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A336836
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Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.
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4
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0, 0, 0, 2, 0, 3, 0, 4, 1, 2, 0, 4, 0, 1, 2, 4, 0, 5, 0, 4, 1, 0, 0, 6, 0, 0, 3, 4, 0, 4, 0, 6, 0, 0, 1, 6, 0, 0, 1, 4, 0, 4, 0, 4, 3, 0, 0, 6, 1, 2, 0, 4, 0, 5, 0, 4, 1, 0, 0, 6, 0, 0, 3, 6, 0, 4, 0, 4, 1, 4, 0, 9, 0, 0, 4, 4, 0, 4, 0, 6, 3, 0, 0, 6, 0, 0, 0, 6, 0, 5, 1, 4, 0, 0, 0, 9, 0, 3, 3, 4, 0, 3, 0, 4, 3
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OFFSET
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1,4
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COMMENTS
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Starting from n, the number of prime shifts needed before a term of A246281 is reached.
It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
Note that in contrast to abundancy used in A336835, the condition [A003961(x) > 2x] (= A252742) is not monotonic when iterating with A003961. For example, we have A003961(9) = 25 > 2*9, A003961(25) = 49 < 2*25, and then again A003961(49) = 121 > 2*49.
Question: Is the escape clause necessary in the definition?
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LINKS
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PROG
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(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336836(n) = for(i=0, oo, my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));
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CROSSREFS
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Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).
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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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