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Statement

Theorem 2 (printed p. 619). There is an absolute constant $C$ such that every finite or infinite sequence of integers $a_1<a_2<\cdots$ in which no term equals a sum of distinct other terms has $\sum_i1/a_i<C$.

The paper calls this "the following old result of P. Erdős [3]" (the 1962 Mat. Lapok paper) and gives "the outline of the proof here" because "the proof appeared in Hungarian" (p. 619); it ends (p. 620): "It is perhaps not quite easy to get the best possible value of $C$. It seems certain that $C<10$." The theorem is applied to property P', defined on p. 619 by "no divisor of $n$ is the distinct sum of other divisors of $n$": $\sigma(n)/n>C$ excludes P'.

Source. S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), no. 126, 617–623, DOI 10.1090/S0025-5718-1974-0347726-9; seven-page scan, printed p. $n$ on PDF p. $n-616$. Theorem 2 and the outline on printed pp. 619–620 (PDF pp. 3–4), read on the page image of p. 619 and in the text layer of p. 620.

Read depth. Claims checked: the statement and the closing remark were read clause by clause. The outline was read for its structure and is not checked here.

Proof pointer

Pages 619–620, the argument of the Hungarian original (Theorem II, where the constant is $103$): with $A(x)=\sum_{a_i\le x}1$, the integers $n$ are split into a first class with $A(2^{n+1})-A(2^n)<2^n/n^2$ (display (2)), whose terms contribute less than $\sum1/n^2<2$ to the reciprocal sum (3), and a second class $n_1<n_2<\cdots$ (4); the sums $a_1+a_2+\cdots+a_r+a_k$, $1\le r<k$, are all distinct (5) because a coincidence would write some $a_k$ as a distinct sum of other terms, and counting them below $2^{n_j+2}$ (displays (6)–(9)) gives $A(2^{n_j+1})<10\cdot2^{n_j}/j^2$ for $j>100$ (10), so the second class also contributes a bounded amount.

Dependencies

None; elementary.

Bears on

• Problem 876: the English source of the bound behind the site's reciprocal-sum question ("Erdős had proved this is $<100$"); the constant is unspecified here, $103$ in the 1962 paper and in Erdős's 1975 restatement, and $100$ in his 1977 restatement, where Sullivan's improvement to $4$ and his conjecture of a maximum "only a little greater than $2$" are reported.