A257696 -id:A257696 - cp's OEIS frontend

A257684 Discard the rightmost digit from the factorial base representation of n and subtract one from all remaining nonzero digits, then convert back to decimal.

0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17

Offset: 0

Antti Karttunen, May 04 2015

In other words, subtract one from all nonzero digits in the factorial base representation (A007623) of n and shift it one step right (i.e., delete the rightmost zero), then convert back to decimal.

			For 4, whose factorial base representation is "20" (as 4 = 2*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 2, we get "1", thus a(4) = 1.
For 18, whose factorial base representation is "300" (as 18 = 3*3! + 0*2! + 0*1!), when we discard the rightmost zero, and subtract 1 from 3, we get "20", thus a(18) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Positions of zeros: A059590.
Cf. A007623, A255411, A257685, A257687.
Can be used to define simple recurrences for sequences like A034968, A246359, A257679, A257694, A257695 and A257696.

nn = 95; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, MixedRadix[Reverse@ Range[2, m]]] &[If[# == 0, 0, # - 1] & /@ Most@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 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==1 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 19 2017

(define (A257684 n) (let loop ((n n) (z 0) (i 2) (f 0)) (cond ((zero? n) z) (else (let ((d (remainder n i))) (loop (quotient n i) (+ z (* f (- d (if (zero? d) 0 1)))) (+ 1 i) (if (zero? f) 1 (* f (- i 1)))))))))

For all n >= 0, a(A255411(n)) = n. [This sequence works as a left inverse of A255411. See A257685 for a "cleaned up" version.]

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.]

A257694 a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 4, 6, 1, 2, 2, 3, 4, 6, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 6, 8, 2, 3, 3, 4, 6, 8, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 4, 6, 6, 8, 9, 12, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 8, 12, 12, 16, 18, 24, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 8, 12, 12, 16, 18, 24, 1

Offset: 0

Antti Karttunen, May 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A034968, A060130, A257684, A257695, A257696, A227153.

a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).
Other identities:
For all n >= 1, a(A033312(n)) = A000142(n-1).

A257695 a(0) = 1; for n >= 1, a(n) = lcm(A060130(n), a(A257684(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 6, 1, 2, 2, 3, 2, 6, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 6, 4, 2, 3, 3, 4, 6, 4, 1, 2, 2, 3, 2, 6, 2, 3, 3, 4, 6, 4, 2, 6, 6, 4, 3, 12, 2, 6, 6, 4, 3, 12, 1, 2, 2, 3, 2, 6, 2, 3, 3, 4, 6, 4, 2, 6, 6, 4, 3, 12, 2, 6, 6, 4, 6, 12, 1, 2, 2, 3, 2, 6, 2, 3, 3, 4, 6, 4, 2, 6, 6, 4, 3, 12, 2, 6, 6, 4, 6, 12, 1

Offset: 0

Antti Karttunen, May 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A060130, A257684, A257694, A257696.

a(0) = 1; for n >= 1, a(n) = lcm(A060130(n), a(A257684(n))).

A278235 Filter-sequence for factorial base (digit levels): Least number with the same prime signature as A275735(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 4, 4, 8, 6, 12, 2, 6, 6, 12, 4, 12, 2, 6, 6, 12, 6, 30, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 12, 36, 6, 12, 12, 24, 30, 60, 2, 6, 6, 12, 4, 12, 6, 12, 12, 24, 12, 36, 4, 12, 12, 36, 8, 24, 6, 30, 30, 60, 12, 60, 2, 6, 6, 12, 6, 30, 6, 12, 12, 24, 30, 60, 6, 30, 30, 60, 12, 60, 4, 12, 12, 36, 12, 60, 2, 6, 6, 12, 6, 30, 6

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain factorial base (A007623) related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A275735(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.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences related to factorial base representation

Cf. A046523, A225901, A275735.
Other factorial base related filter-sequences: A278225, A278234, A278236.
Sequences that partition N into same or coarser equivalence classes: A060130, A257696 (?), A264990, A275806, A275948, A275964 (this is a proper a subset of the sequences that match with A278236).

(define (A278235 n) (A046523 (A275735 n)))

a(n) = A046523(A275735(n)).

a(n) = A278234(A225901(n)).

cp's OEIS Frontend

A257684 Discard the rightmost digit from the factorial base representation of n and subtract one from all remaining nonzero digits, then convert back to decimal.

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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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A257694 a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).

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A257695 a(0) = 1; for n >= 1, a(n) = lcm(A060130(n), a(A257684(n))).

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A278235 Filter-sequence for factorial base (digit levels): Least number with the same prime signature as A275735(n).

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