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A351259
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First noncomposite number reached when iterating the map x -> x', when starting from x = A351255(n). Here x' is the arithmetic derivative of x, A003415.
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6
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1, 2, 3, 5, 5, 7, 5, 7, 31, 7, 41, 71, 191, 2711, 7, 5, 7, 41, 103, 59, 71, 271, 71, 1031, 2887, 439, 5, 5, 7, 631, 251, 401, 3491, 1031, 1319, 17747, 9733, 1931, 16319, 77351, 131, 5, 419, 7079, 22343, 971, 5981, 6861581, 419, 18731, 11903, 33937, 7079, 15287, 15287, 6143, 6944111, 1415651, 11, 13, 5, 61, 103, 401, 631
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OFFSET
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1,2
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COMMENTS
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For the initial 105367 19-smooth terms of A351255, the last 7 occurs here at a(54796), with A351255(54796) = 289993286583 = 3^2 * 7 * 11 * 13^2 * 19^5, and the last 5 occurs here at a(65777), with A351255(65777) = 391899820830375516750 = 2 * 3^2 * 5^3 * 7^3 * 13^3 * 17^3 * 19^6, already a moderately high starting value, in whose vicinity most ending primes for successful iterations are much larger. This observation motivates a conjecture: Even from large numbers with high exponents in their prime factorization it is sometimes possible to reach a small prime. Compare to the conjecture 8 in Ufnarovski & Åhlander paper.
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LINKS
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FORMULA
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EXAMPLE
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From A351255(27) = 2625 it takes 12 iterations of the map x -> A003415(x) to reach zero: 2625 -> 2825 -> 1155 -> 886 -> 445 -> 94 -> 49 -> 14 -> 9 -> 6 -> 5 -> 1 -> 0. Two steps before the final zero is the first and only prime on the path, 5, therefore a(27) = 5.
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PROG
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(PARI)
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i, 2]>=f[i, 1], return(0), s += f[i, 2]/f[i, 1])); (n*s));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351078(n) = { while(n>1&&!isprime(n), n = A003415checked(n)); (n); };
for(n=0, 2^9, u=A276086(n); p = A351078(u); if(p>0, print1(p, ", ")));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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