, 92, 93, 97, 98, 99, 102, 103, 104, 105, 115, 116, 117, 120, 121, 122, 123, 126, 127, 128, 129, 138, 139, 140, 141, 144, 145, 146, 147, 150, 151, 152, 153, 162, 163, 164, 165, 192, 193, 194, 195, 198, 199, 200, 201

Offset: 1

Antti Karttunen, Apr 27 2015

			The factorial base representation (A007623) of 1 is "1", the least nonzero digit missing from that is 2, thus 1 is included in this sequence.
Likewise for 2, whose f.b.r. is "10", and also for  42 whose f.b.r. is "1300", the least nonzero digit missing from them is 2, thus also 2 and 42 are included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..9361
Index entries for sequences related to factorial base representation.

Cf. A007623, A257079, A255411, A257263.
Cf. A059590 (is a subsequence, apart from its zero-term).
Subsequence of A256450.

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; MemberQ[s, 1] && !MemberQ[s, 2]]; Select[Range[200], q] (* Amiram Eldar, Feb 14 2024 *)

A257263 Numbers such that the least missing nonzero digit (A257079) in their factorial base representation is 3.

Original entry on oeis.org

5, 10, 11, 13, 14, 15, 17, 28, 29, 34, 35, 36, 37, 38, 39, 40, 41, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 101, 106, 107, 109, 110, 111, 113, 124, 125, 130, 131, 132, 133, 134, 135, 136, 137, 148, 149, 154, 155, 156, 157, 158, 159, 160, 161, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 220

Offset: 1

Antti Karttunen, Apr 27 2015

			The factorial base representation (A007623) of 5 is "21", the least nonzero digit missing is 3, thus 5 is included.
The f.b.r. of 10 is "120", and likewise, 3 is the least missing digit, thus 10 is included.
The f.b.r. of 101 is "4021", where the least missing digit is 3, thus 101 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10920
Index entries for sequences related to factorial base representation.

Cf. A007623, A257079, A255411, A257262.

Subsequence of A256450.

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; SubsetQ[s, {1, 2}] && !MemberQ[s, 3]]; Select[Range[220], q] (* Amiram Eldar, Feb 14 2024 *)

A255411 Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

Original entry on oeis.org

0, 4, 12, 16, 18, 22, 48, 52, 60, 64, 66, 70, 72, 76, 84, 88, 90, 94, 96, 100, 108, 112, 114, 118, 240, 244, 252, 256, 258, 262, 288, 292, 300, 304, 306, 310, 312, 316, 324, 328, 330, 334, 336, 340, 348, 352, 354, 358, 360, 364, 372, 376, 378, 382, 408, 412, 420, 424, 426, 430, 432, 436, 444

Offset: 0

Antti Karttunen, Apr 16 2015

Nonnegative integers such that the number of ones (A257511) in their factorial base representation (A007623) is zero.
Nonnegative integers such that the least missing nonzero digit (A257079) in their factorial base representation is one.
a(n) can be also directly computed from n by "shifting left" its factorial base representation (that is, by appending one zero to the right, see A153880) and then incrementing all nonzero digits by one, and then converting the resulting (still valid) factorial base number back to decimal. See the examples.
The sequences A227130 and A227132 are closed under a(n), in other words, permutation listed as the a(n)-th entry in tables A060117 & A060118 has the same parity as the n-th entry in those same tables.

			Factorial base representation (A007623) of 1 is "1", shifting it left yields "10", and when we increment all nonzero digits by one, we get "20", which is the factorial base representation of 4 (as 4 = 2*2! + 0*1!), thus a(1) = 4.
F.b.r. of 2 is "10", shifting it left yields "100", and "200" is f.b.r. of 12, thus a(2) = 12.
F.b.r. of 43 is "1301", shifting it left and incrementing all nonzeros by one yields "24020", which is f.b.r of 340, thus a(43) = 340.

Antti Karttunen, Table of n, a(n) for n = 0..40319

Complement: A256450.
Positions of ones in A257079, fixed points of A257080, positions of zeros in A257511, A257081 and A257261.
Cf. A007623, A227157, A153880, A231720.
Cf. A255341, A255342, A255343.
Cf. A257262, A257263.
Cf. also A227130/A227132, A060117/A060118 and also arrays A257503 & A257505.

