| Displaying 1-10 of 10 results found.
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a(n) = (10^(2n+1)-1)/9 + 10^n.
+10
17
2, 121, 11211, 1112111, 111121111, 11111211111, 1111112111111, 111111121111111, 11111111211111111, 1111111112111111111, 111111111121111111111, 11111111111211111111111, 1111111111112111111111111, 111111111111121111111111111, 11111111111111211111111111111, 1111111111111112111111111111111
COMMENTS
a(0) = 2 is the only prime in this sequence, since all other terms factor as a(n) = R(n+1)*(10^n+1), where R(n) = (10^n-1)/9.
FORMULA
G.f.: (2 - 101*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332112 := n -> (10^(2*n+1)-1)/9+10^n;
MATHEMATICA
Array[ (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
PROG
(PARI) apply( { A332112(n)=10^(n*2+1)\9*1+10^n}, [0..15]) (Python) def A332112(n): return 10**(n*2+1)//9+10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332132 .. A332192 (variants with different repeated digit 3, ..., 9). Cf. A332113 .. A332119 (variants with different middle digit 3, ..., 9). Cf. A331860 & A331861 (indices of primes in non-palindromic variants).
a(n) = (10^(2*n+1)-1)/3 + 6*10^n.
+10
9
9, 393, 33933, 3339333, 333393333, 33333933333, 3333339333333, 333333393333333, 33333333933333333, 3333333339333333333, 333333333393333333333, 33333333333933333333333, 3333333333339333333333333, 333333333333393333333333333, 33333333333333933333333333333, 3333333333333339333333333333333
FORMULA
G.f.: (9 - 606*x + 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332139 := n -> (10^(2*n+1)-1)/3+6*10^n;
MATHEMATICA
Array[ (10^(2 # + 1)-1)/3 + 6*10^# &, 15, 0]
LinearRecurrence[{111, -1110, 1000}, {9, 393, 33933}, 20] (* Harvey P. Dale, Sep 17 2020 *)
PROG
(PARI) apply( { A332139(n)=10^(n*2+1)\3+6*10^n}, [0..15]) (Python) def A332139(n): return 10**(n*2+1)//3+6*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8). Cf. A332130 .. A332138 (variants with different middle digit 0, ..., 8).
a(n) = 10^(2n+1) - 1 - 6*10^n.
+10
7
3, 939, 99399, 9993999, 999939999, 99999399999, 9999993999999, 999999939999999, 99999999399999999, 9999999993999999999, 999999999939999999999, 99999999999399999999999, 9999999999993999999999999, 999999999999939999999999999, 99999999999999399999999999999, 9999999999999993999999999999999
FORMULA
G.f.: (3 + 606*x - 1500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332193 := n -> 10^(n*2+1)-1-6*10^n;
MATHEMATICA
Array[ 10^(2 # + 1) - 1 - 6*10^# &, 15, 0]
LinearRecurrence[{111, -1110, 1000}, {3, 939, 99399}, 20] (* Harvey P. Dale, Jan 19 2024 *)
PROG
(PARI) apply( { A332193(n)=10^(n*2+1)-1-6*10^n}, [0..15]) (Python) def A332193(n): return 10**(n*2+1)-1-6*10^n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes). Cf. A332113 .. A332183 (variants with different repeated digit 1, ..., 8).
a(n) = 2*(10^(2n+1)-1)/9 + 10^n.
+10
4
3, 232, 22322, 2223222, 222232222, 22222322222, 2222223222222, 222222232222222, 22222222322222222, 2222222223222222222, 222222222232222222222, 22222222222322222222222, 2222222222223222222222222, 222222222222232222222222222, 22222222222222322222222222222, 2222222222222223222222222222222
FORMULA
G.f.: (3 - 101*x - 100*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332123 := n -> 2*(10^(2*n+1)-1)/9+10^n;
MATHEMATICA
Array[2 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
PROG
(PARI) apply( { A332123(n)=10^(n*2+1)\9*2+10^n}, [0..15]) (Python) def A332123(n): return 10**(n*2+1)//9*2+10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9). Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).
Numbers k such that (10^k - 1)/9 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).
+10
3
COMMENTS
Prime versus probable prime status and proofs are given in the author's table.
The number k = 1 would also correspond to a prime, 3, but not "near-repdigit" or "wing" in a strict sense. - M. F. Hasler, Feb 09 2020
REFERENCES
C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
EXAMPLE
5 is a term because (10^5 - 1)/9 + 2*10^2 = 11311.
