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A278909 Binary Smith numbers: composite numbers n such that sum of bits of n = sum of bits of prime factors of n (counted with multiplicity). 4

%I

%S 15,51,55,85,125,159,185,190,205,215,222,238,246,249,253,287,303,319,

%T 374,407,438,442,469,471,475,489,494,501,507,591,623,639,670,679,687,

%U 699,730,745,755,763,765,771,799,807,822,830,843,867,890,893,917,923,925,935,939,951,970,973,979,986,989,995,1010,1015,1017,1020,1023,1135,1167,1203,1243

%N Binary Smith numbers: composite numbers n such that sum of bits of n = sum of bits of prime factors of n (counted with multiplicity).

%C Binary equivalent of A006753 as well as A176670. (Since bits can only be 0 or 1, having equal sums of bits is logically equivalent to having the same nonzero bits.)

%C There are 615 terms up to 10^4, 6412 up to 10^5, 66369 up to 10^6, 630106 up to 10^7, 6268949 up to 10^8, 62159262 up to 10^9, and 596587090 up to 10^10. - _Charles R Greathouse IV_, Dec 09 2016

%H Ely Golden, <a href="/A278909/b278909.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 15, as 15 (1111) in binary has the same number of 1 bits as its prime factors (11 and 101).

%t Select[Range@ 1250, And[CompositeQ@ #, DigitCount[#, 2, 1] = Total@ Flatten@ Apply[DigitCount[#, 2, 1] & /@ ConstantArray[#1, #2] &, FactorInteger@ #, 1]] &] (* _Michael De Vlieger_, Dec 02 2016 *)

%o (SageMath)

%o def numfactorbits(x):

%o if(x<2):

%o return 0;

%o s=0;

%o f=list(factor(x));

%o #ensures inequality of numfactorbits(x) and bin(x).count("1") if x is prime

%o if((len(f)==1)&(f[0][1]==1)):

%o return 0;

%o for c in range(len(f)):

%o s+=bin(f[c][0]).count("1")*f[c][1]

%o return s;

%o counter=2

%o index=1

%o while(index<=10000):

%o if(numfactorbits(counter)==bin(counter).count("1")):

%o print(str(index)+" "+str(counter))

%o index+=1;

%o counter+=1;

%o (PARI) is(n) = my(f=factor(n)[, 1]~, expo=factor(n)[, 2]~, v=[], s=0); for(k=1, #f, while(expo[k] > 0, expo[k]--; v=concat(v, f[k]))); for(k=1, #v, v[k]=binary(v[k])); my(w=[]); for(y=1, #v, w=concat(w, v[y])); if(vecsum(w)==vecsum(binary(n)), return(1), return(0))

%o terms(n) = my(i=0); forcomposite(c=1, , if(is(c), print1(c, ", "); i++; if(i==n, break)))

%o /* Print initial 70 terms as follows: */

%o terms(70) \\ _Felix Fröhlich_, Dec 01 2016

%o (PARI) is(n)=my(f=factor(n),t=#f~); (t>1 || (t==1 && f[1,2]>1)) && hammingweight(n)==sum(i=1,t, hammingweight(f[i,1])*f[i,2]) \\ _Charles R Greathouse IV_, Dec 02 2016

%Y Cf. A006753, A176670.

%K nonn,base,easy

%O 1,1

%A _Ely Golden_, Nov 30 2016

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Last modified October 14 03:58 EDT 2019. Contains 327995 sequences. (Running on oeis4.)