A370719 Expansion of l.g.f. L(x) = log(G(x)) - x*G(x) where G(x) is the o.g.f. of A370718.

0, 1, 1, 3, 1, 3, 1, 3, 7, 8, 1, 7, 1, 10, 7, 11, 1, 6, 1, 4, 0, 14, 1, 19, 21, 16, 7, 12, 1, 23, 1, 27, 29, 3, 25, 4, 1, 3, 7, 32, 1, 25, 1, 30, 13, 26, 1, 3, 36, 28, 41, 32, 1, 6, 24, 32, 26, 32, 1, 31, 1, 3, 12, 59, 20, 2, 1, 19, 53, 7, 1, 40, 1, 3, 37, 19, 33, 6, 1, 16, 34

Offset

1,4

Comments

It appears that a(p) = 1 for prime p.

Where does the number 5 first appear in this sequence?

Links

Paul D. Hanna, Table of n, a(n) for n = 1..5001

Formula

L.g.f. L(x) = Sum_{n>=1} a(n)*x^n/n satisfies the following formulas, in which G(x) is the o.g.f. of A370718.

(1) L(x) = log(G(x)) - x*G(x).

(2) a(n) = [x^(n-1)] G'(x)/G(x) - n*A370718(n-1) such that 0 <= a(n) < n for n >= 1.

Example

L.g.f.: L(x) = x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 3*x^6/6 + x^7/7 + 3*x^8/8 + 7*x^9/9 + 8*x^10/10 + x^11/11 + 7*x^12/12 + ...
such that L(x) = log(G(x)) - x*G(x), where
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 47*x^6 + 113*x^7 + 279*x^8 + 702*x^9 + 1793*x^10 + ... + A370718(n)*x^n + ...
and
log(G(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 123*x^6/6 + 330*x^7/7 + 907*x^8/8 + 2518*x^9/9 + 7028*x^10/10 + ...
FREQUENCY OF INTEGERS.
a(p) = 1 for prime p holds for at least the initial 5000 terms;
a(n) = 1 when n is nonprime at positions [121, 592, 675, 961, 1548, ...].
These integers do not appear in the initial 5000 terms: [5, 54, 99, 111, 127, 171, 174, 175, 177, 178, 181, 196, 199, 213, 216, 217, 228, 230, 235, 254, 260, 274, 276, 277, 289, 298, ...].
Here is the frequency of some integers in the initial 5000 terms.
0: 4 times, at positions [1, 21, 1106, 1335, ...].
1: 674 times, at positions [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...].
2: 6 times, at positions [66, 153, 333, 1448, 1903, 4239, ...].
3: 198 times, at positions [4, 6, 8, 34, 38, 48, 62, 74, 82, ...].
4: 3 times, at positions [20, 36, 187, ...].
5: 0 times - where does 5 appear in this sequence?
6: 4 times, at positions [18, 54, 78, 312, ...].
7: 77 times, at positions [9, 12, 15, 27, 39, 70, 87, 159, ...].
8: 5 times, at positions [10, 130, 232, 620, 2419, ...].
9: 2 times, at positions [500, 3834, ...].
10: 6 times, at positions [14, 184, 1245, 1352, 1552, 4810, ...].
11: 4 times, at positions [16, 165, 1314, 1491, ...].
12: 5 times, at positions [28, 63, 371, 375, 2964, ...].
13: 3 times, at positions [45, 455, 1599, ...].
14: 2 times, at positions [22, 376, ...].
15: 3 times, at positions [1196, 2225, 4009, ...].
16: 2 times, at positions [26, 80, ...].
19: 60 times, at positions [24, 68, 76, 124, 164, 172, 180, 212, ...].
46: 37 times, at positions [185, 215, 265, 295, 335, 365, 415, ...].
...

Prog

(PARI) {a(n) = my(L=[0], A=[1]); for(i=1, n, L = concat(L, 0);
A = concat(A, 0); F = exp( x*Ser(A) + sum(n=1, #L, L[n]*x^n/n) );
L[#L] = (#L - (#L)*polcoeff(F, #L))%(#L);
A = Vec(F +O(x^#A)); ); L[n]}
for(n=1, 80, print1(a(n), ", "))

Crossrefs

Cf. A370718.

Sequence in context: A023511 A035628 A187562 * A057024 A023892 A085417

Adjacent sequences: A370716 A370717 A370718 * A370720 A370722 A370724

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2024

Status

approved