

A275734


Primefactorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).


21



1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27
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OFFSET

0,2


COMMENTS

These are primefactorization representations of singlevariable polynomials where the coefficient of term x^(k1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.


LINKS



FORMULA

Other identities and observations. For all n >= 0:


EXAMPLE

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 a(23) = prime(1)^3 = 2^3 = 8.
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 "subsubsubmaximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the submaximal, and the most significant 1 is on the "subsubsubmaximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the subsubmaximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.


PROG

(Scheme, with memoizationmacro definec)
(Python)
from operator import mul
from sympy import prime, factorial as f
def a007623(n, p=2): return n if n<p else a007623(int(n/p), p+1)*10 + n%p
def a275732(n):
x=str(a007623(n))[::1]
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'])
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))


CROSSREFS

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).


KEYWORD

nonn,base


AUTHOR



STATUS

approved



