Search: a332179 -id:a332179
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A332171
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a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.
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+10
9
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1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777
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internal format)
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OFFSET
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0,2
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COMMENTS
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For n == 0 or n == 2 (mod 6), there is no obvious divisibility pattern.
According to M. Kamada, n = 116 is the only index of a prime up to n = 10^5.
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LINKS
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FORMULA
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For n == 1 (mod 3), 3 | a(n) and a(n)/3 = 259*(10^(2n+1)-1)/999 - 2*10^n;
for n == 3 or 5 (mod 6), 13 | a(n) and a(n)/13 = (A(n)-1)*10^n + B(n), where A(n) (resp. B(n)) are the n leftmost (resp. rightmost) digits of 59829*(10^(ceiling(n/6)*6)-1)/(10^6-1).
G.f.: (1 + 606*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.
(End)
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 54*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 - 6*10^# &, 15, 0] (* or *)
CoefficientList[Series[(1 + 606 x - 1300 x^2)/((1 - x) (1 - 10 x) (1 - 100 x)), {x, 0, 15}], x] (* Michael De Vlieger, Feb 08 2020 *)
Table[FromDigits[Join[PadRight[{}, n, 7], {1}, PadRight[{}, n, 7]]], {n, 0, 20}] (* or *) LinearRecurrence[ {111, -1110, 1000}, {1, 717, 77177}, 20] (* Harvey P. Dale, Apr 04 2024 *)
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PROG
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(PARI) apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])
(PARI) Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020
(Python) def A332171(n): return 10**(n*2+1)//9*7-6*10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332178
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a(n) = 7*(10^(2n+1)-1)/9 + 10^n.
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+10
8
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8, 787, 77877, 7778777, 777787777, 77777877777, 7777778777777, 777777787777777, 77777777877777777, 7777777778777777777, 777777777787777777777, 77777777777877777777777, 7777777777778777777777777, 777777777777787777777777777, 77777777777777877777777777777, 7777777777777778777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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See A183182 = {1, 3, 39, 54, 168, 240, ...} for the indices of primes.
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LINKS
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FORMULA
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G.f.: (8 - 101*x - 600*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.
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MAPLE
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A332178 := n -> 7*(10^(n*2+1)-1)/9 + 10^n;
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 + 10^# &, 15, 0]
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PROG
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(PARI) apply( {A332178(n)=10^(n*2+1)\9*7+10^n}, [0..15])
(Python) def A332178(n): return 10**(n*2+1)//9*7+10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332170
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a(n) = 7*(10^(2n+1)-1)/9 - 7*10^n.
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+10
2
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0, 707, 77077, 7770777, 777707777, 77777077777, 7777770777777, 777777707777777, 77777777077777777, 7777777770777777777, 777777777707777777777, 77777777777077777777777, 7777777777770777777777777, 777777777777707777777777777, 77777777777777077777777777777, 7777777777777770777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: 7*x*(101 - 200*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.
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MAPLE
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A332170 := n -> 7*(10^(2*n+1)-1)/9-7*10^n;
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MATHEMATICA
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Array[7 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
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PROG
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(PARI) apply( {A332170(n)=(10^(n*2+1)\9-10^n)*7}, [0..15])
(Python) def A332170(n): return (10**(n*2+1)//9-10^n)*7
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332172
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a(n) = 7*(10^(2n+1)-1)/9 - 5*10^n.
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+10
1
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2, 727, 77277, 7772777, 777727777, 77777277777, 7777772777777, 777777727777777, 77777777277777777, 7777777772777777777, 777777777727777777777, 77777777777277777777777, 7777777777772777777777777, 777777777777727777777777777, 77777777777777277777777777777, 7777777777777772777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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Indices of prime terms: {0, 1, 3, 7, 10, 12, 480, 949, ...} = A183178.
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LINKS
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FORMULA
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G.f.: (2 + 505*x - 1200*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.
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MAPLE
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A332172 := n -> 7*(10^(n*2+1)-1)/9 -5*10^n;
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MATHEMATICA
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Array[7 (10^(2 # +1)-1)/9 -5*10^# &, 15, 0]
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PROG
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(PARI) apply( {A332172(n)=10^(n*2+1)\9*7-5*10^n}, [0..25])
(Python) def A332172(n): return 10**(n*2+1)//9*7-5*10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 (analog with middle digit 1).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332173
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a(n) = 7*(10^(2n+1)-1)/9 - 4*10^n.
