A275811 - OEIS

%I #26 Apr 09 2021 14:58:26

%S 0,1,1,1,1,2,1,1,1,1,1,2,1,1,2,2,1,2,1,2,1,2,2,3,1,1,1,1,1,2,1,1,1,1,

%T 1,2,1,1,2,2,1,2,1,2,1,2,2,3,1,1,1,1,1,2,2,2,2,2,2,2,1,1,2,2,1,2,1,2,

%U 1,2,2,3,1,1,2,2,1,2,1,1,2,2,1,2,2,2,3,3,2,2,1,2,2,2,2,3,1,2,1,2,2,3,1,2,1,2,2,3,1,2,2,2,2,3,2,3,2,3,3,4,1

%N Number of nonzero digits on a maximally occupied slope of factorial base representation of n: a(n) = A051903(A275734(n)). See comments for the definition.

%C Digit slopes are called "maximal", "sub-maximal", "sub-sub-maximal", etc. For digit-positions we employ one-based indexing, thus we say that the least significant digit of factorial base expansion of n is in position 1, etc. The maximal digit slope is occupied when there is at least one digit-position k that contains digit k (maximal digit allowed in that position), so that A260736(n) > 0, and n is thus a term of A273670. The sub-maximal digit slope is occupied when there is at least one nonzero digit k in a digit-position k+1. The sub-sub-maximal slope is occupied when there is at least one nonzero digit k in a digit-position k+2, etc. This sequence gives the number of nonzero digits on a slope (of possibly several) for which there exists no other slopes with more nonzero digits. See the examples.

%C In other words: a(n) gives the number of occurrences of a most common element in the multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present in factorial base representation of n and i_x is that digit's position from the right].

%C Involution A225901 maps this metric to another metric A264990 which gives the maximal number of equal nonzero digits occurring in factorial base representation (A007623) of n. See also A060502.

%H Antti Karttunen, <a href="/A275811/b275811.txt">Table of n, a(n) for n = 0..40320</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F a(n) = A051903(A275734(n)).

%F a(n) = A264990(A225901(n)).

%e 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 the "maximal slope" is also the "maximally occupied slope" (as there are no other occupied slopes present), and a(23) = 3.

%e 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 here the "maximal slope" is also the "maximally occupied slope" (with 2 nonzero digits present), and a(29) = 2.

%e 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 there are three occupied slopes in total, all with just one nonzero digit present, and a(37) = 1.

%e 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 here the sub-sub-maximal slope is the "maximally occupied slope" with its two occupiers, and a(55) = 2.

%o (Scheme, two versions)

%o (define (A275811 n) (A051903 (A275734 n)))

%o (define (A275811 n) (A264990 (A225901 n)))

%o (Python)

%o from sympy import prime, factorint

%o from operator import mul

%o from functools import reduce

%o from sympy import factorial as f

%o def a051903(n): return 0 if n==1 else max(factorint(n).values())

%o def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p

%o def a275732(n):

%o     x=str(a007623(n))[::-1]

%o     return 1 if n==0 or x.count("1")==0 else reduce(mul, [prime(i + 1) for i in range(len(x)) if x[i]=='1'])

%o def a257684(n):

%o     x=str(a007623(n))[:-1]

%o     y="".join([str(int(i) - 1) if int(i)>0 else '0' for i in x])[::-1]

%o     return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])

%o def a275734(n): return 1 if n==0 else a275732(n)*a275734(a257684(n))

%o def a(n): return 0 if n==0 else a051903(a275734(n))

%o print([a(n) for n in range(201)]) # _Indranil Ghosh_, Jun 20 2017

%Y Cf. A051903, A060502, A225901, A264990, A275734.

%Y Cf. A275804 (gives the indices of 0 and 1's), A275805 (gives the indices of terms > 1).

%K nonn,base

%O 0,6

%A _Antti Karttunen_, Aug 10 2016

%E Signs in comment corrected and clarification added by _Antti Karttunen_, Aug 16 2016