0,2

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Paul D. Hanna, Table of n, a(n) for n = 0..200

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 0 = Sum_{n=-oo..+oo} x^(2*n) * (x^n - 2*A(x))^(3*n+1).

(2) 0 = Sum_{n=-oo..+oo} x^(3*n*(n-1)) / (1 - 2*A(x)*x^n)^(3*n-1).

a(n) ~ c * d^n / n^(3/2), where d = 13.043520100475... and c = 0.432996977380... - Vaclav Kotesovec, Dec 08 2022

G.f.: A(x) = 1 + 2*x + 18*x^2 + 124*x^3 + 1244*x^4 + 11652*x^5 + 122153*x^6 + 1281722*x^7 + 14009973*x^8 + 154993908*x^9 + 1748602308*x^10 + ...

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);

A[#A] = polcoeff( sum(n=-#A, #A, x^(2*n) * (x^n - 2*Ser(A))^(3*n+1) ), #A-1)/2); A[n+1]}

for(n=0, 20, print1(a(n), ", "))

Cf. A355865, A358953, A358954, A358955, A358956, A358957, A358958, A358959.

nonn,new

Paul D. Hanna, Dec 07 2022

approved

0,5

Table of n, a(n) for n=0..23.

E.g.f.: (-Li_2(2*x-x^2)-2*Li_2(-x))/(2*x-2).

(Maxima)

a(n):=n!*sum(1/(m*(binomial(n-1, 2*m-1))*n), m, 1, floor(n/2));

nonn,new

Vladimir Kruchinin, Dec 07 2022

approved

1,1

Table of n, a(n) for n=1..88.

Dimitri Tishchenko, Hexagon Prime Spiral

The spiral begins

13--12--11

/ \

14 4---3 10

/ / \ \

15 5 1---2 9

\ \ /

16 6---7---8

\

17--18--19--...

For n=4, prime(4) = 7 in the spiral has a(3) = 2 primes adjacent (2 and 19).

Cf. A307011, A307013 (spiral coordinates), A056105 (spiral first spoke).

nonn,easy,new

Wade Reece Eberly, Sep 23 2022

approved

5,1

Conjecture: the sequence is bounded.

Table of n, a(n) for n=5..95.

If a(n) > 1 then a(pi(k)) = a(n) - 1 where k = 100*floor(p/10) + p mod 10 and p = prime(n) (i.e., k is the result when a single 0 is inserted to the left of the ones digit of p).

For n = 8, prime(8) is 19. 109, 1009, and 10009 are all primes, while 100009 is not, thus a(8) = 4.

For n = 30, prime(30) is 113. 1103 and 11003 are prime, while 110003 is not, thus a(30) = 3.

a[n_] := Module[{p = Prime[n], c = 1, q, r}, r = Mod[p, 10]; q = 10*(p - r); While[PrimeQ[q + r], q *= 10; c++]; c]; Array[a, 100, 5] (* Amiram Eldar, Nov 27 2022 *)

(Python)

from sympy import isprime, prime

def a(n):

s, c = str(prime(n)), 1

while isprime(int(s[:-1] + '0'*c + s[-1])): c += 1

return c

print([a(n) for n in range(5, 92)]) # Michael S. Branicky, Nov 09 2022

Cf. A000040, A344637.

nonn,base,new

Rida Hamadani, Nov 09 2022

approved

1,2

Like n!^(n-1!)^...^2!^1!, but with all power operators nested from the left.

This sequence grows roughly doubly-exponentially. - Charles R Greathouse IV, Dec 07 2022

Table of n, a(n) for n=1..4.

a(n) = n!^A000178(n-1). - Charles R Greathouse IV, Dec 07 2022

log log a(n) ~ 0.5*n^2*log^2 n. - Charles R Greathouse IV, Dec 07 2022

a(4) = 4!^3!^2!^1! = 24^6^2 = 191102976^2 = 36520347436056576

Cf. A067039.

nonn,new

Arsen Vardanyan, Dec 07 2022

approved

1,2

Table of n, a(n) for n=1..62.

Petr Kucheriaviy, On numbers not representable as n + ω(n), arXiv preprint (2022). arXiv:2203.12006 [math.NT]

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density.

(PARI) list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+omega(n)); if(t<=lim, listput(v, t))); Set(v)

Cf. A337455.

nonn,new

Charles R Greathouse IV, Dec 07 2022

approved

1,2

This sequence lists the distinct values in A358893, in ascending order.

For any n > 0, there are A191755(n)/2 terms with binary length 2*n.

All terms are evil (A001969) and have an even number of binary digits (A053754).

Table of n, a(n) for n=1..56.

Rémy Sigrist, PARI program

Index entries for sequences related to binary expansion of n

The binary expansion of 204 is "11001100" and can be obtained by self-shuffling the binary expansion of 10 ("1010") or 12 ("1100"), so 204 is a term.

(PARI) See Links section.

Cf. A001969, A053754, A191755, A358893.

nonn,base,new

Rémy Sigrist, Dec 05 2022

approved

0,2

See A358892 for the distinct values.

n and T(n, k) have the same parity.

Rémy Sigrist, Table of n, a(n) for n = 0..4768 (rows for n = 0..127 flattened)

Rémy Sigrist, PARI program

Index entries for sequences related to binary expansion of n

T(n, 1) = A330940(n).

T(n, A193020(n)) = A330941(n).

Triangle T begins (in decimal):

n n-th row

-- --------

0 0,

1 3,

2 10, 12,

3 15,

4 36, 40, 48,

5 43, 45, 51, 53,

6 54, 58, 60,

7 63,

8 136, 144, 160, 192,

9 147, 149, 153, 163, 165, 169, 195, 197, 201,

...

Triangle T begins (in binary):

n n-th row

---- --------

0 0,

1 11,

10 1010, 1100,

11 1111,

100 100100, 101000, 110000,

101 101011, 101101, 110011, 110101,

110 110110, 111010, 111100,

111 111111,

1000 10001000, 10010000, 10100000, 11000000,

...

(PARI) See Links section.

Cf. A193020 (row lengths), A330940, A330941, A358892.

nonn,base,tabf,new

Rémy Sigrist, Dec 05 2022

approved

1,1

Equivalently, numbers m such that A020946(m) > A020946(m*k) for some k > 1.

For the first 38 terms, k = 2.

Table of n, a(n) for n=1..33.

Rémy Sigrist, C program

Rémy Sigrist, C++ program

Index entries for sequences related to Stern's sequences

A020946(54) = 1275, but A020946(2*54) = 1173 is smaller, so 54 belongs to this sequence.

(C) See Links section.

(C++) See Links section.

Cf. A002487, A020946.

nonn,new

Rémy Sigrist, Dec 06 2022

approved

1,3

Every positive integer appears infinitely many times in A002487, hence the sequence is well defined.

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Rémy Sigrist, PARI program

Index entries for sequences related to Stern's sequences

a(2^k) = 2^(k-1) for any k > 0.

a(n) = 2 iff n belongs to A097581 \ {2}.

The first terms, alongside fusc(n) and the direction where to find the same value, are:

n a(n) fusc(n) dir

-- ---- ------- ---

1 1 1 +

2 1 1 -

3 3 2 +

4 2 1 -

5 2 3 +

6 3 2 -

7 2 3 -

8 4 1 -

9 6 4 +

10 3 3 -

11 2 5 +

12 6 2 -

(PARI) See Links section.

Cf. A002487, A097581.

nonn,new

Rémy Sigrist, Dec 07 2022

approved