Displaying 1-10 of 33 results found.
Triangle T(n,k) giving the number of terms of A219666 which have n digits ( A084558) in their factorial base expansion and whose most significant digit ( A099563) in that base is k.
+20
7
1, 1, 1, 2, 2, 1, 6, 5, 4, 4, 22, 19, 16, 14, 12, 94, 82, 73, 65, 59, 55, 479, 432, 395, 362, 336, 314, 293, 2886, 2667, 2482, 2324, 2189, 2073, 1971, 1881, 20276, 19123, 18124, 17249, 16473, 15775, 15140, 14555, 14011, 164224, 156961, 150389, 144378, 138828, 133664, 128831, 124289, 120010, 115974
COMMENTS
See A007623 for the factorial number system representation.
EXAMPLE
The first rows of this triangular table are:
1;
1, 1;
2, 2, 1;
6, 5, 4, 4;
22, 19, 16, 14, 12;
94, 82, 73, 65, 59, 55;
...
T(4,2) = 5 as only the terms 48, 52, 57, 63 and 70 of A219666 (with factorial base representations 2000, 2020, 2111, 2211 and 2320) have four significant digits in the factorial base, with the most significant digit being 2.
a(n) = A099563( A001813(n)); the most significant digit in factorial base representation of (2n)! / n!.
+20
7
1, 1, 2, 1, 2, 6, 1, 4, 1, 2, 7, 1, 3, 10, 1, 3, 11, 1, 3, 10, 1, 3, 8, 25, 2, 6, 19, 1, 4, 13, 38, 2, 7, 23, 1, 4, 13, 39, 2, 6, 20, 1, 3, 9, 29, 1, 4, 13, 40, 1, 5, 16, 51, 2, 6, 20, 62, 2, 7, 23, 70, 2, 8, 25, 77, 2, 8, 25, 79, 2, 8, 25, 78, 2, 7, 23, 73, 2, 6, 21, 66, 1, 6, 18, 57, 1, 4, 15, 47, 1, 3, 12, 38, 118, 3, 9
EXAMPLE
1, 10, 200, 10000, 220000, 6000000, 174000000, 4760000000, 110000000000, ...
Taking the first digit (actually: a place holder value) of each gives the terms a(0) .. a(8) of this sequence: 1, 1, 2, 1, 2, 6, 1, 4, 1, ...
MATHEMATICA
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]; (* taken from A007623, Alonso del Arte, May 03 2006 *) f[n_] := factBaseIntDs[(2 n)!/n!][[1]]; Array[f, 96, 0] (* Robert G. Wilson v, Dec 25 2015 *)
PROG
(PARI)
allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2, dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
(Scheme)
(Scheme)
(define ( A265894 n) (A265890bi (+ 1 n) n)) ;; Code for A265890bi given in A265890.
CROSSREFS
Cf. A265898 (positions of ones), A265899 (of descents), A266120 (local maxima just before those descents).
a(n+1) gives the number of occurrences of the first digit of a(n) in factorial base (i.e., A099563(a(n))) so far amongst the factorial base representations of all the terms up to and including a(n), with a(0)=0.
+20
6
0, 1, 1, 2, 3, 5, 1, 7, 9, 12, 2, 13, 3, 16, 5, 6, 18, 1, 19, 2, 21, 3, 25, 27, 30, 32, 35, 38, 40, 41, 43, 45, 48, 13, 14, 15, 16, 18, 6, 53, 20, 7, 57, 21, 8, 64, 24, 65, 27, 69, 28, 72, 10, 73, 11, 76, 12, 33, 80, 13, 34, 85, 14, 37, 89, 15, 41, 94, 17, 46, 96, 1, 97, 2, 99, 3, 103, 4, 48, 49, 50
EXAMPLE
a(0) = 0 (by definition)
a(1) = 1 ('1' in factorial base), as 0 has occurred once in all the preceding terms.
a(2) = 1 as 1 has occurred once in all the preceding terms.
a(3) = 2 ('10' in factorial base), as digit '1' has occurred two times in total in all the preceding terms.
a(4) = 3 ('11' in factorial base), as '1' occurs once in each a(1) and a(2) and a(3).
