A060130 -id:A060130 - cp's OEIS frontend

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

2, 4, 12, 8, 12, 8, 60, 36, 24, 16, 24, 16, 60, 24, 24, 16, 36, 16, 60, 24, 36, 16, 24, 16, 420, 180, 180, 72, 180, 72, 120, 72, 48, 32, 48, 32, 120, 48, 48, 32, 72, 32, 120, 48, 72, 32, 48, 32, 420, 180, 120, 48, 120, 48, 120, 72, 48, 32, 48, 32, 180, 72, 48, 32, 72, 32, 180, 72, 72, 32, 48, 32, 420, 120, 120, 48, 180, 48, 180, 72, 48, 32, 72, 32, 120, 48, 48

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain sequences related to cycle-structures in finite permutations as ordered by lists A060117 / A060118 (and thus also related to factorial base representation, A007623) because it matches only with any such sequence b that can be computed as b(n) = f(A275725(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.

Cf. A007623, A060117, A060118, A046523, A275725.
Other filter-sequences related to factorial base: A278234, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A048764, A048765, A060129, A060130, A060131, A084558, A275803, A275851, A257510.

(define (A278225 n) (A046523 (A275725 n)))

a(n) = A046523(A275725(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)).

A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 08 2024

See A376885 for details about this factorization.
First differs from A371090 at n = 2^18 = 262144.
Differs from A064547 at n = 64, 128, 192, 256, 320, 384, 448, 512, ... .
Differs from A058061 at n = 128, 384, 512, 640, 896, ... .

			For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007623, A058061, A060130, A077761, A371090.

Similar sequences: A064547, A318464, A376885.

fdignum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, s++]; m++]; s]; f[p_, e_] := fdignum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

fdignum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 0, s ++); m++); s;}
a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdignum(e[i]));}

Additive with a(p^e) = A060130(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.12589120926760155013..., where f(x) = -x + (1-x) * Sum_{k>=1} A060130(k) * x^k.

A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 10 2013

			This triangular table begins:
  0;
  1, 0;
  1, 2, 0;
  2, 1, 1, 0;
  1, 2, 1, 2, 0;
  2, 1, 2, 1, 1, 0;
  1, 2, 2, 3, 2, 3, 0;
  ...
Please see A230415 for examples showing how the terms are computed.

Antti Karttunen, Rows n = 0..120 of triangle, flattened

This is a lower, or equivalently, an upper triangular subregion of symmetric square array A230415.

Cf. A231714, A060130, A055881.

(define (A230417 n) (A230415bi (A003056 n) (A002262 n)))
(define (A230415bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (if (= (modulo x i) (modulo y i)) 0 1)))))))

a(n) = A230415bi(A003056(n),A002262(n)). [As a sequence, this is obtained by taking a subsection from array A230415.]
T(n,0) = A060130(n) [the leftmost column].
For n >= 1, T(n,n-1) = A055881(n) [the last nonzero column].
Each entry T(n,k) <= A231714(n,k).

A276002 Numbers n for which A060502(n) = 2; numbers with exactly two occupied slopes in their factorial representation.

Original entry on oeis.org

3, 7, 8, 10, 11, 13, 15, 16, 17, 20, 21, 25, 26, 28, 29, 30, 36, 38, 42, 43, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 60, 62, 66, 67, 70, 71, 73, 75, 76, 77, 78, 80, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 98, 99, 102, 103, 106, 107, 108, 109, 110, 111, 112, 113, 116, 117, 121, 122, 124, 125, 126, 132, 134, 138, 139, 142, 143, 144, 168, 174, 192, 194

Offset: 1

Antti Karttunen, Aug 16 2016

Also numbers n such that A060498(n) is a two-ball juggling pattern.

Antti Karttunen, Table of n, a(n) for n = 1..14609
Index entries for sequences related to factorial base representation

Cf. A060130, A060498, A060502, A276001, A276003.

Other identities. For all n >= 1:

A060130(a(n)) >= 2.

A276003 Numbers n for which A060502(n) = 3; numbers with exactly three occupied slopes in their factorial representation.

