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A275805
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).
7
5, 11, 14, 15, 17, 19, 21, 22, 23, 29, 35, 38, 39, 41, 43, 45, 46, 47, 53, 54, 55, 56, 57, 58, 59, 62, 63, 65, 67, 69, 70, 71, 74, 75, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 91, 92, 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
OFFSET
1,1
COMMENTS
Numbers n for which A008683(A275734(n)) = 0.
Numbers n for which A275811(n) > 1.
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).
EXAMPLE
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.
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.
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.
PROG
(Scheme) ;; With Antti Karttunen's IntSeq-library, two alternatives.
(define A275805 (ZERO-POS 1 1 (COMPOSE A008683 A275734)))
(define A275805 (MATCHING-POS 1 1 (lambda (n) (< 1 (A275811 n)))))
(Python)
from operator import mul
from sympy import prime, factorial as f, mobius
from functools import reduce
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a275732(n):
x=str(a007623(n))[::-1]
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'])
def a257684(n):
x=str(a007623(n))[:-1]
y="".join(str(int(i) - 1) if int(i)>0 else '0' for i in x)[::-1]
return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
print([n for n in range(201) if mobius(a(n))==0]) # Indranil Ghosh, Jun 19 2017
CROSSREFS
Complement: A275804.
Cf. A275809 (a subsequence apart from its initial 0-term).
Subsequence of A115945.
Sequence in context: A275118 A385451 A275640 * A340605 A313993 A297251
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 10 2016
STATUS
approved