A144095 a(n) = number of exponents of the prime-factorization of n that occur somewhere in n when the exponents and n are represented in base 2.

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

Offset

1,6

Comments

a(n) = A001221(n) if n is not included in sequence A144096.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to binary expansion of n

Example

40 has the prime-factorization 2^3 * 5^1, so the exponents are 3 and 1. 40 in binary is 101000. 3 = 11 in binary. 11 does not occur anywhere in 101000. 1 is 1 in binary. 1 does occur (twice) in 101000. So a(40) = 1, since one exponent occurs in the binary representation of n.
From Antti Karttunen, Nov 01 2017: (Start)
For n = 6 = 2^1 * 3^1, the binary representation "1" of exponent 1 (of 2) is found from the binary representation "110" of 6, like is found also the exponent of 3 (which is also 1), thus a(6) = 2.
For n = 8 = 2^3, the binary representation "11" of the only exponent 3 is not found from the binary representation "1000" of 8, thus a(8) = 0.
For n = 24 = 2^3 * 3^1, both the binary representation "11" of exponent 3 and the binary representation "1" of exponent 1 are found from the binary representation "11000" of 24, thus a(24) = 2.
(End)

Maple

A144095 := proc(n) local n2, a, ifa, e2, p ; n2 := convert(n, base, 2) ; ifa := ifactors(n)[2] ; a := 0 ; for p in ifa do e2 := convert( op(2, p), base, 2) ; if verify(n2, e2, 'superlist') then a := a+1 ; fi; od: RETURN(a) ; end: for n from 1 to 200 do printf("%d, ", A144095(n)) ; od: # R. J. Mathar, Sep 17 2008

Prog

(PARI)
is_vecsuffix(va, vb) = { my(ka=#va, kb=#vb, i=kb); if(ka < kb, 0, while(i>0, if(va[(ka-kb)+i] != vb[i], return(0), i = i-1)); (1)); };
is_base2infix(a, b) = { my(va=binary(a), vb=binary(b)); while(#va >= #vb, if(is_vecsuffix(va, vb), return(1), a \= 2; va=binary(a))); (0); };
A144095(n) = vecsum(apply(e -> is_base2infix(n, e), factorint(n)[, 2])); \\ Antti Karttunen, Nov 01 2017

Crossrefs

Cf. A144096.

Differs from A293439 for the first time at n=24.

Sequence in context: A344652 A366074 A293439 * A362719 A076092 A080468

Adjacent sequences: A144092 A144093 A144094 * A144096 A144097 A144098

Keyword

base,nonn

Author

Leroy Quet, Sep 10 2008

Extensions

More terms from R. J. Mathar, Sep 17 2008

Status

approved