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, 1350, 47520, 1995840, 59376240

Offset: 1

Author

Paolo P. Lava, Mar 29 2019

Keywords

• nonn
• base
• more

Comments

If 0's instead of 1's are considered, then the sequence starts with 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, ... (see A161324).

Next value greater than 10^12. - Giovanni Resta, Mar 29 2019

Examples

			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.
		

Crossrefs

Cf. A000030, A007088, A010879, A161324, A306286.

Programs

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);

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

Formula

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