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After 2, terms of A003961(A019565(A339816(i))) [or equally, of A019565(2*A339816(i))], for i = 1.., sorted into ascending order.

Natural numbers n that satisfy equation y * phi(n) = n - 1 (with y an integer) all occur in this sequence. Lehmer conjectured that there are no solutions such that n is composite (and thus y > 1).

From Antti Karttunen, Dec 22-26 2020: (Start)

Composite terms in this sequence are all of the form 4u+1 (A016813, A091113).

Generally, if any term k > 2 here has x prime divisors (which are all odd and distinct, i.e., A001221(k) = A001222(k) = x), then k is of the form 2^x * u + 1 (where u maybe even or odd), because each prime divisor of k contributes at least one instance of 2 in phi(k). Specifically, each prime factor of the form 4u+3 (A002145) contributes one instance of 2 (+1 to the 2-adic valuation), while primes of the form 4u+1 (A002144) contribute at least +2 to the 2-adic valuation. There must be an even number of 4u+3 primes, as otherwise the product would be of the form 4u+3. On the other hand, although all the terms of A016105 occur here, none of them occurs in A339870.

If the only terms this sequence shares with A339879 are the primes (A000040), then Lehmer's conjecture certainly holds. Similarly if the sequences A339818 and A339869 do not have any common terms.

(End)

Lehmer conjectured that the equation k * phi(n) = n - 1 (with k integer) has no solutions for any composite n (i.e., when k > 1). If this sequence has no common terms with A339818, then the conjecture certainly holds.

From Antti Karttunen, Dec 26 2020: (Start)

Equally, squarefree composite numbers k of the form 4u+1 for which A336466(k) divides k-1. This follows because on squarefree n, A336466(n) = A053575(n).

No common terms with A016105, because 4xy + 2(x+y) + 1 does not divide 4xy + 3(x+y) + 2 for any distinct x, y >= 0 (where 4x+3 and 4y+3 are the two prime factors of Blum integers).

This can also seen by another way: If this sequence contained any Blum integers, then, because A016105 is a subsequence of A339817, we would have found a composite number n satisfying Lehmer's totient problem y * phi(n) = n-1, for some integer y > 1. But Lehmer proved that such solutions should have at least 7 distinct prime factors, while Blum integers have only two.

Moreover, it seems that none of the terms of A167181 may occur here, and a few of A137409 (i.e., of A125667). See A339875 for those terms.

(End)

Numbers k such that A339971(k) divides A339809(2k).

Union of {0}, A000079 and the terms of (1/2)*A048675(A339880(i)), for i >= 1, sorted into ascending order.

Sequence A003961(A340076(i)), i = 1.., sorted into ascending order. In other words, this sequence consists of such odd numbers k that A064989(k) is in A340076.

Odd primes that do not occur as the greatest prime divisor (A006530) of any of the terms of A339880.

Naive way of computing (essentially an exhaustive search): apply A000523 to the terms of A339973, select unique values, add +2, and take the corresponding prime.

Questions: Is this sequence finite? If infinite, are there still only a finite number of 4k+1 primes (A002144) like 37 and 61?

a(13) >= 149, if it exists.

Sequence A003961(A340151(i)), for i >= 1, sorted into ascending order.

By definition, this has no common terms with A340077 nor any of its subsequences like A339869 or A339880.