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A276091
Numbers obtained by reinterpreting base-2 representation of n in A001563-base (A276326): a(n) = Sum_{k>=0} A030308(n,k)*A001563(k+1).
13
0, 1, 4, 5, 18, 19, 22, 23, 96, 97, 100, 101, 114, 115, 118, 119, 600, 601, 604, 605, 618, 619, 622, 623, 696, 697, 700, 701, 714, 715, 718, 719, 4320, 4321, 4324, 4325, 4338, 4339, 4342, 4343, 4416, 4417, 4420, 4421, 4434, 4435, 4438, 4439, 4920, 4921, 4924, 4925, 4938, 4939, 4942, 4943, 5016, 5017, 5020, 5021, 5034, 5035, 5038, 5039, 35280, 35281
OFFSET
0,3
COMMENTS
Numbers that are sums of distinct terms of A001563.
A number is included if and only if all the nonzero digits in its factorial base representation (A007623) are maximal allowed in those digit positions, thus this sequence gives all numbers n for which A060130(n) = A260736(n).
Numbers n for which A276328(n) = A276337(n), thus from 1 onward the positions of ones in A276336.
Conjectured also to give all numbers n for which A255411(n) = A276340(n) (thus zeros of A276339).
FORMULA
a(0) = 0, a(2n) = A255411(a(n)), a(2n+1) = 1+A255411(a(n)).
a(0) = 0, a(2n) = A276340(a(n)), a(2n+1) = 1+A276340(a(n)).
Other identities. For all n >= 0:
a(n) = A225901(A059590(n)).
a(n) = A276090(A275959(n)).
A276328(a(n)) = A276337(a(n)) = A000120(n).
MATHEMATICA
Table[Total[Times @@@ Transpose@ {Map[# #! &, Range@ Length@ #], Reverse@ #}] &@ IntegerDigits[n, 2], {n, 64}] (* Michael De Vlieger, Aug 31 2016 *)
PROG
(Scheme)
;; This is a standalone program:
(define (A276091 n) (let loop ((n n) (s 0) (f 1) (i 2)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) s (* i f) (+ 1 i))) (else (loop (/ (- n 1) 2) (+ s (* (- i 1) f)) (* i f) (+ 1 i))))))
;; This implements one of the given recurrences:
(definec (A276091 n) (cond ((zero? n) n) ((even? n) (A255411 (A276091 (/ n 2)))) (else (+ 1 (A255411 (A276091 (/ (- n 1) 2)))))))
;; Alternatively, we can use A276340 in place of A255411:
(definec (A276091 n) (cond ((zero? n) n) ((even? n) (A276340 (A276091 (/ n 2)))) (else (+ 1 (A276340 (A276091 (/ (- n 1) 2)))))))
(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 a255411(n):
x=(str(a007623(n)) + '0')
y="".join(str(int(i) + 1) if int(i)>0 else '0' for i in x)[::-1]
return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
def a(n): return 0 if n==0 else a255411(a(n//2)) if n%2==0 else 1 + a255411(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 19 2016
EXTENSIONS
Name changed (to emphasize the functional nature of the sequence) with the original definition moved to the comments by Antti Karttunen, Sep 01 2016
STATUS
approved