a(5) = 5 ('21' in factorial base), as '1' occurs once in each of a(1), a(2) and a(3) and twice at a(4).
a(6) = 1 as '2' so far occurs only once at a(5)
a(7) = 7 = '101'
a(8) = 9 = '111'
a(9) = 12 = '200'
a(10) = 2 = '2'
a(11) = 13 = '201'
a(12) = 3 = '11'
a(12) = 3 = '11'
a(13) = 16 = '220'
a(14) = 5 = '21'
a(15) = 6 = '100'
a(16) = 18 = '300'
a(17) = 1 = '1'
a(18) = 19 = '301'
a(19) = 2 = '10'
a(20) = 21 = '311'
a(21) = 3 = '11'
a(22) = 25 = '1001'
a(23) = 27 = '1011'
a(24) = 30 = '1100'
a(25) = 32 = '1110'
a(26) = 35 = '1121'
a(27) = 38 = '1210' as the leftmost digit '1' has occurred 38 times in total in the factorial base expansions of the preceding terms a(0) - a(26).
etc.
PROG
(MIT/GNU Scheme with memoizing definec-macro from Antti Karttunen's IntSeq-library) (definec ( A249069 n) (if (zero? n) n (vector-ref (A249069aux_digit_counts (- n 1)) ( A099563 ( A249069 (- n 1)))))) (definec (A249069aux_digit_counts n) (cond ((zero? n) (vector 1)) (else (let* ((start_n ( A249069 n)) (copy-of-prevec (vector-copy (A249069aux_digit_counts (- n 1)))) (newsize (max (vector-length copy-of-prevec) (+ 1 ( A246359 start_n)))) (digcounts-for-n (vector-grow copy-of-prevec newsize))) (let loop ((n start_n) (i 2)) (cond ((zero? n) digcounts-for-n) (else (vector-set! digcounts-for-n (modulo n i) (+ 1 (or (vector-ref digcounts-for-n (modulo n i)) 0))) (loop (floor->exact (/ n i)) (+ i 1)))))))))
CROSSREFS
Cf. A249009 (analogous sequence in base-10). Differs from a variant A249070 for the first time at n=27, where a(27) = 38, while A249070(27) = 7.
Array read by ascending antidiagonals: A(n,k) = A099563( A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.
+20
6
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 2, 2, 2, 1, 4, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 4, 1, 1, 0, 1, 1, 3, 1, 1, 6, 3, 1, 5, 1, 1, 0, 1, 1, 4, 1, 1, 1, 8, 4, 1, 5, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 5, 2, 6, 1, 1, 0, 1, 2, 1, 2, 3, 3, 3, 2, 1, 6, 2, 6, 1, 1, 0
COMMENTS
Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
A265609(n,k) is the rising factorial, also known as Pochhammer symbol and A099563(n) is the most significant "digit" (place holder) in the factorial representation ( A007623) of n.
EXAMPLE
The top left corner of the array A265609 with its terms shown in factorial base ( A007623) looks like this: 1, 0, 0, 0, 0, 0, 0, 0, 0
1, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000
1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000
1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000
1, 20, 310, 10000, 110000, 1220000, 14000000, 160000000, 1830000000
1, 21, 1100, 13300, 220000, 3000000, 36000000, 452000000, 5500000000
1, 100, 1300, 24000, 411000, 6000000, 82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Taking the most significant "digit" (placeholder that may get arbitrarily large values) gives us the top left corner of this array:
-
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11
1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3
1, 2, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 17, 19, 21, 1
1, 1, 1, 2, 4, 6, 8, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5
1, 1, 2, 4, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 9, 11, 12, 14, 16, 19, 21
1, 1, 3, 1, 1, 2, 3, 4, 6, 8, 11, 14, 1, 1, 1, 1, 2, 2, 2, 3, 3
1, 1, 3, 1, 2, 3, 5, 8, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 12
1, 1, 4, 1, 3, 5, 9, 1, 2, 2, 3, 5, 6, 8, 11, 14, 17, 21, 1, 1, 1
1, 1, 1, 2, 4, 8, 1, 2, 3, 5, 7, 10, 14, 1, 1, 1, 2, 2, 3, 3, 4
1, 2, 1, 3, 6, 1, 2, 4, 6, 9, 14, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13
1, 2, 1, 3, 1, 2, 3, 6, 10, 1, 1, 2, 3, 5, 6, 9, 12, 16, 21, 1, 1
1, 2, 1, 4, 1, 2, 5, 9, 1, 2, 3, 4, 7, 10, 14, 20, 1, 1, 2, 2, 3
1, 2, 2, 5, 1, 3, 7, 1, 2, 3, 5, 8, 13, 1, 1, 1, 2, 3, 4, 6, 8
...
