%I #24 Mar 24 2021 09:51:49

%S 0,1,2,3,4,6,7,8,9,10,12,13,16,18,20,24,25,26,27,28,30,31,32,33,34,36,

%T 37,40,42,44,48,49,50,51,52,60,61,64,66,68,72,73,76,78,79,82,90,96,98,

%U 102,104,108,120,121,122,123,124,126,127,128,129,130,132,133,136,138,140,144,145,146,147,148,150,151,152,153,154,156,157,160

%N Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n.

%C Indexing starts from zero, because a(0) = 0 is a special case in this sequence.

%C Numbers n for which A275947(n) = 0 or equally, for which A275811(n) <= 1.

%C Numbers n for which A008683(A275734(n)) <> 0, that is, indices of squarefree terms in A275734.

%C Numbers n for which A060130(n) = A060502(n).

%C Numbers with at most one nonzero digit on each digit slope of the factorial base representation of n (see A275811 and A060502 for the definition of slopes in this context). More exactly: numbers n in whose factorial base representation (A007623) there does not exist a 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="/A275804/b275804.txt">Table of n, a(n) for n = 0..10000</a>

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

%o (Scheme, with _Antti Karttunen_'s IntSeq-library, various implementations)

%o (define A275804 (ZERO-POS 0 0 A275947))

%o (define A275804 (MATCHING-POS 0 0 (lambda (n) (>= 1 (A275811 n)))))

%o (define A275804 (NONZERO-POS 0 0 (COMPOSE A008683 A275734)))

%o (define A275804 (MATCHING-POS 0 0 (lambda (n) (= (A060130 n) (A060502 n)))))

%o (Python)

%o from operator import mul

%o from sympy import prime, factorial as f

%o from sympy.ntheory.factor_ import core

%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 a(n): return 1 if n==0 else a275732(n)*a(a257684(n))

%o def ok(n): return 1 if n==0 else core(a(n))==a(n)

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

%Y Complement: A275805.

%Y Indices of zeros in A275947 and A275962.

%Y Intersection with A276005 gives A261220.

%Y Cf. A059590 (a subsequence).

%Y Cf. A007623, A008683, A060130, A060502, A275734, A275811.

%K nonn,base

%O 0,3

%A _Antti Karttunen_, Aug 10 2016