cp's OEIS Frontend

For all nonzero terms a(n), A263083(n) = a(n) + A262509(n) and A155043(A263083(n)) < A155043(A262509(n)) because at each A262509(n) the "distance to zero", A155043 obtains a unique value A262508(n), thus no A049820-iteration trajectory starting from any k larger than A262509(n) and using a greater or equal number of steps to reach zero may bypass A262509(n) [i.e., without going through A262509(n)], because then A262508(n) would not be unique anymore. See also comments in A262909.

Primes p that there is at least one such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Let conditions 0 and 1 respectively pertain to novel a(n) and extant a(n) respectively. For n>3 condition 0 results in a fall from a(n) to a(n+1), whereas condition 1 results in an increase, which may be sustained over several consecutive terms as the least unused term is applied to the result of a sum which is the same as a prior term, therefore prompting a further application of condition 1, until a novel term is reached.

A distinguishing feature of the scatterplot of this sequence is that the data are highly concentrated around several quasi-radial, apparently straight line "spokes" emanating from the origin, which appear to be equally spaced. The lower spoke relates to applications of condition 0, producing the values d(k) of novel terms k, which being small by comparison to k, are distributed widely across the lower zone of the plot. Higher spokes relate to consecutive terms generated by condition 1, namely repeated addition of the least unused term m.

Since m appears in multiples and increments by small amounts, successive additions result in roughly equally spaced consecutive terms. Repeated application of condition 1 continues until a novel term k is reached, followed by a fall to d(k), from which the process repeats, carrying forward any remaining copies of m until all are used, at which point m increments, causing the spokes to diverge gradually as the sequence advances.

Records beyond a(1) arise consequent only to repetitions of condition 1, and not to the appearance of a novel term. With expansion of the data the spokes are gradually obscured from the bottom up by horizontal striations resulting from the fact that many different numbers have the same number of divisors, and that with greater values of a(n), greater values of d(a(n)) can appear. As the lower spokes are obscured, so new spokes appear above, corresponding to new records.

The greatest fall from a(n) to a(n+1) is when a(n) is a prime record, then a(n+1)=2. This sometimes creates a remarkable disturbance in the sequence, involving many chaotic reversals before recovery to the more stable pattern described above is achieved. The first conspicuous example of this is at a(243)=89. A repeat of condition 0 can only be consequent to a novel highly divisible term k for which d(k) is also novel. For n<=2^19 the only occurrence of this is at a(422)=144, followed by d(144)=15. For more details see links.

Except for the first three terms, all others are multiples of 3.

Square roots of perfect squares in A055927. [Juri-Stepan Gerasimov, Apr 06 2011]

Which of the terms of A002183 occur here, and for how many times each?

Is a(n + 1) / a(n) ~ 1 for large n?

Every term in this sequence also appears in A002183, where every element of this sequence occurs exactly once.

In A067128 it is asked if A034287 = A067128. If that is the case then this sequence is also the number of divisors of A034287.

Records in A319442.

Analog of A002183 and A302249, which list the records of number of divisors in rational integers and Gaussian integers respectively.

It seems that 15 is the largest odd term.

First differs from A002183 at n = 20.

The first position n where a(n+1) = a(n) is n = 49: a(49) = a(50) = 1440.

The first position n where a(n+1) < a(n) is n = 173: a(173) = 5308416 and a(174) = 5160960.