Original entry on oeis.org

9, 27, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 51, 57, 61, 63, 64, 65, 68, 69, 79, 81, 82, 83, 104, 105, 123, 127, 128, 130, 131, 133, 135, 136, 137, 140, 141, 145, 146, 148, 149, 150, 156, 158, 162, 163, 166, 167, 169, 170, 172, 173, 175, 176, 178, 179, 180, 182, 186, 187, 190, 191, 193, 195, 196, 197, 198, 200, 205, 207, 208, 209, 210, 211, 212

Offset: 1

Antti Karttunen, Aug 16 2016

Also numbers n such that A060498(n) is a three-ball juggling pattern.

			27 ("1011" in factorial base) is included as there are three distinct values attained by the difference digit_position - digit_value when computed for its nonzero digits: 4-1 = 3, 2-1 = 1 and 1-1 = 0.
51 ("2011" in factorial base) is included as there are three distinct values attained by the difference digit_position - digit_value when computed for its nonzero digits: 4-2 = 2, 2-1 = 1 and 1-1 = 0.
57 ("2111" in factorial base) is included as there are three distinct values attained by the difference digit_position - digit_value when computed for its nonzero digits: 4-2 = 3-1 = 2, 2-1 = 1 and 1-1 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..15620
Index entries for sequences related to factorial base representation

Cf. A060130, A060498, A060502, A276001, A276002.

Other identities. For all n >= 1:

A060130(a(n)) >= 3.

A331171 a(n) = min(n, A225901(n)), where A225901 is factorial base flip.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 6, 7, 10, 11, 8, 9, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 48, 49, 52, 53, 50, 51, 66, 67, 70, 71, 68, 69, 60, 61, 64, 65, 62, 63, 54, 55, 58, 59

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A060130(i) = A060130(j).
For all i, j > 0:
  a(i) = a(j) => A055881(i) = A055881(j).

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation

Cf. A055881, A060130, A225901.

Cf. also A330740, A331172.

A225901(n) = { my(s=0, d, k=2); while(n, d=n%k; n=n\k; if(d, s += (k-d)*(k-1)!); k=k+1); (s); };
A331171(n) = min(n, A225901(n));

a(n) = min(n, A225901(n)).

A348138 For any number n with factorial base representation (d_1, ..., d_k), for i = 1..k, let f_i be the number of i's in (d_1, ..., d_k); the factorial base representation of a(n) is (f_1, ..., f_k).

Original entry on oeis.org

0, 1, 2, 4, 1, 3, 6, 12, 12, 18, 8, 14, 2, 8, 8, 14, 4, 10, 1, 7, 7, 13, 3, 9, 24, 48, 48, 72, 30, 54, 48, 72, 72, 96, 54, 78, 30, 54, 54, 78, 36, 60, 26, 50, 50, 74, 32, 56, 6, 30, 30, 54, 12, 36, 30, 54, 54, 78, 36, 60, 12, 36, 36, 60, 18, 42, 8, 32, 32, 56

Offset: 0

Rémy Sigrist, Oct 02 2021

Leading zeros in factorial base representation of n are ignored.

			The first terms, in decimal and in factorial base, are:
  n   a(n)  f(n)  f(a(n))
  --  ----  ----  -------
   0     0     0        0
   1     1     1        1
   2     2    10       10
   3     4    11       20
   4     1    20        1
   5     3    21       11
   6     6   100      100
   7    12   101      200
   8    12   110      200
   9    18   111      300
  10     8   120      110

Cf. A000142, A108731, A034968, A060130, A153880.

a(n) = { my (f=[]); for (r=2, oo, if (n==0, return (sum(k=1, #f, f[k]*(#f-k+1)!)), f=concat(f, 0); my (d=n%r); n\=r; if (d, f[d]++))) }

a(n) = n iff n = 0 or n is a factorial number (A000142).
A034968(a(n)) = A060130(n).
a(A153880(n)) = A153880(a(n)).

cp's OEIS Frontend

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

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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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A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

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A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

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A276002 Numbers n for which A060502(n) = 2; numbers with exactly two occupied slopes in their factorial representation.

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A276003 Numbers n for which A060502(n) = 3; numbers with exactly three occupied slopes in their factorial representation.

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A331171 a(n) = min(n, A225901(n)), where A225901 is factorial base flip.

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A348138 For any number n with factorial base representation (d_1, ..., d_k), for i = 1..k, let f_i be the number of i's in (d_1, ..., d_k); the factorial base representation of a(n) is (f_1, ..., f_k).

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