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

1,2

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

Links

Table of n, a(n) for n=1..5.

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.

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

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

Sequence in context: A307586 A224743 A260542 * A278383 A174638 A154375

Adjacent sequences: A307215 A307216 A307217 * A307219 A307220 A307221

Keyword

nonn,base,more

Author

Paolo P. Lava, Mar 29 2019

Status

approved