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Suggested by the fact that the converse to Lagrange's theorem does not hold. These numbers might be called "Non-Converse Lagrange Theorem Orders".

A subsequence of A066085. The first difference between them is that 224 is missing from the present sequence (see MacHale-Manning, 2016). The sequence of terms of A066085 not in the present sequence is infinite, and begins 224, 2464, ... [This sequence is now A341048. - Bernard Schott, Feb 15 2021]

From Jianing Song, Dec 06 2021: (Start)

If k is a term, gcd(k,m) = 1, then k*m is again a term. Proof: If G is a group of order k without a subgroup of order k', then G X C_m has no subgroup of order k'*m. Suppose that it has, let G' be that subgroup. For every (a,b) in G', let m_0 be a multiple of m congruent to 1 modulo k, then (a,b)^(m_0) = (a,1) in G'; let k_0 be a multiple of k congruent to 1 modulo m, then (a,b)^(k_0) = (1,b) in G'. This shows that G' itself is of the form H X C_{m'}, where H is a subgroup of G and m' divides m. We have |H|*m' = k'*m, so |H| = k' and m' = m, contradicting with our assumption that G has no subgroup of order k'.

On the other hand, if gcd(k,m) > 1, then k*m need not be a term, as 56 is here but 224 is missing. In fact, N has a proper divisor here but N itself is not in this sequence if and only if N is in A341048. For the "only if" part, if N = k*m is a CLT order and k is a NCLT order, then k is a NSS order. Since every multiple of a NSS order is a NSS order, N is a NSS order, so by definition N is in A341048. The "if" part follows from MacHale-Manning, 2016, Corollary 13, Page 5.

Conjecture: If k = p^a*q^b, where p, q are primes, q !== 1 (mod p), b >= ord(q,p), then k is a term of this sequence, unless k is an NSS-CLT order of the form described in MacHale-Manning, 2016, Theorem 8, Page 5. Here ord(q,p) is the multiplicative order of q modulo p. Moreover, if k satisfies this condition, it seems that for each NCLT group of order k, the missing orders of subgroups are of the form p^a'*q^b' where either a' = a or b' = b, and a' = a if p == 1 (mod q) or a < ord(p,q). (End)

Primitive terms of A066085.

A closely related sequence would be the primitive terms in A340511, but for that we need more terms of A340511.

A finite group is supersolvable if it has a normal series with cyclic factors. Huppert showed that a finite group is supersolvable iff the index of any maximal subgroup is prime.

The converse to Lagrange's theorem does not hold. A340511 lists the numbers n such that there exists a group of order n which has no subgroup of order d, for some divisor d of n; they are called "non-converse Lagrange theorem" (NCLT) orders.

A finite group is supersolvable (SS) if it has a normal series of subgroups with cyclic factors; A066085 lists the numbers for which there exists a group of order n that is not supersolvable; they are called a "non-supersolvable" (NSS) order.

Theorem: Every NCLT order is an NSS order (see MacHale-Manning, 2016, Theorem 3, page 2); hence A340511 is a subsequence of A066085.

However, there exist infinitely many NSS orders that are not NCLT orders (see MacHale-Manning, 2016, Corollary 19, page 6) and these NSS-CLT orders are listed in this sequence.

Theorem: The number 224*p is an NSS-CLT order for all primes p <> 2, 3, 5, 7, 31 (see MacHale-Manning, 2016, Theorem 18, page 6). So, 10528, 11872, 13216, 13664, 15008, 15904, ... are other terms.

Number of supersolvable groups of order <= n. Diverges from A063756 after n=11. The subsequence of primes in this sequence begins: 2, 3, 5, 19, 23, 41, 47, 53, 79, 83, 89, 181, 191, 233, 241, 281, 293, 571, 577. The subsequence of perfect powers in this sequence begins: 1, 8, 9, 16, 27, 144, 625.