A275805 - OEIS

%I #24 Apr 26 2021 11:19:05

%S 5,11,14,15,17,19,21,22,23,29,35,38,39,41,43,45,46,47,53,54,55,56,57,

%T 58,59,62,63,65,67,69,70,71,74,75,77,80,81,83,84,85,86,87,88,89,91,92,

%U 93,94,95,97,99,100,101,103,105,106,107,109,110,111,112,113,114,115,116,117,118,119,125,131,134,135,137,139,141,142,143,149,155

%N Indices of nonsquarefree terms in A275734; numbers with at least one digit slope (in their factorial base representation) with multiple nonzero digits. (See comments for the exact definition).

%C Numbers n for which A008683(A275734(n)) = 0.

%C Numbers n for which A275811(n) > 1.

%C Numbers n in whose factorial base representation (A007623) there exists at least one pair of digit positions i_1 and i_2 with nonzero digits d_1 and d_2 such that (i_1 - d_1) = (i_2 - d_2).

%H Antti Karttunen, <a href="/A275805/b275805.txt">Table of n, a(n) for n = 1..10000</a>

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

%e For n=5, "21" in factorial base (A007623), the pair 2 (in position 2) and 1 (in position 1) satisfies the condition, as (2-2) = (1-1), thus 5 is included.

%e For n=55, "2101" in factorial base, the pair 2 (in position 4) and 1 (in position 3) satisfies the condition, as (4-2) = (3-1), thus 55 is included.

%e For n=67, "2301" in factorial base, the pair 3 (in position 3) and 1 (in position 1) satisfies the condition, as (3-3) = (1-1), thus 67 is included in the sequence.

%o (Scheme, with _Antti Karttunen_'s IntSeq-library, two alternatives)

%o (define A275805 (ZERO-POS 1 1 (COMPOSE A008683 A275734)))

%o (define A275805 (MATCHING-POS 1 1 (lambda (n) (< 1 (A275811 n)))))

%o (Python)

%o from operator import mul

%o from sympy import prime, factorial as f, mobius

%o from functools import reduce

%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 "1" not in x 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 a(n): return 1 if n==0 else a275732(n)*a(a257684(n))

%o print([n for n in range(201) if mobius(a(n))==0]) # _Indranil Ghosh_, Jun 19 2017

%Y Complement: A275804.

%Y Cf. A007623, A008683, A275734, A275811.

%Y Cf. A275809 (a subsequence apart from its initial 0-term).

%Y Subsequence of A115945.

%K nonn,base

%O 1,1

%A _Antti Karttunen_, Aug 10 2016