A275725 -id:A275725 - cp's OEIS frontend

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27

Offset: 0

Antti Karttunen, Aug 08 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus a(23) = prime(1)^3 = 2^3 = 8.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A257684, A275732.
Cf. A275811.
Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).
Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

from operator import mul
from sympy import prime, 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)))
def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).
Other identities and observations. For all n >= 0:
a(n) = A275735(A225901(n)).
a(A007489(n)) = A002110(n).
A001221(a(n)) = A060502(n).
A001222(a(n)) = A060130(n).
A007814(a(n)) = A260736(n).
A051903(a(n)) = A275811(n).
A048675(a(n)) = A275728(n).
A248663(a(n)) = A275808(n).
A056169(a(n)) = A275946(n).
A056170(a(n)) = A275947(n).
A275812(a(n)) = A275962(n).

A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 5, 10, 10, 20, 15, 30, 10, 20, 20, 40, 30, 60, 15, 30, 30, 60, 45, 90, 25, 50, 50, 100, 75

Offset: 0

Antti Karttunen, Aug 09 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the factorial base representation of n. See the examples.

			For n = 0 whose factorial base representation (A007623) is also 0, there are no nonzero digits at all, thus there cannot be any prime present in the encoding, and a(0) = 1.
For n = 1 there is just one 1, thus a(1) = prime(1) = 2.
For n = 2 ("10"), there is just one 1-digit, thus a(2) = prime(1) = 2.
For n = 3 ("11") there are two 1-digits, thus a(3) = prime(1)^2 = 4.
For n = 18 ("300") there is just one 3, thus a(18) = prime(3) = 5.
For n = 19 ("301") there is one 1 and one 3, thus a(19) = prime(1)*prime(3) = 2*5 = 10.
For n = 141 ("10311") there are three 1's and one 3, thus a(141) = prime(1)^3 * prime(3) = 2^3 * 5^1 = 40.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A000079, A001221, A001222, A003961, A007623, A008683, A181819, A225901, A257511, A257684, A265349, A265350, A264990, A275729, A275806, A351954.
Cf. also A275725, A275733, A275734 for other such prime factorization encodings of A060117/A060118-related polynomials, and also A276076.
Differs from A227154 for the first time at n=18, where a(18) = 5, while A227154(18) = 4.

A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A275735(n) = A181819(A276076(n)); \\ Antti Karttunen, Apr 03 2022

from sympy import prime
from operator import mul
import collections
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(n) = A275734(A225901(n)).
A001221(a(n)) = A275806(n).
A001222(a(n)) = A060130(n).
A048675(a(n)) = A275729(n).
A051903(a(n)) = A264990(n).
A008683(a(A265349(n))) = -1 or +1 for all n >= 0.
A008683(a(A265350(n))) = 0 for all n >= 1.
From Antti Karttunen, Apr 03 2022: (Start)
A342001(a(n)) = A351954(n).
a(n) = A181819(A276076(n)). (End)

A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 3, 4, 4, 4, 2, 3, 4, 4, 3, 4, 2, 4, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 2, 3, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 2, 3, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 4, 4, 5, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. also A000142, A055093, A060117, A060118, A060120, A060128, A060130, A084558, A275851.

a(n) = A060128(n) + A060130(n).
From Antti Karttunen, Aug 11 2016: (Start)
a(n) = A275812(A275725(n)).
a(n) = 1 + A084558(n) - A275851(n).
Other identities. For all n >= 0:
a(n) = A055093(A060120(n)).
a(A000142(n)) = 2.
(End)

A060128 a(n) is the number of disjoint cycles (excluding 1-cycles, i.e., fixed elements) in the n-th permutation of A060117 and A060118.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A055090, A056170, A060118, A060120, A060129, A060130, A275725, A278225, A276004.

Cf. A276005 (positions where coincides with A060502).

A060128(n) = nops(convert(PermUnrank3L(n), 'disjcyc')); # Code for function PermUnrank3L given in A060118.

a(n) = A060129(n) - A060130(n).
From Antti Karttunen, Aug 07 2017: (Start)
a(n) = A056170(A275725(n)).
a(n) = A055090(A060120(n)).
a(n) = A060502(n) - A276004(n).
(End)

A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42

Offset: 0

Antti Karttunen, Aug 08 2016

a(n) = product of primes whose indices are positions of nonzero-digits in factorial base representation of n (see A007623). Here positions are one-based, so that the least significant digit is the position 1, the next least significant the position 2, etc.

			For n=19, A007623(19) = 301, thus a(19) = prime(3)*prime(1) = 5*2 = 10.
For n=52, A007623(52) = 2020, thus a(52) = prime(2)*prime(4) = 3*7 = 21.

