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Cohen 1996 iterating sum divisors function
conjecture_p98: Records the paper's computation that the iterated sum-of-divisors sequences starting at 2 to 200 fall into 21 classes that do not meet below 10^200, and its conjecture that they never meet, which it offers as evidence against statement (vi).
remark_p98: Records the paper's computed values of the iterated sum-of-divisors function bearing on whether sigma^m(n)^{1/m} tends to infinity, and its observation that this statement together with eventual monotonicity of the sequence sigma^i(n)^{1/i} implies that sigma^{i+1}(n)/sigma^i(n) tends to infinity.
theorem_2_1: States that if l is an odd (2,k)-perfect number, 2^a divides k sigma(sigma(2^a)) and sigma(2^a) is coprime to sigma(l), then 2^a l is (2, 2^{-a} k sigma(sigma(2^a)))-perfect.
theorem_2_2: States that the equation sigma(sigma(2n)) = 2 sigma(sigma(n)) has infinitely many solutions, in contrast with sigma(2n) = 2 sigma(n), which has none.
theorem_3_1: States that if the least m with sigma^m(n)/n integral is finite, t divides the resulting multiplier and sigma^{M+a}(n) = sigma^M(tn) with M below that least m minus a, then the least m for tn is at most that least m minus a, with an exact or bounded multiplier for tn.
Graeme L. Cohen, Herman J. J. te Riele, Iterating the Sum-of-Divisors Function. Experimental Mathematics 5 (1996), no. 2, 91-100. doi:10.1080/10586458.1996.10504580. The copy read for this card prints "© A K Peters, Ltd. 1058-6458/96 $0.50 per page" in its first-page footer and "© A K Peters, Ltd. 1058-6458/1997 $0.50 per page" on the appended 1997 errata page (Experimental Mathematics 6 (1997), no. 2, 177), every other right reserved.
Cohen and te Riele call n (m,k)-perfect when sigma_m(n)=kn, tabulate all (2,k)-perfect numbers below 10^9 (Table 1, which omits the (2,15)-perfect number 506967552 that the 1997 errata restores) and all (3,k)- and (4,k)-perfect numbers up to 2*10^8 (given in their 1995 report), and compute the least m for which n is (m,k)-perfect for every n up to 1000. Theorem 2.1 builds new (2,k)-perfect numbers as 2^a times an odd one, Theorem 2.2 shows sigma(sigma(2n))=2sigma(sigma(n)) has infinitely many solutions, and Theorem 3.1 allows exact values of the least iterate count and its multiplier to be predicted from earlier values in many cases; the computations required factoring a 104-digit number by the special number field sieve. The method is large-scale computation with sigma-iteration trees plus elementary multiplicative arguments. On the Erdos-Granville-Pomerance-Spiro list of six statements (p. 92) the paper reports h(401)=1.1146, h(461)=1.1276 and h(659)=1.1658, suggesting sigma_m(n)^(1/m) grows at least like log m, and observes that if statement (iii) holds and the sequence sigma_i(n)^(1/i) is eventually monotone then statement (ii) follows, with the computations strongly suggesting that monotonicity for every n. Against statement (vi) it reports that the sigma sequences from 2 to 200 form 21 trees that do not meet below 10^200 and conjectures that they never meet (p. 98). It also records Maier's 1984 proof that the liminf of sigma_3(n)/n is finite.
Bears on. #410: the problem is statement (iii) of the paper's p. 92 list. The paper gives numerical evidence for it (the values of h(n) on p. 98 and Table 4 on p. 99) and proves nothing about it; its observation that (iii) with eventual monotonicity of sigma_i(n)^(1/i) implies statement (ii) does not bear on the problem's answer. #412: the problem is statement (vi) of the same list, which the paper says it does not believe (p. 97). It reports that the sigma sequences from 2 to 200 fall into 21 trees that do not meet below 10^200 and conjectures that they never meet (p. 98), which would give a negative answer; the computation itself decides nothing.
Results. Theorem 2.1 (p. 93); Theorem 2.2 (p. 94); Theorem 3.1 (p. 94, proof p. 96); the evidence for statement (iii) (p. 98, unnumbered, with Table 4, p. 99); the tree conjecture (pp. 97-98, unnumbered).
Read status. Claims checked for the five results above, read clause by clause on the print; the proofs were read for their structure, and the computations were not rerun.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.