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+10
1
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3, 737, 77377, 7773777, 777737777, 77777377777, 7777773777777, 777777737777777, 77777777377777777, 7777777773777777777, 777777777737777777777, 77777777777377777777777, 7777777777773777777777777, 777777777777737777777777777, 77777777777777377777777777777, 7777777777777773777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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According to M. Kamada, n = 0 and n = 2 are the only indices of a prime up to n = 2*10^4.
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LINKS
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FORMULA
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G.f.: (1 + 404*x - 1100*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.
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MAPLE
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A332173 := n -> 7*(10^(n*2+1)-1)/9 - 4*10^n;
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 - 4*10^# &, 15, 0]
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PROG
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(PARI) apply( {A332173(n)=10^(n*2+1)\9*7-4*10^n}, [0..15])
(Python) def A332173(n): return 10**(n*2+1)//9*7-4*10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332174
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a(n) = 7*(10^(2n+1)-1)/9 - 3*10^n.
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+10
1
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4, 747, 77477, 7774777, 777747777, 77777477777, 7777774777777, 777777747777777, 77777777477777777, 7777777774777777777, 777777777747777777777, 77777777777477777777777, 7777777777774777777777777, 777777777777747777777777777, 77777777777777477777777777777, 7777777777777774777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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See A183179 = {2, 3, 6, 23, 36, 69, 561, ...} for the indices of primes.
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LINKS
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FORMULA
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G.f.: (4 + 303*x - 1000*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.: (1/9)*exp(x)*(70*exp(99*x) - 27*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020
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MAPLE
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A332174 := n -> 7*(10^(n*2+1)-1)/9 - 3*10^n;
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 - 3*10^# &, 15, 0]
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PROG
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(PARI) apply( {A332174(n)=10^(n*2+1)\9*7-3*10^n}, [0..15])
(Python) def A332174(n): return 10**(n*2+1)//9*7-3*10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332175
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a(n) = 7*(10^(2n+1)-1)/9 - 2*10^n.
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+10
1
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5, 757, 77577, 7775777, 777757777, 77777577777, 7777775777777, 777777757777777, 77777777577777777, 7777777775777777777, 777777777757777777777, 77777777777577777777777, 7777777777775777777777777, 777777777777757777777777777, 77777777777777577777777777777, 7777777777777775777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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See A183180 = {0, 1, 7, 13, 58, 129, 253, ...} for the indices of primes.
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LINKS
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FORMULA
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G.f.: (5 + 202*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.
E.g.f.: (1/9)*exp(x)*(70*exp(99*x) - 18*exp(9*x) - 7). - Stefano Spezia, Feb 08 2020
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MAPLE
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A332175 := n -> 7*(10^(n*2+1)-1)/9 - 2*10^n;
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 - 2*10^# &, 15, 0]
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PROG
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(PARI) apply( {A332175(n)=10^(n*2+1)\9*7-2*10^n}, [0..15])
(Python) def A332175(n): return 10**(n*2+1)//9*7-2*10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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A332176
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a(n) = 7*(10^(2n+1)-1)/9 - 10^n.
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+10
1
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6, 767, 77677, 7776777, 777767777, 77777677777, 7777776777777, 777777767777777, 77777777677777777, 7777777776777777777, 777777777767777777777, 77777777777677777777777, 7777777777776777777777777, 777777777777767777777777777, 77777777777777677777777777777, 7777777777777776777777777777777
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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See A183181 = {4, 5, 8, 11, 1244, 1685, ...} for the indices of primes.
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LINKS
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FORMULA
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G.f.: (6 + 101*x - 800*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.
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MAPLE
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A332176 := n -> 7*(10^(n*2+1)-1)/9 - 10^n;
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MATHEMATICA
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Array[7 (10^(2 # + 1) - 1)/9 - 10^# &, 15, 0]
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PROG
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(PARI) apply( {A332176(n)=10^(n*2+1)\9*7-10^n}, [0..15])
(Python) def A332176(n): return 10**(n*2+1)//9*7-10^n
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CROSSREFS
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Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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