The first non-square odd composite is 15. The greatest prime < sqrt(15) = 3. Since 15=3*5, 3 is the desired factor and a(1)=0 (no iterations required). The same applies to 21, so a(2)=0.
The same applies to 21, so a(2)=0.
Iterations to the nearest prime factor < sqrt(A082686(n)).
The first odd composite is 9 but it is a square and so is skipped.
The next first non-square odd composite is 15. The greatest prime < sqrt(15) = 3. Since 15=3*5, 3 is the desired factor and a(1)=0 (no iterations required).
The next treated odd composite is 27, with the greatest prime < sqrt(27) = 5. Since 27=3*9, 3 is one prime preceding 5 and so a(3)=1.
When A082686(n)=7923 is evaluated, the starting prime factor for evaluation is 89, and we see the actual factor as 19. a(n) is the iteration count to the resulting term, or 16 (evaluations from the 24th prime down to the 8th prime).
Cf. A082686.
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Iterations downwards from to the nearest prime < sqrt(Q) to an encountered factor, < sqrt(Q), Q being a non-square odd composite.
Every integer should be seen in this infinite sequence. New records appear to be in consecutive numerical order, suggesting every integer should be seen in this infinite sequence. Considering a(n)=0, empirically a power fit Y=k*x^c correlates well with the "xth" occurrence. For example, the 491st 0 value is at n=99808.
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(PARI) sumtwins(n) = { local(x, j, s, sr, p10x); for(x=1, n, s=0; p10x=10^x; forstep(j=3, 10^x, 2, if(j+2 < p10x & & ispseudoprime(j) & & ispseudoprime(j+2), s+=j+j+2); ); print1(s", "); ) }
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Iterations downwards from the nearest prime < sqrt(Q) to an encountered factor, Q being a non-square odd composite.
0, 0, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 3, 2, 3, 1, 2, 3, 0, 2, 1, 3, 2, 3, 1, 0, 4, 2, 4, 4, 3, 1, 2, 0, 4, 2, 3, 4, 1, 4, 3, 2, 4, 0, 1, 3, 4, 4, 2, 0, 4, 1, 3, 2, 4, 3, 4, 0, 1, 4, 3, 2, 5, 4, 2, 1, 3, 5, 4, 5, 3
1,8
Every integer should be seen in this infinite sequence. New records appear to be in consecutive numerical order. Considering a(n)=0, empirically a power fit Y=k*x^c correlates well with the "xth" occurrence. For example, the 491st 0 value is at n=99808.
The first odd composite is 9 but it is a square and so is skipped. The next odd composite is 15. The greatest prime < sqrt(15) = 3. Since 15=3*5, 3 is the desired factor and a(1)=0. The same applies to 21, so a(2)=0. The next treated odd composite is 27, with the greatest prime < sqrt(27) = 5. Since 27=3*9, 3 is one prime preceding 5 and so a(3)=1.
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Bill McEachen, Oct 15 2024
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allocated for Bill McEachen
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