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
Examples
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)
Links
Programs
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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
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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
Extensions
More terms from R. J. Mathar, Sep 17 2008
Comments