A265898 -id:A265898 - cp's OEIS frontend

A265894 a(n) = A099563(A001813(n)); the most significant digit in factorial base representation of (2n)! / n!.

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

Offset: 0

Antti Karttunen, Dec 24 2015

			The terms A001813(0) .. A001813(8) in factorial base representation (A007623) look as:
  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, ...

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

Submain diagonal of A265890.
Cf. A001813, A099563.
Cf. A265898 (positions of ones), A265899 (of descents), A266120 (local maxima just before those descents).
Cf. also A265891.

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 *)

allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265894 = n -> A099563((2*n)! / n!);

(define (A265894 n) (A099563 (A001813 n)))

(define (A265894 n) (A265890bi (+ 1 n) n)) ;; Code for A265890bi given in A265890.

a(n) = A099563(A001813(n)).

a(n) = A265890(n+1, n).

A265334 Numbers that are >= k! but < 2*k! for some k; numbers whose factorial base representation (A007623) begins with digit "1".

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150

Offset: 1

Antti Karttunen, Dec 25 2015

nonn

Numbers k for which A099563(k) = 1.

Numbers k for which A265333(k) = 1.

			1 is present as 1 >= 1! but < 2*1!
2 is present as 2 >= 2! but < 2*2!
3 is present as 3 >= 2! but < 2*2!
4 is not present as 4 >= 2! but not < 2*2! (and not >= 3!).

Antti Karttunen, Table of n, a(n) for n = 1..5915
Index entries for sequences related to factorial base representation

Cf. A007623, A099563, A265333 (characteristic function).
Cf. A059590, A060112 (subsequences).
Cf. also A265897, A265898.

A265899 After a(1) = 1, positions of descents in A265894.

Original entry on oeis.org

1, 3, 6, 8, 11, 14, 17, 20, 24, 27, 31, 34, 38, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 94, 98, 102, 106, 111, 115, 120, 124, 128, 133, 137, 142, 146, 151, 156, 160, 165, 169, 174, 179, 184, 188, 193, 198, 202, 207, 212, 217, 222, 227, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281, 286, 291, 296, 301

Offset: 1

Antti Karttunen, Dec 24 2015

nonn

Numbers n for which A099563(A001813(n)) <= A099563(A001813(n-1)), where A001813(n) = (2n)! / n!, and A099563 gives the most significant digit in the factorial base representation (A007623) of n.

Antti Karttunen, Table of n, a(n) for n = 1..1503

Cf. A001813, A007623, A099563, A265894.
Cf. A265898 (a subsequence), A266119 (first differences), A266120 (terms immediately before descents).
Cf. also A031435.

A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265894 = n -> A099563((2*n)! / n!);
my(i=0, p=1, n=0); while(i < 60, n++; my(k = A265894(n)); if(k <= p, i++; print1(n, ", ")); p = k; );

;; With Antti Karttunen's IntSeq-library.
(define A265899 (MATCHING-POS 1 1 (lambda (n) (<= (A265894 n) (A265894 (- n 1))))))

A265897 Numbers n for which (2n+1)! / n! [= A000407(n)] is >= k! but < 2*k! for some k; positions of ones in A265891.

Original entry on oeis.org

0, 1, 3, 8, 11, 14, 17, 20, 23, 30, 37, 41, 52, 56, 60, 64, 68, 72, 76, 89, 93, 97, 106, 110, 119, 128, 132, 137, 146, 155, 164, 169, 178, 183, 197, 202, 216, 221, 226, 231, 260, 265, 270, 275, 280, 285, 290, 295, 300, 331, 336, 341, 346, 351, 367, 372, 377, 393, 398, 403, 414, 419, 435, 440, 451, 456, 467, 472, 483, 494, 499, 510

Offset: 0

Antti Karttunen, Dec 24 2015

nonn

a(0) = 0 is a special case in this sequence, thus the indexing starts from zero.

Numbers n such that A000407(n) is in A265334.

Antti Karttunen, Table of n, a(n) for n = 0..845

Cf. A000407, A265891, A265334.

Cf. also A265898.

cp's OEIS Frontend

A265894 a(n) = A099563(A001813(n)); the most significant digit in factorial base representation of (2n)! / n!.

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A265334 Numbers that are >= k! but < 2*k! for some k; numbers whose factorial base representation (A007623) begins with digit "1".

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A265899 After a(1) = 1, positions of descents in A265894.

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A265897 Numbers n for which (2n+1)! / n! [= A000407(n)] is >= k! but < 2*k! for some k; positions of ones in A265891.

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