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 500}]; {0}~Join~Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 0](* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
Select[Range[0, 444], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@Range[2, 12]]; Table[FromDigits[Map[If[# == 0, 0, # + 1] &, IntegerDigits[n, r]]~Join~{0}, r], {n, 0, 60}] (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

A257679 The smallest nonzero digit present in the factorial base representation (A007623) of n, 0 if no nonzero digits present.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1

Offset: 0

Antti Karttunen, May 04 2015

a(0) = 0 by convention, because "0" has no nonzero digits present.

a(n) gives the row index of n in array A257503 (equally, the column index for array A257505).

			Factorial base representation (A007623) of 4 is "20", the smallest digit which is not zero is "2", thus a(4) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Positions of records: A001563.
Cf. A256450, A257692, A257693 (positions of 1's, 2's and 3's in this sequence).
Cf. A007623, A051683, A099563, A257680, A257684, A257687.
Cf. also A257079, A246359 and arrays A257503, A257505.

a[n_] := Module[{k = n, m = 2, rmin = n, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[0 < r < rmin, rmin = r]; m++]; rmin]; Array[a, 100, 0] (* Amiram Eldar, Jan 23 2024 *)

def A(n, p=2):
    return n if nIndranil Ghosh, Jun 19 2017

(define (A257679 n) (let loop ((n n) (i 2) (mind 0)) (if (zero? n) mind (let ((d (modulo n i))) (loop (/ (- n d) i) (+ 1 i) (cond ((zero? mind) d) ((zero? d) mind) (else (min d mind))))))))
;; Alternative implementations based on given recurrences, using memoizing definec-macro:
(definec (A257679 n) (if (zero? (A257687 n)) (A099563 n) (min (A099563 n) (A257679 (A257687 n)))))
(definec (A257679 n) (cond ((zero? n) n) ((= 1 (A257680 n)) 1) (else (+ 1 (A257679 (A257684 n))))))

If A257687(n) = 0, then a(n) = A099563(n), otherwise a(n) = min(A099563(n), a(A257687(n))).
In other words, if n is either zero or one of the terms of A051683, then a(n) = A099563(n) [the most significant digit of its f.b.r.], otherwise take the minimum of the most significant digit and a(A257687(n)) [value computed by recursing with a smaller value obtained by discarding that most significant digit].
a(0) = 0, and for n >= 1: if A257680(n) = 1, then a(n) = 1, otherwise 1 + a(A257684(n)).
Other identities:
For all n >= 0, a(A001563(n)) = n. [n * n! gives the first position where n appears. Note also that the "digits" (placeholders) in factorial base representation may get arbitrarily large values.]
For all n >= 0, a(2n+1) = 1 [because all odd numbers end with digit 1 in factorial base].

A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 2, 6, 6, 30, 12, 60, 4, 12, 12, 60, 36, 180, 8, 24, 24, 120, 72, 360, 2, 6, 6, 30, 12, 60, 6, 30, 30, 210, 60, 420, 12, 60, 60, 420, 180, 1260, 24, 120, 120, 840, 360, 2520, 4, 12, 12, 60, 36, 180, 12, 60, 60, 420, 180, 1260, 36, 180, 180, 1260, 900, 6300, 72, 360, 360, 2520, 1800, 12600, 8, 24, 24, 120, 72, 360, 24, 120, 120, 840, 360, 2520

Offset: 0

Antti Karttunen, Nov 16 2016

This sequence can be used for filtering certain factorial base related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A276076(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Any such sequence should match where the result is computed from the nonzero digits (that may also be > 9) in the factorial base representation of n, but does not depend on their order. Some of these are listed on the last line of the Crossrefs section.
Note that as A275735 is present in that list it means that the sequences matching to its filter-sequence A278235 form a subset of the sequences matching to this sequence. Also, for A275735 there is a stronger condition that for any i, j: a(i) = a(j) <=> A275735(i) = A275735(j), which if true, would imply that there is an injective function f such that f(A275735(n)) = A278236(n), and indeed, this function seems to be A181821.

