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A302790
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Number of runs of consecutive Fermi-Dirac factors of n (the runs are separated by gaps between indices of factors): a(n) = A069010(A052331(n)).
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5
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0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2
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OFFSET
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1,8
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LINKS
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FORMULA
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For all n >= 1, a(n) <= A064547(n).
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EXAMPLE
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n = 84 has Fermi-Dirac factorization as A050376(2) * A050376(3) * A050376(5) = 3*4*7. Because there is a gap between A050376(3) and A050376(5), the factors occur in two separate runs (3*4 and 7), thus a(84) = 2.
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PROG
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(PARI)
up_to = 65537;
v050376 = vector(up_to);
ispow2(n) = (n && !bitand(n, n-1));
i = 0; for(n=1, oo, if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to, break));
A052331(n) = { my(s=0, e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A069010(n) = ((1 + (hammingweight(bitxor(n, n>>1)))) >> 1); \\ From A069010
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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