

A307218


Numbers x with k digits in base 2 (MSD(x)_2 = d_1, LSD(x)_2 = d_k) that are equal to the product of the positions of 1's (see examples and formula).


1




OFFSET

1,2


COMMENTS

If 0's instead of 1's are considered, then the sequence starts with 1, 2, 12, 504, ...
If the positions are counted with MSD(x)_2 = d_k and LSD(x)_2 = d_1 then the sequence starts with 1, 2, 6, 12, 576000, ...


LINKS



FORMULA

Solutions of the equation x = Product_{j=1..k} ((1/2)*(1+j+(1)^d_j*(1j))), where d_j are the digits of x in base 2.


EXAMPLE

1350 in base 2 is 10101000110. The 1's are in positions 1, 3, 5, 9, 10 and 1*3*5*9*10 = 1350.
47520 in base 2 is 1011100110100000. The 1's are in positions 1, 3, 4, 5, 8, 9, 11 and 1*3*4*5*8*9*11 = 47520.
1995840 in base 2 is 111100111010001000000. The 1's are in positions 1, 2, 3, 4, 7, 8, 9, 11, 15 and 1*2*3*4*7*8*9*11*15 = 1995840.


MAPLE

P:=proc(q) local a, b, k, n; for n from 1 to q do
a:=convert(n, base, 2); b:=1; for k from 1 to nops(a) do
if a[k]=1 then b:=b*(nops(a)k+1); fi; od; if b=n then print(n);
fi; od; end: P(10^9);


PROG

(PARI) b(n)=fromdigits(binary(n), 10); \\ A007088
is(n)={k=1; v=digits(b(n)); for(j=2, #v, if(v[j]==1, k=k*j)); k==n; } \\ Jinyuan Wang, Mar 29 2019


CROSSREFS



KEYWORD

nonn,base,more


AUTHOR



STATUS

approved



