0,2

Table of n, a(n) for n=0..90.

From Alois P. Heinz, Mar 20 2023: (Start)

a(n) = pi(2^n+n) - pi(2^n-1), pi = A000720.

a(n) = = 1, 1 <= k <= n.">A143537(2^n+n,2^n-1). (End)

In the interval [2^1, 2^1 + 1] there are 2 primes (2 and 3). So a(1) = 2.

a:= n-> nops(select(isprime, [$2^n..2^n+n])):

seq(a(n), n=0..100); # Alois P. Heinz, Mar 20 2023

Array[PrimePi[2^# + #] - PrimePi[2^# - 1] &, 50, 0] (* Michael De Vlieger, Mar 27 2023 *)

(PARI) a(n)=#primes([2^n, 2^n+n])

Cf. A000720, A036378, = 1, 1 <= k <= n.">A143537.

nonn,new

Jean-Marc Rebert, Mar 20 2023

approved

0,3

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds for all y as a formal power series in x.

Paul D. Hanna, Table of n, a(n) for n = 0..300

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

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

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

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 57*x^6 + 123*x^7 + 280*x^8 + 666*x^9 + 1614*x^10 + 3955*x^11 + 9733*x^12 + ...

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

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

for(n=0, 35, print1(a(n), ", "))

Cf. A358952, A355866.

nonn,new

Paul D. Hanna, Mar 26 2023

approved

1,6

Although each path is self-avoiding, the different paths are allowed to intersect.

Table of n, a(n) for n=1..25.

Ivaylo Kortezov, Sets of Paths between Vertices of a Polygon, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1).

E.g.f.: x^3*exp(x)*(exp(2*x) - 1)^3/384. - Andrew Howroyd, Mar 07 2023

a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.

(PARI) a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1)} \\ Andrew Howroyd, Mar 07 2023

If there is only one path, we get A261064. If there is are two paths, we get A360716. If all n points need to be used, we get A359404.

nonn,easy,new

Ivaylo Kortezov, Mar 07 2023

approved

1,1

Table of n, a(n) for n=1..58.

Eric Weisstein's World of Mathematics, Prime Triplet

Wikipedia, Prime triplet

a(n) = A000720(A007529(n)). - Alois P. Heinz, Mar 06 2023

q:= n-> is(ithprime(n+2)-ithprime(n)=6):

select(q, [$1..400])[]; # Alois P. Heinz, Mar 06 2023

Select[Range[400], Prime[# + 2] - Prime[#] == 6 &] (* Michael De Vlieger, Mar 06 2023 *)

(Clojure)

(defn next-prime [n]

(if (= n 2)

3

(let [m (+ n 2)

t (-> n Math/sqrt int (+ 2))]

(if (some #(zero? (mod m %)) (range 2 t))

(next-prime m)

m))))

(def primes (lazy-seq (iterate next-prime 2)))

(defn triplet-primes-positions [n]

(->> primes

(take n)

(partition 3 1)

(map list (range))

(filter (fn [[i xs]] (= 6 (- (last xs) (first xs)))))

(map #(-> % first inc))))

(println (triplet-primes-positions 2000))

Cf. A000040, A000720, A007529, A022004, A022005.

nonn,new

Atabey Kaygun, Mar 06 2023

approved

2,1

Widely used as a cutoff line of web maps which use the web Mercator projection.

Table of n, a(n) for n=2..128.

Equals arctan(sinh(Pi))*180/Pi.

Equals 360/Pi*arctan(exp(Pi)) - 90.

85.05112877980659237779671552192469206698259126842068...

RealDigits[ArcTan[Sinh[Pi]]/Degree, 10, 100][[1]] (* Amiram Eldar, Mar 06 2023 *)

(PARI) atan(sinh(Pi))*180/Pi \\ Michel Marcus, Mar 06 2023

Cf. A334401.

nonn,cons,new

Donghwi Park, Mar 06 2023

approved

2,1

Main diagonal of A360184.

Table of n, a(n) for n=2..14.

(PARI)

strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;

strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k, e, r) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1 && strong_check(p, base, e, r), my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; strong_check(p, base, e, r) || next; my(z=znorder(Mod(base, p))); gcd(m, z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1, e, r)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); my(res=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 2, k, v, -1))); vecsort(Set(res));

a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);

Cf. A001262, A180065, A271874, A360184.

nonn,new

Daniel Suteu, Mar 06 2023

approved

3,1

The two-player impartial {0,1}-Toggle game is played on a simple connected graph G where each vertex is assigned an initial weight of 0 or 1.

