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M(k) = (k+1)*(1 - zeta(2)/2) - 1 - ( Sum_{j=1..k} k mod j )/k is a measure of either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer k. The measure follows from the known facts that Sum_{j=1..k} (sigma(j) + k mod j) = k^2 and that the average order of sigma(k)/k is Pi^2/6 = zeta(2) (see derivation below).

M(k) ~ 0 when sigma(k) ~ 2*k and for sufficiently large k, M(k) is positive when k is an abundant number (A005101) and negative when k is a deficient number (A005100).

The terms of this sequence are the deficient k for which M(k) < M(m) for all m < k and may be thought of as "superdeficient", contra-analogous to the superabundant numbers A004394 utilizing sigma(k)/k as the measure of abundance, which is otherwise not particularly meaningful as a deficiency measure.

15119=13*1163 is the first term that is composite and subsequently, up to 1000000000, roughly half of the terms are composite.

The residue-based quantifier function, M(k), measures either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer k. The measure is defined by M(k) = (k+1)*(1 - zeta(2)/2) - 1 - (Sum_{j=1..k} k mod j)/k. It follows from the known facts that Sum_{j=1..k} (sigma(j) + k mod j) = k^2 and that the average order of sigma(k)/k is Pi^2/6 = zeta(2) (see derivation below).

M(k) ~ 0 when sigma(k) ~ 2*k and for sufficiently large k, M(k) is positive when k is an abundant number (A005101) and negative when k is a deficient number (A005100). The terms of this sequence are the deficient k such that k+1 is abundant and abs(M(k)) + abs(M(k+1)) achieves a new maximum, somewhat analogous to A335067 and A326393.

Compactification of A135507 akin to A000705 with respect to A002201.

Proper subset of the intersection A025487 and A379336.

There are three kinds of pairs (d, k/d), d | k, such that gcd(d, k/d) does not equal 1, d, or k/d:

Type A: rad(d) does not divide k/d, and rad(k/d) does not divide d (see A379752), where rad = A007947.

Type B: the squarefree kernel of one divisor divides the other but the reverse is not true (see A379772).

Type C: rad(d) = rad(k/d), i.e., d, k/d, and k are coreful (see A379552).

Conjecture: Numbers k that set records in A376281 do not have type C divisor pairs, i.e., those that are coreful but neither divides the other. This, since type C requires k to be powerful and divisible by cubes of 2 distinct primes (i.e., in A376936). Therefore the record is achieved only through large numbers of type A and B.

Since type A divisor pairs are common for composite k in A375055, this sequence is resembles A379752.

Since d and k/d are both composite, this sequence resembles A059992.

This sequence, to a lesser extent A379752, and a greater extent A059992, contains many highly composite numbers. (See plot of S(n) = union of this sequence and A002182 below, and corresponding graphs in respective other sequences.)

Proper subset of the intersection of A025487 and A375055.

Conjecture: subset of A332785 = A126706 \ A286708.

This sequence seems to be rich in highly composite numbers, the prime shape of a(n) resembles that of highly composite numbers, with long tails of large prime factors with multiplicity 1.

Terms not in A002182 are not all of the form 2^5 * prime(i..j), 1 < i < j, for example, a(24) = 443520 = 2^7 * 3^2 * 5 * 7 * 11.

The numbers contain the starred entries on pp. 187-190 of Nicolas. It is a subsequence of A002809 by selecting only elements of a set/property "G" (page 150). G contains all N such that a real, strictly positive rho exists such that for all strictly positive integers A we have l(A)-l(N) >= rho*log(A/N). The function l()=A008475() is defined on page 139. - R. J. Mathar, Mar 23 2012

These numbers were named superior l-composite numbers (nombres l-composes superieurs, the function l(n) is A002809) by Massias, in analogy to Ramanujan's superior highly composite numbers (A002201). Deléglise and Nicolas named these numbers l-superchampion numbers. They are used by Deléglise et al. in calculating values of Landau's function g(n) (A000793). - Amiram Eldar, Aug 23 2019

Sequence A002201 gives the values of the n-th superior highly composite number N(n) and A000705 gives the values of the (prime) ratio N(n)/N(n-1).

(1+1/a(n)) appears in the denominators of the log arguments of the denominators of the numbers in the table of the reference, pp. 115-117.

The SHC numbers are a subset of the HC numbers. Is there a formula for a(n) that depends on the two consecutive SHC numbers A002201(n) and A002201(n+1)?

Number of superior highly composite numbers (A002201) with at most n digits.