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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). 11
 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 FrÃ¶hlich, 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 (Python) from sympy import factorint def sbd(n): return bin(n).count('1') def ok(n):   f = factorint(n)   return sum(f[p] for p in f) > 1 and sbd(n) == sum(sbd(p)*f[p] for p in f) print(list(filter(ok, range(1244)))) # Michael S. Branicky, Apr 22 2021 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 August 5 19:49 EDT 2021. Contains 346488 sequences. (Running on oeis4.)