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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
15, 51, 55, 85, 125, 159, 185, 190, 205, 215, 222, 238, 246, 249, 253, 287, 303, 319, 374, 407, 438, 442, 469, 471, 475, 489, 494, 501, 507, 591, 623, 639, 670, 679, 687, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.)

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

LINKS

Ely Golden, Table of n, a(n) for n = 1..10000

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(SageMath)

def numfactorbits(x):

    if(x<2):

        return 0;

    s=0;

    f=list(factor(x));

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

    if((len(f)==1)&(f[0][1]==1)):

        return 0;

    for c in range(len(f)):

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

    return s;

counter=2

index=1

while(index<=10000):

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

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

        index+=1;

    counter+=1;

(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))

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

/* Print initial 70 terms as follows: */

terms(70) \\ Felix Fröhlich, Dec 01 2016

(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

CROSSREFS

Cf. A006753, A176670.

Sequence in context: A318084 A191746 A029941 * A194851 A075928 A020214

Adjacent sequences:  A278906 A278907 A278908 * A278910 A278911 A278912

KEYWORD

nonn,base,easy

AUTHOR

Ely Golden, Nov 30 2016

STATUS

approved

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Last modified September 20 10:10 EDT 2019. Contains 327229 sequences. (Running on oeis4.)