A Toggle move consists of selecting a vertex v and switching its weight as well as the weights of each of its neighbors. This move is only legal provided the weight of vertex v is 1 and the total sum of the vertex weights decreases.

In the special case G=GP(n,2), a (1,0)-weight assignment is one in which each vertex of the outer polygon is assigned weight 1 and each vertex of the inner polygon(s) is assigned weight 0.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, CRC Press, 2001.

Table of n, a(n) for n=3..39.

E. Fiorini, M. Lind, A. Woldar, and T. W. H. Wong, Characterizing Winning Positions in the Impartial Two-Player Pebbling Game on Complete Graphs, Journal of Integer Sequences, 24(6), 2021.

Katherine Levandosky, CGSuite Program.

For n = 3, the {0,1}-Toggle game on GP(3,2) with a (1,0)-weight assignment is a next-player winning game.

For n = 5, the {0,1}-Toggle game on GP(5,2) with a (1,0)-weight assignment is a next-player winning game.

(CGSuite) # See Levandosky link

Cf. A071426, A340631, A346197, A346401, A346637.

nonn,more,new

Eugene Fiorini, Maxwell Fogler, Katherine Levandosky, Bryan Lu, Jacob Porter and Andrew Woldar, Mar 14 2023

approved

10,1

Similar to the concept of (but not limited to) Keith numbers, form a sequence {b(i)} whose initial terms are the t digits of n, later terms given by the rule that b(i) = sum of t previous terms, until b(i) >= n.

Originally, the concept of Keith numbers did not include n < 10. This sequence follows this rule; however, a(n) is mathematically possible for n < 10: a(n) = n.

Table of n, a(n) for n=10..65.

For n = 15, the sequence is {1, 5, 6, 11, 17} (the first two terms being each of the two digits of 15 and the sequence stops at 17 because this is the first number that is at least n). So, a(15) = 1 + 5 + 6 + 11 + 17 = 40.

(Ruby)

def a(n)

digits = n.to_s.chars.map(&:to_i)

countDigits = digits.size

until digits.last >= n do

sum = digits.last(countDigits).sum

digits.push(sum)

end

return digits.sum # Terms of this OEIS sequence

end # Diego V. G. Silva, Mar 24 2023

(PARI) a(n)={my(v=digits(n), s=vecsum(v)); while(v[#v] < n, v=concat(v[2..#v], vecsum(v)); s+=v[#v]); s} \\ Andrew Howroyd, Mar 23 2023

Cf. A007629.

nonn,base,new

Diego V. G. Silva, Mar 23 2023

approved

0,5

Here, a source-like component of a digraph D is a strongly connected component of D that corresponds to a node of in-degree 0 in the condensation of D.

Table of n, a(n) for n=0..44.

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.

R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Wikipedia, Strongly connected component

Triangle begins:

1;

0, 1;

0, 3, 1;

0, 51, 12, 1;

0, 3614, 447, 34, 1;

0, 991930, 53675, 2885, 85, 1;

...

nn = 6; B[n_] := n! 2^Binomial[n, 2]; strong =Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]];

ggfz[egfx_] := Normal[Series[egfx, {x, 0, nn}]] /.Table[x^i -> z^i/2^Binomial[i, 2], {i, 0, nn}]; Table[B[n], {n, 0, nn}] CoefficientList[Series[ggfz[Exp[(u - 1) s[x]]]/ggfz[Exp[- s[x]]], {z, 0, nn}], {z u}] // Grid

Cf. A003028 (column k=1), A053763 (row sums).

nonn,tabl,new

Geoffrey Critzer, Mar 16 2023

approved

1,1

There are 948729 possible patterns for centuries having 18 primes.

Brian Kehrig, Table of n, a(n) for n = 1..19

1228537713709 is in the sequence because there are 18 primes between 122853771370900 and 122853771370999: 122853771370900 + x, where x is one of (1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, or 99).

(PARI) isok(k) = sum(i=0, 99, isprime(100*k + i)) == 18; \\ Michel Marcus, Mar 23 2023

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).

Cf. A181098 (no primes), A186393-A186408 (1 to 16 primes), A186509 (17 primes).

nonn,new

Brian Kehrig, Mar 21 2023

approved