Cf. A007623, A046523, A181821, A276076.
Similar sequences: A278222 (base-2 related), A069877 (base-10), A278226 (primorial base), A278225, A278234, A278235 (other variants for factorial base),
Differs from A278226 for the first time at n=24, where a(24)=2, while A278226(24)=16.
Sequences that partition N into same or coarser equivalence classes: A275735 (<=>), A034968, A060130, A227153, A227154, A246359, A257079, A257511, A257679, A257694, A257695, A257696, A264990, A275729, A275806, A275948, A275964, A278235.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = ReverseSort[s]; Times @@ (Prime[Range[Length[s]]] ^ s)]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

(define (A278236 n) (A046523 (A276076 n)))

a(n) = A046523(A276076(n)).

a(n) = A181821(A275735(n)). [Empirical formula found with the help of equivalence class matching. Not yet proved.]

A329028 The least missing nonzero digit in the primorial base expansion of n.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 3, 1, 4, 4, 4, 1, 4, 1, 3, 3, 3, 1, 3, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 4, 1, 4, 4, 4

Offset: 0

Antti Karttunen, Nov 03 2019

			19 in primorial base (A049345) is written as "301". The least missing nonzero digit is 2, thus a(19) = 2.
809 in primorial base is written as "35421". The least missing nonzero digit is 6, thus a(809) = 6, and this is also the first position where 6 appears in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..32786
Index entries for sequences related to primorial base.

Cf. A049345, A134193, A257993, A276086, A328835, A329027, A329030.
Cf. A328840 (the positions of ones in this sequence).
Cf. A257079 for analogous sequence.

a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Min[Complement[Range[Max[s] + 1], s]]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

A329028(n) = { my(m=Map(), p=2); while(n, mapput(m,(n%p),1); n = n\p; p = nextprime(1+p)); for(k=1,oo,if(!mapisdefined(m,k),return(k))); };

a(n) = A134193(A276086(n)) = A257993(A328835(n)).

a(A276086(n)) = A329030(n).

A257081 a(n) = Number of iterations of A257080 needed, starting from n, before a fixed point is reached.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 15 2015

nonn

Note: when at some point of iteration we reach some k whose factorial representation (A007623) does not contain any 1's, then at next step A257080(k) = 1*k, and thus a fixed point has been reached.

			For n = 5, with factorial representation A007623(5) = "21", the least missing nonzero digit is 3, thus A257080(5) = 3*5 = 15. 15 has factorial representation "211", so again we multiply by 3, resulting 3*15 = 45, with factorial representation "1311", thus the least missing nonzero digit is now 2, and 2*45 = 90, "3300" in factorial base, for which the least missing digit is 1, resulting 1*90 = 90 forever after, thus we have reached a fixed point after three iteration steps (5 -> 15 -> 45 -> 90) and a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10080
Eric Angelini, et al, "Multiply by the fantom digit", Discussion on Seqfan-list

A255411 gives the positions of zeros.

Cf. A007623, A257079, A257080.

(define (A257081 n) (let loop ((oldn n) (n (A257080 n)) (s 1)) (if (= oldn n) s (loop n (A257080 n) (+ 1 s)))))
;; Alternative, recursive version, optionally using the memoizing definec-macro:
(definec (A257081 n) (if (= 1 (A257079 n)) 0 (+ 1 (A257081 (A257080 n)))))

If A257079(n) = 1, a(n) = 0, otherwise, a(n) = 1 + a(A257080(n)).

cp's OEIS Frontend

A257080 n multiplied by the least nonzero digit missing from its factorial base representation: a(n) = n * A257079(n).

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A257262 Numbers such that the least missing nonzero digit (A257079) in their factorial base representation is 2.

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A257263 Numbers such that the least missing nonzero digit (A257079) in their factorial base representation is 3.

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A255411 Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

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A257679 The smallest nonzero digit present in the factorial base representation (A007623) of n, 0 if no nonzero digits present.

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A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

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A329028 The least missing nonzero digit in the primorial base expansion of n.

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A257081 a(n) = Number of iterations of A257080 needed, starting from n, before a fixed point is reached.

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