A019513 -id:A019513 - cp's OEIS frontend

A307102 Numbers written in base of double factorial numbers (A006882).

1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, 1100, 1101, 1110, 1200, 10000, 10001, 10010, 10100, 10101, 10110, 10200, 10201, 11000, 11001, 11010, 11100, 11101, 11110, 11200, 20000, 20001, 20010, 20100, 20101, 20110, 20200, 20201, 21000

Offset: 1

Sean A. Irvine, Mar 24 2019

a(1122755752855713895623244049306709034778906250) is the first term which cannot be included in the OEIS since it includes a non-decimal digit. - Charles R Greathouse IV, Sep 19 2012

Numbers in this mixed-radix number system can have multiple representations, so to avoid ambiguity this sequence assumes a greedy approach where leading digits are made as high as possible; thus we choose a(30) = 20000 rather than a(30) = 11201. - Sean A. Irvine, Mar 24 2019

			The digits (from right to left) have values 1, 2, 3, 8, 15, etc. (A006882), hence a(29) = 11200 because 29 = 1*15 + 1*8 + 2*3 + 0*2 + 0*1.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
Sean A. Irvine, Java program (github).
F. Smarandache, Collected Papers, Vol. II.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Index entries for sequences related to factorial numbers.

Cf. A006882 (double factorial numbers), A007623 (factorial base), A019513 (erroneous version).

a[n_] := FromDigits[NumberDecompose[n, Range[n, 1, -1]!!]]; Array[a, 40] (* Amiram Eldar, May 11 2024 *)

A181521 Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.

Original entry on oeis.org

1, 2, 3, 13, 23, 33, 133, 4, 14, 24, 34, 134, 234, 334, 5, 15, 25, 35, 135, 235, 335, 1335, 45, 145, 245, 345, 1345, 2345, 3345, 55, 155, 255, 355, 1355, 2355, 3355, 13355, 455, 1455, 2455, 3455, 13455, 23455, 33455, 555, 1555, 2555, 6, 16, 26, 36, 136, 236, 336

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 26 2010

The encoding of n is similar to A111095 but uses a double-factorial base A006882 to define the expansion coefficients.
The expansion coefficients b_k in n = sum_{k>=1} b_k * A006882(k) are defined "greedily" by taking the largest A006882(k) which is <=n, choosing b_k as large as possible such that b_k*A006882(k) remains <=n, subtracing b_k*A006882(k) from n to define a remainder, and recursively slicing the remainder to generate b_{k-1}, then b_{k-2} etc until the remainder reduces to zero. This produces the b_k for each n equivalent to A019513(n).
This representation A019513 is then scanned from the least to the most-significant b_k, i.e., along increasing k, and for each nonzero b_k, b_k copies of k are appended to a string representation -- starting from an empty string. This final representation is interpreted as a base-10 number a(n).

			a(39) = 1455 because 1!!+4!!+5!!+5!! = 1+8+15+15 = 39

Cf. A111095

dblfactfloor := proc(n) local j ; for j from 1 do if doublefactorial(j) > n then return j-1 ; end if; end do: end proc:
dblfbase := proc(n) local nshf,L,f; nshf := n ; L := [] ; while nshf > 0 do f := dblfactfloor(nshf) ; L := [f,op(L)] ; nshf := nshf-doublefactorial(f) ; end do: L ; end proc:
read("transforms") ; A181521 := proc(n) digcatL(dblfbase(n)) ; end proc:
seq(A181521(n),n=1..70) ; # R. J. Mathar, Dec 06 2010

cp's OEIS Frontend

A307102 Numbers written in base of double factorial numbers (A006882).

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