PROG
(Scheme)
(define (A265890bi row col) ( A099563 (A265609bi row col))) ;; Code for A265609bi given in A265609.
CROSSREFS
Main diagonal: A265891 (essentially, without the initial 1 from the corner of this array).
a(n) = A099563( A000407(n)); the most significant digit in factorial base representation of (2n+1)! / n!.
+20
6
1, 1, 2, 1, 3, 8, 2, 6, 1, 3, 10, 1, 5, 14, 1, 5, 16, 1, 5, 15, 1, 4, 12, 1, 3, 9, 28, 2, 6, 19, 1, 3, 11, 35, 2, 6, 19, 1, 3, 10, 30, 1, 4, 14, 44, 2, 6, 20, 61, 2, 8, 25, 1, 3, 10, 31, 1, 3, 11, 35, 1, 4, 12, 38, 1, 4, 12, 39, 1, 4, 12, 39, 1, 3, 11, 36, 1, 3, 10, 33, 102, 3, 9, 28, 89, 2, 7, 23, 74, 1, 6, 19, 59
EXAMPLE
1, 100, 2200, 110000, 3000000, 82000000, 2374000000, 65500000000, 1550000000000, ...
Taking the first digit (actually: a place holder value) of each gives the terms a(0) .. a(8) of this sequence: 1, 1, 2, 1, 3, 8, 2, 6, 1, ...
MATHEMATICA
a[n_] := Module[{k = (2*n+1)!/n!, m = 2, r, d=0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, d = r]; m++]; d]; Array[a, 100, 0] (* Amiram Eldar, Feb 14 2024 *)
PROG
(PARI)
allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2, dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
(Scheme)
(Scheme)
(define ( A265891 n) (A265890bi (+ 1 n) (+ 1 n))) ;; Code for A265890bi given in A265890.
CROSSREFS
Main diagonal of A265890 (apart from the corner term).
Triangle T(n,k) giving the smallest term in "the infinite trunk of factorial beanstalk" ( A219666) whose factorial base representation contains n digits ( A084558) and the most significant such digit ( A099563) is k.
+20
4
1, 2, 5, 7, 12, 23, 25, 48, 74, 97, 121, 240, 362, 481, 605, 721, 1440, 2162, 2881, 3605, 4326, 5041, 10080, 15122, 20161, 25205, 30246, 35288, 40321, 80640, 120962, 161281, 201605, 241926, 282248, 322568, 362881, 725760, 1088642, 1451521, 1814405, 2177286, 2540168, 2903048, 3265923
EXAMPLE
The first rows of this triangular table are:
1;
2, 5;
7, 12, 23;
25, 48, 74, 97;
121, 240, 362, 481, 605;
...
T(3,1) = 7 as 7 has factorial base representation 101, which is the smallest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1). T(3,2) = 12 as 12 has factorial base representation 200, which is the smallest such three digit term in A219666 beginning with factorial base digit 2. T(3,3) = 23 as 23 has factorial base representation 321, which is the smallest such three digit term in A219666 beginning with factorial base digit 3.
Triangle T(n,k) giving the largest member of "the infinite trunk of factorial beanstalk" ( A219666) whose factorial base representation contains n digits ( A084558) and the most significant such digit ( A099563) is k.
+20
4
1, 2, 5, 10, 17, 23, 46, 70, 92, 119, 238, 358, 476, 597, 719, 1438, 2158, 2876, 3597, 4319, 5039, 10078, 15118, 20156, 25197, 30239, 35279, 40319, 80638, 120958, 161276, 201597, 241919, 282239, 322558, 362879, 725758, 1088638, 1451516, 1814397, 2177279, 2540159, 2903038, 3265912, 3628799
COMMENTS
See A007623 for the factorial number system representation.
