A033680 -id:A033680 - cp's OEIS frontend

A046255 a(1) = 5; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

5, 9, 9, 21, 53, 67, 71, 87, 87, 91, 117, 161, 187, 213, 363, 419, 501, 537, 543, 739, 879, 1101, 1329, 1391, 1641, 1939, 2093, 2109, 2331, 2557, 2639, 2697, 2863, 3441, 3441, 4413, 4461, 4479, 4557, 5489, 6033, 6267, 6351, 6973, 7181, 7459, 7679, 8113, 8241

Offset: 1

Patrick De Geest, May 15 1998

nonn

Robert Israel, Table of n, a(n) for n = 1..300

Cf. A069607, A074341, A033680, A033679, A033681, A046254, A046256, A046257, A046258, A046259, A111524.

R:= 5: p:= 5: x:= 5:
for count from 2 to 100 do
  for y from x by 2 do
    if isprime(10^(1+ilog10(y))*p+y) then
      R:= R, y; p:= 10^(1+ilog10(y))*p+y; x:= y;
      break
    fi
od od:
R; # Robert Israel, Nov 22 2020

a[1] = 5; a[n_] := a[n] = Block[{k = a[n - 1], c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ !PrimeQ[ FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 49}] (* Robert G. Wilson v, Aug 05 2005 *)

from sympy import isprime
def aupton(terms):
  alst, astr = [5], "5"
  while len(alst) < terms:
    an = alst[-1]
    while an%5 ==0 or not isprime(int(astr + str(an))): an += 2
    alst, astr = alst + [an], astr + str(an)
  return alst
print(aupton(49)) # Michael S. Branicky, May 09 2021

A046259 a(1) = 9; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

Original entry on oeis.org

9, 11, 21, 21, 23, 33, 37, 93, 119, 129, 133, 147, 293, 321, 429, 433, 497, 627, 661, 897, 1161, 1187, 1197, 1711, 1769, 1807, 2097, 2099, 4143, 4149, 4197, 4587, 4587, 5629, 5711, 5889, 6153, 6351, 6399, 6511, 6651, 7179, 7563, 7661, 8071, 8163, 9663

Offset: 1

Patrick De Geest, May 15 1998

nonn

Cf. A069611, A074345, A033680, A033679, A033681, A046254, A046255, A046256, A046257, A046258, A111524.

a[1] = 9; a[n_] := a[n] = Block[{k = a[n - 1], c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ !PrimeQ[ FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 48}] (* Robert G. Wilson v, Aug 05 2005 *)

A069602 a(1) = 1; a(n) = smallest composite number such that the juxtaposition a(1)a(2)...a(n) is a prime.

Original entry on oeis.org

1, 9, 9, 9, 21, 9, 51, 21, 9, 57, 301, 51, 51, 33, 209, 111, 87, 153, 121, 87, 63, 39, 77, 27, 57, 81, 129, 147, 111, 21, 147, 321, 69, 93, 153, 621, 817, 129, 81, 803, 129, 153, 451, 171, 717, 801, 959, 459, 187, 291, 231, 533, 399, 291, 289, 869, 489, 171, 381, 667, 21

Offset: 1

Amarnath Murthy, Mar 26 2002

			a(5) = 21 and the number 199921 is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A033680, A074336, A092528.

a[1] = 1; a[n_] := a[n] = Block[{k = 3, c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[PrimeQ[k] || !PrimeQ[FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 61}] (* Robert G. Wilson v, Aug 05 2005 *)
nxt[{jx_,a_}]:=Module[{c=9},While[PrimeQ[c]||CompositeQ[jx*10^IntegerLength[c]+c],c+=2];{jx*10^IntegerLength[c]+c,c}]; NestList[nxt,{1,1},60][[;;,2]] (* Harvey P. Dale, Feb 08 2025 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 31 2003

A069604 a(1) = 1; for n>1, a(n) = smallest number with all odd digits giving a prime in concatenation with the previous terms.

