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Search: a332179 -id:a332179
Displaying 1-8 of 8 results found.
page 1
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A332171
a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.
+0
9
1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77177...77.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 10^n.
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).
From Colin Barker, Feb 07 2020: (Start)
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332170 .. A332179 (variants with different middle digit 2, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 06 2020
STATUS
approved
A332178
a(n) = 7*(10^(2n+1)-1)/9 + 10^n.
+0
8
8, 787, 77877, 7778777, 777787777, 77777877777, 7777778777777, 777777787777777, 77777777877777777, 7777777778777777777, 777777777787777777777, 77777777777877777777777, 7777777777778777777777777, 777777777777787777777777777, 77777777777777877777777777777, 7777777777777778777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
See A183182 = {1, 3, 39, 54, 168, 240, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77877...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 8*10^n.
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.
MAPLE
A332178 := n -> 7*(10^(n*2+1)-1)/9 + 10^n;
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 + 10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077793-1)/2 = A183182: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332176
a(n) = 7*(10^(2n+1)-1)/9 - 10^n.
+0
1
6, 767, 77677, 7776777, 777767777, 77777677777, 7777776777777, 777777767777777, 77777777677777777, 7777777776777777777, 777777777767777777777, 77777777777677777777777, 7777777777776777777777777, 777777777777767777777777777, 77777777777777677777777777777, 7777777777777776777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
See A183181 = {4, 5, 8, 11, 1244, 1685, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 6*10^n.
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.
MAPLE
A332176 := n -> 7*(10^(n*2+1)-1)/9 - 10^n;
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 - 10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077788-1)/2 = A183181: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332175
a(n) = 7*(10^(2n+1)-1)/9 - 2*10^n.
+0
1
5, 757, 77577, 7775777, 777757777, 77777577777, 7777775777777, 777777757777777, 77777777577777777, 7777777775777777777, 777777777757777777777, 77777777777577777777777, 7777777777775777777777777, 777777777777757777777777777, 77777777777777577777777777777, 7777777777777775777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
See A183180 = {0, 1, 7, 13, 58, 129, 253, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77577...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 5*10^n.
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
MAPLE
A332175 := n -> 7*(10^(n*2+1)-1)/9 - 2*10^n;
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 - 2*10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. (A077785-1)/2 = A183180: indices of primes.
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332174
a(n) = 7*(10^(2n+1)-1)/9 - 3*10^n.
+0
1
4, 747, 77477, 7774777, 777747777, 77777477777, 7777774777777, 777777747777777, 77777777477777777, 7777777774777777777, 777777777747777777777, 77777777777477777777777, 7777777777774777777777777, 777777777777747777777777777, 77777777777777477777777777777, 7777777777777774777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
See A183179 = {2, 3, 6, 23, 36, 69, 561, ...} for the indices of primes.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77477...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 4*10^n.
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
MAPLE
A332174 := n -> 7*(10^(n*2+1)-1)/9 - 3*10^n;
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 - 3*10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only).
Cf. (A077781-1)/2 = A183179: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
A332173
a(n) = 7*(10^(2n+1)-1)/9 - 4*10^n.
+0
1
3, 737, 77377, 7773777, 777737777, 77777377777, 7777773777777, 777777737777777, 77777777377777777, 7777777773777777777, 777777777737777777777, 77777777777377777777777, 7777777777773777777777777, 777777777777737777777777777, 77777777777777377777777777777, 7777777777777773777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
According to M. Kamada, n = 0 and n = 2 are the only indices of a prime up to n = 2*10^4.
LINKS
Table of n, a(n) for n=0..15.
Makoto Kamada, Factorization of 77...77377...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 3*10^n.
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.
MAPLE
A332173 := n -> 7*(10^(n*2+1)-1)/9 - 4*10^n;
MATHEMATICA
Array[7 (10^(2 # + 1) - 1)/9 - 4*10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 06 2020
STATUS
approved
A332172
a(n) = 7*(10^(2n+1)-1)/9 - 5*10^n.
+0
1
2, 727, 77277, 7772777, 777727777, 77777277777, 7777772777777, 777777727777777, 77777777277777777, 7777777772777777777, 777777777727777777777, 77777777777277777777777, 7777777777772777777777777, 777777777777727777777777777, 77777777777777277777777777777, 7777777777777772777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Indices of prime terms: {0, 1, 3, 7, 10, 12, 480, 949, ...} = A183178.
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) + 2*10^n.
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.
MAPLE
A332172 := n -> 7*(10^(n*2+1)-1)/9 -5*10^n;
MATHEMATICA
Array[7 (10^(2 # +1)-1)/9 -5*10^# &, 15, 0]
PROG
(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
CROSSREFS
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332171 (analog with middle digit 1).
Cf. (A077777-1)/2 = A183178: indices of primes.
Cf. A002275 (repunits R_n = [10^n/9]), A002281 (7*R_n), A011557 (10^n).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 06 2020
STATUS
approved
A332170
a(n) = 7*(10^(2n+1)-1)/9 - 7*10^n.
+0
2
0, 707, 77077, 7770777, 777707777, 77777077777, 7777770777777, 777777707777777, 77777777077777777, 7777777770777777777, 777777777707777777777, 77777777777077777777777, 7777777777770777777777777, 777777777777707777777777777, 77777777777777077777777777777, 7777777777777770777777777777777
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 7*A138148(n) = A002281(2n+1) - 7*A011557(n).
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.
MAPLE
A332170 := n -> 7*(10^(2*n+1)-1)/9-7*10^n;
MATHEMATICA
Array[7 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
PROG
(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
CROSSREFS
Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
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).
KEYWORD
nonn,base,easy
AUTHOR
M. F. Hasler, Feb 08 2020
STATUS
approved
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