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

Cf. A001221, A002110, A003961, A005117, A007489, A007623, A048675, A060130, A061395, A084558, A257684, A275732.
Subsequence of A005117.
Cf. A275727.
Cf. also A275725, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(A007489(n)) = A002110(n).
A001221(a(n)) = A001222(a(n)) = A060130(n).
A048675(a(n)) = A275727(n).
A061395(a(n)) = A084558(n).

A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A060117, A060118.
Cf. A056169, A060129, A084558, A275725.
Cf. A275852 (indices of zeros).

a(n) = A056169(A275725(n)).

a(n) = 1 + A084558(n) - A060129(n).

A275723 Square array A(n,k) = prime factorization of n (= 1..) completely reduced by factorial base representation of k (= 0..), read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc. See the Comments section for the meaning of reduction in this context.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 09 2016

The reduction is done by scanning the factorial base representation of k [see A007623] from its most significant end [where the most significant digit occurs at the one-based position A084558(k)], by adding the exponent of prime(1+A084558(k)) of n to the current exponent (possibly zero) of prime(1+A084558(k)-A099563(k)) in the prime factorization of n, after which the exponent of prime(1+A084558(k)) is changed to zero. Thus the total number of prime factors of n [A001222(n)] never changes. This single step of reduction is performed with a bivariate function A273673. The reduction then proceeds to the next digit to the right, effectively skipping any zeros until all factorial base digits of k have been scanned through and the prime factorization of n has been changed accordingly. See the examples.

This bivariate function is used to compute A275725.

			The top left 7 X 15 corner of the array:
   1,  1,  1,  1,  1,  1,  1
   2,  2,  2,  2,  2,  2,  2
   3,  2,  3,  2,  3,  2,  3
   4,  4,  4,  4,  4,  4,  4
   5,  5,  3,  2,  2,  2,  5
   6,  4,  6,  4,  6,  4,  6
   7,  7,  7,  7,  7,  7,  5
   8,  8,  8,  8,  8,  8,  8
   9,  4,  9,  4,  9,  4,  9
  10, 10,  6,  4,  4,  4, 10
  11, 11, 11, 11, 11, 11, 11
  12,  8, 12,  8, 12,  8, 12
  13, 13, 13, 13, 13, 13, 13
  14, 14, 14, 14, 14, 14, 10
  15, 10,  9,  4,  6,  4, 15
For row 15 (above), we have 15 = 3*5 = prime(2)*prime(3) and the terms for columns 0 - 6 (in factorial base: 0, 1, 10, 11, 20, 21, 100, see A007623) are computed as:
When k=0, we do nothing and n stays as n (thus column 0 of array is A000027).
When k=1 (with the length 1), we transfer the exponent of prime(2) to prime(1), to get prime(1)*prime(3) = 2*5 = 10.
When k=2, in factorial base "10", with the length 2, we transfer (add) the exponent of prime(3) to prime(2), to get prime(2)*prime(2) = 9.
When k=3, in factorial base "11", we first do as above, to get 9 = prime(2)^2, and for the least significant one, we transfer (add) the exponent of prime(2) to prime(1), to get prime(1)*prime(1) = 4.
When k=4, in factorial base "20", with the length 2, we transfer (add) the exponent of prime(3) to prime(1), to get prime(2)*prime(1) = 6.
When k=5, in factorial base "21", we first do as above, to get 6 = prime(2)*prime(1), and for the remaining "1" in factorial base representation of k, we transfer (add) the exponent of prime(2) to prime(1), to get prime(1)*prime(1) = 4.
When k=6, in factorial base "100", with the length 3, we transfer (add) the exponent of prime(4) to prime(3), but prime(4) = 7 whose exponent is zero in 15, thus the result is also 15.

Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of array
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Transpose: A275724.
Column 0: A000027.
Cf. A001222, A002260, A004736, A007623, A257687, A273673, A275725.

(define (A275723 n) (A275723bi (A002260 n) (- (A004736 n) 1)))
(define (A275723bi n fex) (let loop ((n n) (fex fex)) (cond ((zero? fex) n) (else (loop (A273673bi n fex) (A257687 fex))))))
;; Code for A273673bi given in A273673.

A(n,0) = n, and for k > 0, A(n,k) = A(A273673(n,k), A257687(k)).
Other identities. For all n >= 1 and k >= 0:
A001222(A(n,k)) = A001222(n). [This reduction doesn't change the total number of prime factors of n.]

cp's OEIS Frontend

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

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A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

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A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

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A060128 a(n) is the number of disjoint cycles (excluding 1-cycles, i.e., fixed elements) in the n-th permutation of A060117 and A060118.

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A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

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A275851 a(n) = number of elements in range [1..(1+A084558(n))] fixed by the permutation with rank n of permutation list A060117 (or A060118).

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A275723 Square array A(n,k) = prime factorization of n (= 1..) completely reduced by factorial base representation of k (= 0..), read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc. See the Comments section for the meaning of reduction in this context.

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