Original entry on oeis.org

1, 1, 3, 11, 1, 3, 3, 53, 13, 39, 9, 3, 399, 11, 9, 133, 3, 11, 51, 111, 13, 53, 31, 3, 173, 1, 317, 519, 579, 1, 573, 357, 5111, 39, 51, 73, 3317, 1977, 5173, 579, 357, 359, 9, 57, 3991, 959, 951, 7, 111, 1959, 39, 191, 3357, 3151, 3137, 577, 117, 1353, 951, 153, 99

Offset: 1

Amarnath Murthy, Mar 26 2002

			a(5) = 1 and the number 113111 is a prime.

Cf. A033680, A074336, A092528, A069602.

a[1] = 1; a[n_] := a[n] = Block[{k = 1, c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ Union[ Mod[ IntegerDigits[k], 2]] != {1} || !PrimeQ[ FromDigits[ Join[ Flatten[c], IntegerDigits[k]]]], k = k + 1]; k]; Table[ a[n], {n, 61}] (* corrected by Jason Yuen, Jun 22 2025 *)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 30 2003

A242042 Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -6, 9, 14, -54, 36, 65, -162, 126, 148, -438, 252, 344, -756, 513, 546, -1458, 756, 1022, -2064, 1332, 1352, -3510, 1764, 2198, -4374, 2808, 2710, -6804, 3276, 4161, -7992, 4914, 4816, -11826, 5616, 6860, -13188, 8190, 7658, -18576, 8892, 10804, -20412

Offset: 2

Michael Somos, Aug 12 2014

sign

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

In McKay and Sebbar on page 274 in equation (8.2) the last term on the right side is a multiple of the g.f.

			G.f. = q^2 - 6*q^3 + 9*q^4 + 14*q^5 - 54*q^6 + 36*q^7 + 65*q^8 - 162*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275. MR1795562 (2001m:11063)

Cf. A033680, A106401, A198956.

A := Basis( ModularForms( Gamma0(9), 4), 19); A[3] - 6*A[4] + 9*A[5];

a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q] QPochhammer[ q^9])^6 / QPochhammer[ q^3]^4, {q, 0, n}];

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

Expansion of (eta(q) * eta(q^9))^6 / eta(q^3)^4 in powers of q.
Euler transform of period 9 sequence [ -6, -6, -2, -6, -6, -2, -6, -6, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: x^2 * Product_{k>0} (1 - x^k)^6 * (1 - x^(9*k))^6 / (1 - x^(3*k))^4.
Convolution square of A106401.
a(3*n) = -6 * A198956(n). a(3*n + 1) = 9 * A033690(n).

A239547 a(1) = 1; a(n) is smallest number > a(n-1) such that the juxtaposition a(n)a(n-1)...a(1) is a prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 14, 21, 43, 56, 96, 141, 178, 180, 198, 263, 271, 315, 347, 352, 471, 530, 565, 588, 707, 711, 793, 812, 850, 887, 952, 1083, 1214, 1218, 1266, 1564, 1661, 1686, 1744, 1976, 2047, 2066, 2166, 2268, 2412, 2740, 2777, 2895, 2905, 3056, 3058, 3293

Offset: 1

Paolo P. Lava, Mar 21 2014

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A033680, A074336.

with(numtheory);
S:=proc(s) local w; w:=convert(s, base, 10); sum(w[j], j=1..nops(w)); end:
T:=proc(t) local w, x, y; x:=t; y:=0; while x>0 do x:=trunc(x/10); y:=y+1; od; end:
P:=proc(q) local a, b, c, j, n; a:=1; j:=2; print(1);
for n from 1 to q do b:=T(a); c:=j*10^b+a;
if isprime(c) then a:=j*10^b+a; print(j); fi;
j:=j+1; od; print(); end: P(10^10);

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A046255 a(1) = 5; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

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A046259 a(1) = 9; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

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A069602 a(1) = 1; a(n) = smallest composite number such that the juxtaposition a(1)a(2)...a(n) is a prime.

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A069604 a(1) = 1; for n>1, a(n) = smallest number with all odd digits giving a prime in concatenation with the previous terms.

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A242042 Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.

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A239547 a(1) = 1; a(n) is smallest number > a(n-1) such that the juxtaposition a(n)a(n-1)...a(1) is a prime.

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