cp's OEIS Frontend

With V=A181391, a(n) is the smallest number m such that V(m) = V(m-1) = n.

Since van Eck's sequence is generated by considering the gap between identical terms reappearing, it is of interest to consider terms of value n which repeat with a gap of length n.

When this happens the term is repeated in succession.

Some observations that follow from the definition of V:

V(a(n)-1-n) = n. The value n has to appear exactly n terms apart in V to make the following term equal to n, e.g., for n=3: "..., 3, 8, 0, 3, 3, ...".

V(a(n)+1) = 1. Since the term n appeared twice in a row, the following term of V must be 1.

V(a(n)-2) = V(a(n)-n-2) = V(a(n)-2*n-2). The number preceding the repeated terms appears three times with gaps of n.

V(a(n)+2) = the number of terms since the previous repeated value of some number (though it may not be the first time it is repeated). So V(a(n)-V(a(n)+2)) = V(a(n)-V(a(n)+2)-1).

Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).

The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.

Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:

-- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.

-- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.

-- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.

Subsequence of A083359.

Overlap of the distinct primes is allowed per A324257. Terms without overlap form the subsequence A121342.

Includes all squarefree terms in A324257 by definition.

Generalization of A083361 without the restriction of limited multiplicity inherited from A083359.

Generalization of A083360 with multiplicity of the distinct prime factors (not limited by the number of times a prime factor appears in the factorization of the number).

There can be a large number of terms in A324261 with 2n digits. For example, there are 227 terms with length 66. Selecting only the smallest term for each length allows terms to be listed for larger values of n.

The terms of A324261 with length 2n are formed by concatenating two copies of prime p, where p has length n and the decimal representation of p contains all the prime factors of 10^n + 1 as described in A324261 and A083359.

Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).

"Conceited Numbers" (they are full of themselves).

The decimal representation of these numbers can be formed typographically from their prime factors. Every distinct prime factor must appear at least once. Generalization of the sequence A083359.

Subsequences:

--No overlap: A324258

--Every prime factor appears in number (not just distinct prime factors): A324259

--No multiplicity: A324260

--Multiplicity only up to the exponent of the distinct prime factor: A083359

Other subsequences are formed by more than one constraint; e.g., A121342 is the intersection of A324258 and A324260, terms with no overlap and no multiplicity.