MATHEMATICA
Do[ If[ PrimeQ[(10^n + 18*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 20000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)
CROSSREFS
See A332113 for the (prime and composite) near-repunit palindromes 1..131..1.
a(n) = 2*(10^(2n+1)-1)/9 + 4*10^n.
+10
3
6, 262, 22622, 2226222, 222262222, 22222622222, 2222226222222, 222222262222222, 22222222622222222, 2222222226222222222, 222222222262222222222, 22222222222622222222222, 2222222222226222222222222, 222222222222262222222222222, 22222222222222622222222222222, 2222222222222226222222222222222
FORMULA
G.f.: (6 - 404*x + 200*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: 2*exp(x)*(10*exp(99*x) + 18*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024
MAPLE
A332126 := n -> 2*(10^(2*n+1)-1)/9+4*10^n;
MATHEMATICA
Array[2 (10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{}, n, 2], {6}, PadRight[{}, n, 2]]], {n, 0, 20}] (* or *) LinearRecurrence[{111, -1110, 1000}, {6, 262, 22622}, 20] (* Harvey P. Dale, Oct 17 2021 *)
PROG
(PARI) apply( { A332126(n)=10^(n*2+1)\9*2+4*10^n}, [0..15]) (Python) def A332126(n): return 10**(n*2+1)//9*2+4*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332116 .. A332196 (variants with different repeated digit 1, ..., 9). Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).
a(n) = 4*(10^(2*n+1)-1)/9 - 10^n.
+10
2
3, 434, 44344, 4443444, 444434444, 44444344444, 4444443444444, 444444434444444, 44444444344444444, 4444444443444444444, 444444444434444444444, 44444444444344444444444, 4444444444443444444444444, 444444444444434444444444444, 44444444444444344444444444444, 4444444444444443444444444444444
FORMULA
G.f.: (3 + 101*x - 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332143 := n -> 4*(10^(2*n+1)-1)/9-10^n;
MATHEMATICA
Array[4 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
PROG
(PARI) apply( { A332143(n)=10^(n*2+1)\9*4-10^n}, [0..15]) (Python) def A332143(n): return 10**(n*2+1)//9*4-10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9). Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).
a(n) = 8*(10^(2n+1)-1)/9 - 5*10^n.
+10
2
3, 838, 88388, 8883888, 888838888, 88888388888, 8888883888888, 888888838888888, 88888888388888888, 8888888883888888888, 888888888838888888888, 88888888888388888888888, 8888888888883888888888888, 888888888888838888888888888, 88888888888888388888888888888, 8888888888888883888888888888888
FORMULA
G.f.: (3 + 505*x - 1300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332183 := n -> 8*(10^(2*n+1)-1)/9-5*10^n;
MATHEMATICA
Array[8 (10^(2 # + 1)-1)/9 - 5*10^# &, 15, 0]
PROG
(PARI) apply( { A332183(n)=10^(n*2+1)\9*8-5*10^n}, [0..15]) (Python) def A332183(n): return 10**(n*2+1)//9*8-5*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only). Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9). Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).
a(n) = 5*(10^(2*n+1)-1)/9 - 2*10^n.
+10
1
3, 535, 55355, 5553555, 555535555, 55555355555, 5555553555555, 555555535555555, 55555555355555555, 5555555553555555555, 555555555535555555555, 55555555555355555555555, 5555555555553555555555555, 555555555555535555555555555, 55555555555555355555555555555, 5555555555555553555555555555555
FORMULA
G.f.: (3 + 202*x - 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332153 := n -> 5*(10^(2*n+1)-1)/9-2*10^n;
MATHEMATICA
Array[5 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
PROG
(PARI) apply( { A332153(n)=10^(n*2+1)\9*5-2*10^n}, [0..15]) (Python) def A332153(n): return 10**(n*2+1)//9*5-2*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9). Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).
a(n) = 6*(10^(2*n+1)-1)/9 - 3*10^n.
+10
1
3, 636, 66366, 6663666, 666636666, 66666366666, 6666663666666, 666666636666666, 66666666366666666, 6666666663666666666, 666666666636666666666, 66666666666366666666666, 6666666666663666666666666, 666666666666636666666666666, 66666666666666366666666666666, 6666666666666663666666666666666
FORMULA
G.f.: (3 + 303*x - 900*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
MAPLE
A332163 := n -> 6*(10^(2*n+1)-1)/9-3*10^n;
MATHEMATICA
Array[6 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
PROG
(PARI) apply( { A332163(n)=10^(n*2+1)\9*6-3*10^n}, [0..15]) (Python) def A332163(n): return 10**(n*2+1)//9*6-3*10**n
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes). Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9). Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).
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