A262651 -id:A262651 - cp's OEIS frontend

A262652 Base-10 representation of the primes at A262651.

3, 1627, 2298073, 12055033499, 5135831051749, 132374942978911, 684810967798954343, 27800772498401810543, 31705538882266542665633, 10426504028032396567753387, 1495118023958546538597068807, 425143541608166955051684643069, 15918355788492735798148953834991

Offset: 1

Clark Kimberling, Oct 27 2015

			n   A262651(n)    base-10 representation
1    3                 3
2    11311             1627
3    121131121         2298073

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A262651.

s = {3}; base = 6; z = 20; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#, base]], base]] &];
AppendTo[s, FromDigits[IntegerDigits[tmp, base]]], {z}]; s  (* A262651 *)
Map[FromDigits[ToString[#], base] &, s]  (* A262652 *)
(* Peter J. C. Moses, Sep 01 2015 *)

A262627 Minimal nested base-2 palindromic primes with seed 0.

Original entry on oeis.org

0, 101, 11001010011, 101100101001101, 10101011001010011010101, 111010101100101001101010111, 1111101010110010100110101011111, 101111111010101100101001101010111111101, 110101111111010101100101001101010111111101011

Offset: 1

Clark Kimberling, Oct 02 2015

Using only base-2 digits 0 and 1, let s be a palindrome and put a(1) = s. Let a(2) be the least palindromic prime having s in the middle; for n > 2, let a(n) be the least palindromic prime have a(n-1) in the middle. Then (a(n)) is the sequence of minimal nested base-2 palindromic primes with seed s -- a(1) being not prime, of course.
Guide to related sequences
base  seed   base-b repr.    base-10 repr.
       0      A262627        A262628
       1      A262629        A262630
       1      A262631        A262632
       0      A262633        A262634
       1      A262635        A262636
       2      A262637        A262638
       3      A262639        A262640
       1      A262641        A262642
       3      A262643        A262644
       0      A262645        A262646
       1      A262647        A262648
       2      A262649        A262650
       3      A262651        A262652
       4      A262653        A262654
       5      A262655        A262656
       1      A262657        A262658
       0      A262659        A262660
       1      A262661        A262662
      0      A261881        A261881
      1      A261818        A261818

			a(3) = 11001010011 =A117697(15) is the least prime having a(2) = 101 in its middle. Triangular format:
               0
              101
          11001010011
        101100101001101
    10101011001010011010101
  111010101100101001101010111
1111101010110010100110101011111

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A117697, A261881 (base 10), A262628-A262662.

s = {0}; base = 2; z = 20; Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s]], Reverse[#]] &[IntegerDigits[#, base]], base]] &];
AppendTo[s, FromDigits[IntegerDigits[tmp, base]]], {z}]; s  (* A262627 *)
Map[FromDigits[ToString[#], base] &, s]  (* A262628 *)
(* Peter J. C. Moses, Sep 01 2015 *)

A263773 Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 6, 9, -12, -42, -18, 36, 48, 45, -12, -108, -36, 84, 84, 72, -72, -186, -54, 36, 120, 126, -96, -216, -72, 180, 186, 126, -12, -336, -90, 216, 192, 189, -144, -324, -144, 84, 228, 180, -168, -540, -126, 288, 264, 252, -72, -432, -144, 372, 342, 279, -216

Offset: 0

Michael Somos, Oct 27 2015

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 6*x + 9*x^2 - 12*x^3 - 42*x^4 - 18*x^5 + 36*x^6 + 48*x^7 + 45*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A134079, A226535, A262651.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^6 / QPochhammer[ -q^3]^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^3))^2, n))};

Expansion of f(q)^6 / f(q^3)^2 in powers of q where f() is a Ramanujan theta function.
Expansion of (eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^3))^2 in powers of q.
Euler transform of period 12 sequence [ 6, -12, 4, -6, 6, -8, 6, -6, 4, -12, 6, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134079.
G.f.: Product_{k>0} (1 - (-x)^k)^6 / (1 - (-x)^(3*k))^2.
a(2*n + 1) = 6 * A252651(n). a(3*n + 2) = 9 * A134079(n).
Convolution square of A226535.

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A262652 Base-10 representation of the primes at A262651.

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A262627 Minimal nested base-2 palindromic primes with seed 0.

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A263773 Expansion of b(-q)^2 in powers of q where b() is a cubic AGM theta function.

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