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a(n) is the number of necklaces with 4 black beads and n white beads.

Also nonnegative integer 2 X 2 matrices with sum of elements equal to n, up to rotational symmetry.

The g.f. is Z(C_4,x), the 4-variate cycle index polynomial for the cyclic group C_4, with substitution x[i]->1/(1-x^i), i=1,...,4. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 4-necklaces whose 4 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_4,x). - Wolfdieter Lang, Feb 15 2005

a(n) is the number of necklaces with 5 black beads and n white beads.

The g.f. is Z(C_5,x), the 5-variate cycle index polynomial for the cyclic group C_5, with substitution x[i]->1/(1-x^i), i=1,...,5. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 5-necklaces whose 5 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_5,x). - Wolfdieter Lang, Feb 15 2005

Since Sum_{d|k} phi(d) = k, these are numbers k such that the set {phi(d) | d|k} is a partition of k into distinct parts.

Includes all the odd prime numbers, since an odd prime p has 2 divisors, 1 and p, whose phi values are 1 and p-1.

If k is a term, then all the divisors of k are also terms. If k is not a term, then all its multiples are not terms. The primitive terms of the complementary sequence are 2, 63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, ....

In particular, all the terms are odd since 2 is not a term (phi(1) = phi(2)).

The number of terms below 10^k for k = 1, 2, ... are 5, 49, 488, 4860, 48598, 485807, 4857394, 48572251, 485716764, 4857144075, ...

Apparently the sequence has an asymptotic density of 0.4857...

n is phi-practical if and only if each natural number up to n is a subsum of the multiset {phi(d) : d | n} (see Pomerance link p. 2).

There are 6 terms up to 10, 28 up to 10^2, 174 up to 10^3, 1198 up to 10^4, 9301 up to 10^5, 74461 up to 10^6, 635528 up to 10^7, 5525973 up to 10^8, and 48386047 up to 10^9. - Charles R Greathouse IV, Nov 13 2015

There are 431320394 terms up to 10^10. - Charles R Greathouse IV, Sep 19 2017

Let F(x) denote the number of phi-practical numbers up to x. F(x) has order of magnitude x/log(x) (See Thompson 2012). Moreover, we have F(x) = c*x/log(x) + O(x/(log(x))^2), where 0.945 < c < 0.967 (See Pomerance, Thompson & Weingartner 2016 and Weingartner 2019). As a result, a(n) = k*n*log(n*log(n)) + O(n), where k = 1/c and 1.034 < k < 1.059. - Andreas Weingartner, Jun 29 2021

The g.f. is Z(C_6,x)/x^6, the 6-variate cycle index polynomial for the cyclic group C_6, with substitution x[i]->1/(1-x^i), i=1,...,6. Therefore by Polya enumeration a(n+6) is the number of cyclically inequivalent 6-necklaces whose 6 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_6,x). Note the equivalence of this formulation with the one given as this sequence's name: start with a black 6-necklace (all 6 beads have labels 0). Insert after each of the 6 black beads k white ones if the label was k and then disregard the labels. - Wolfdieter Lang, Feb 15 2005

The g.f. of the CIK[k] transform of the sequence (b(n): n>=1), which has g.f. B(x) = Sum_{n>=1} b(n)*x^n, is CIK[k](x) = (1/k)*Sum_{d|k} phi(d)*B(x^d)^{k/d}. Here, k = 6, b(n) = 1 for all n >= 1, and B(x) = x/(1-x), from which we get another proof of the g.f.s given below. - Petros Hadjicostas, Jan 07 2018

"CIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1, ...

The g.f. is Z(C_7,x)/x^7, the 7-variate cycle index polynomial for the cyclic group C_7, with substitution x[i]->1/(1-x^i), i=1,...,7. Therefore by Polya enumeration a(n+7) is the number of cyclically inequivalent 7-necklaces whose 7 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_7,x) and the comment in A032191 on the equivalence of this problem with the one given in the 'Name' line. - Wolfdieter Lang, Feb 15 2005

From Petros Hadjicostas, Dec 08 2017: (Start)

For p prime, if a_p(n) is the number of necklaces with p black beads and n-p white beads, then (a_p(n): n>=1) = CIK[p](1, 1, 1, 1, ...). Since CIK[k](B(x)) = (1/k)*Sum_{d|k} phi(d)*B(x^d)^{k/d} with k = p and B(x) = x + x^2 + x^3 + ... = x/(1-x), we get Sum_{n>=1} a_p(n)*x^n = ((p-1)/(1 - x^p) + 1/(1 - x)^p)*x^p/p, which is Herbert Kociemba's general formula for the g.f. when p is prime.

We immediately get a_p(n) = ((p-1)/p)*I(p|n) + (1/p)*C(n-1,p-1) = ((p-1)/p)*I(p|n) + (1/n)*C(n,p) = ceiling(C(n,p)/n), which is a generalization of the conjecture made by N. J. A. Sloane and Wolfdieter Lang. (Here, I(condition) = 1 if the condition holds, and 0 otherwise. Also, as usual, for integers n and k, C(n,k) = 0 if 0 <= n < k.)

Since the sequence (a_p(n): n>=1) is column k = p of A047996(n,k) = T(n,k), we get from the documentation of the latter sequence that a_p(n) = T(n, p) = (1/n)*Sum_{d|gcd(n,p)} phi(d)*C(n/d, p/d), from which we get another proof of the formulae for a_p(n).

(End)

The sum of the distinct values of the n-th row of A102190.

Apparently, all the terms are odd.

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).

The row lengths sequence is A000041. The number of nonzero entries in row nr. n is 1 for n=1, 2 for n=2 and A000005(n)+1 otherwise. This is the sequence A212356.

The cycle index (multivariate polynomial) for the dihedral group D_n (of order 2*n), called Z(D_n), is for odd n given by (Z(C_n) + x[1]*x[2]^((n-1)/2))/2 and for even n by (2*Z(C_n) + x[2] ^(n/2) + x[1]^2*x[2]^((n-2)/2))/4, where Z(C_n) is the cycle index for the cyclic group C_n. For the coefficients of Z(C_n) see A054523 or A102190. See the Harary and Palmer reference.