EXAMPLE
The first rows of this triangular table are:
1;
2, 5;
10, 17, 23;
46, 70, 92, 119;
238, 358, 476, 597, 719;
...
T(3,1) = 10 as 10 has factorial base representation 120, which is the largest such three digit term in A219666 beginning with factorial base digit 1 (in other words, for which A084558(x)=3 and A099563(x)=1). T(3,2) = 17 as 17 has factorial base representation 221, which is the largest such three digit term in A219666 beginning with factorial base digit 2. T(3,3) = 23 as 23 has factorial base representation 321, which is the largest such three digit term in A219666 beginning with factorial base digit 3.
0, 1, 2, 2, 4, 4, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 72
COMMENTS
For n >= 1, a(n) = the smallest term of A051683 >= n. Can also be obtained by replacing with zeros all other digits except the first (the most significant) in the factorial base representation of n ( A007623), then converting back to decimal.
FORMULA
Other identities:
For all n >= 0, a(n) = n - A257687(n).
EXAMPLE
Factorial base representation ( A007623) of 2 is "10", zeroing all except the most significant digit does not change anything, thus a(2) = 2. Factorial base representation ( A007623) of 3 is "11", zeroing all except the most significant digit gives "10", thus a(3) = 2. Factorial base representation of 23 is "321", zeroing all except the most significant digit gives "300" which is factorial base representation of 18, thus a(23) = 18.
PROG
(Python)
from sympy import factorial as f
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a(n):
x=str(a007623(n))
return int(x[0])*f(len(x))
CROSSREFS
Cf. also A053644 (analogous sequence for base-2).
Square array A(n,k) = (n / prime(1+ A084558(k))^e) * prime(1+ A084558(k)- A099563(k))^e, where e = A249344((1+ A084558(k)), n) = the exponent of the largest power of prime(1+ A084558(k)) which divides n. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
+20
2
1, 1, 2, 1, 2, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 3, 6, 7, 1, 2, 3, 4, 2, 6, 7, 8, 1, 2, 3, 4, 2, 6, 7, 8, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 11, 1, 2, 3, 4, 5, 6, 5, 8, 9, 6, 11, 8, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 10
COMMENTS
Informally: "clear" the exponent of prime(1+ A084558(k)) and add it (the old value of exponent) to the exponent of prime(1+ A084558(k)- A099563(k)) in the prime factorization of n. Auxiliary function for computing array A275723.
EXAMPLE
The top left 6 x 15 corner of the array:
1, 1, 1, 1, 1, 1
2, 2, 2, 2, 2, 2
2, 3, 3, 3, 3, 3
4, 4, 4, 4, 4, 4
5, 3, 3, 2, 2, 5
4, 6, 6, 6, 6, 6
7, 7, 7, 7, 7, 5
8, 8, 8, 8, 8, 8
4, 9, 9, 9, 9, 9
10, 6, 6, 4, 4, 10
11, 11, 11, 11, 11, 11
8, 12, 12, 12, 12, 12
13, 13, 13, 13, 13, 13
14, 14, 14, 14, 14, 10
10, 9, 9, 6, 6, 15
PROG
(Scheme)
;; Code for A249344bi given in A249344.
0, 1, 4, 4, 2, 2, 18, 18, 18, 18, 18, 18, 12, 12, 12, 12, 12, 12, 6, 6, 6, 6, 6, 6, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 48
COMMENTS
Auxiliary function for computing A225901: the most significant digit in factorial base representation of n is "inverted", the rest of digits are "cleared" (replaced with zeros).
MATHEMATICA
{0}~Join~Table[# (1 + ((m = 1; While[m! <= n, m++]; m - 1) - Floor[n/#])) &[k = 1; While[(k + 1)! <= n, k++]; k!], {n, 96}] (* Michael De Vlieger, Aug 30 2016, after Jayanta Basu at A084558 *)
Search completed in 0.019 seconds
| 
