A143792 a(n) = the number of distinct prime divisors, p, of n that, when p is represented in binary, each p occurs at least once in the binary representation of n.

0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 2, 1, 1, 0, 2, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 2, 2, 2, 0, 2, 1, 2, 0, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2

Offset: 1

Author

Leroy Quet, Sep 01 2008

Keywords

• base
• nonn

Comments

a(2^k * p) = 2, where k = any positive integer and p = any odd prime.

a(p) = 1, where p = any prime.

a(2^k) = 1, where k = any positive integer.

a(n) <= A078826(n). - Reinhard Zumkeller, Sep 08 2008

Size of intersection of n-th rows of tables A225243 and A027748. - Reinhard Zumkeller, Aug 14 2013

Examples

			60 in binary is 111100. The distinct primes dividing 60 are 2 (which is 10 in binary), 3 (11 in binary) and 5 (101) in binary. The string 10 does occur within 111100 like so: 111(10)0. The string 11 also occurs (multiple times) within 111100, in one way like so: (11)1100. But the string 101 does not occur in 111100. Since 2 and 3 occur within 60 (when each of these numbers is written in binary), but 5 does not, then a(60) = 2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A143791.

Programs

Haskell

import Data.List (intersect)
a143792 n = length $ a225243_row n `intersect` a027748_row (fromIntegral n)
-- Reinhard Zumkeller, Aug 14 2013

Mathematica

f[n_] := Block[{nb = ToString@ FromDigits@ IntegerDigits[n, 2], psb = ToString@ FromDigits@ IntegerDigits[ #, 2] & /@ First@ Transpose@ FactorInteger@ n, c = 0, k = 1}, lmt = 1 + Length@ psb; While[k < lmt, If[ StringCount[nb, psb[[k]]] > 0, c++ ]; k++ ]; c]; f[1] = 0; Array[f, 105] (* Robert G. Wilson v, Sep 22 2008 *)

Extensions

More terms from Robert G. Wilson v, Sep 22 2008