A332179 -id:A332179 - cp's OEIS frontend

A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

1, 717, 77177, 7771777, 777717777, 77777177777, 7777771777777, 777777717777777, 77777777177777777, 7777777771777777777, 777777777717777777777, 77777777777177777777777, 7777777777771777777777777, 777777777777717777777777777, 77777777777777177777777777777, 7777777777777771777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

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.

Makoto Kamada, Factorization of 77...77177...77.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

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 *)

apply( {A332171(n)=10^(n*2+1)\9*7-6*10^n}, [0..15])

Vec((1 + 606*x - 1300*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)) + O(x^15)) \\ Colin Barker, Feb 07 2020

def A332171(n): return 10**(n*2+1)//9*7-6*10^n

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

A332178 a(n) = 7*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

8, 787, 77877, 7778777, 777787777, 77777877777, 7777778777777, 777777787777777, 77777777877777777, 7777777778777777777, 777777777787777777777, 77777777777877777777777, 7777777777778777777777777, 777777777777787777777777777, 77777777777777877777777777777, 7777777777777778777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183182 = {1, 3, 39, 54, 168, 240, ...} for the indices of primes.

Makoto Kamada, Factorization of 77...77877...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332178 := n -> 7*(10^(n*2+1)-1)/9 + 10^n;

Array[7 (10^(2 # + 1) - 1)/9 + 10^# &, 15, 0]

apply( {A332178(n)=10^(n*2+1)\9*7+10^n}, [0..15])

def A332178(n): return 10**(n*2+1)//9*7+10^n

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.

A332170 a(n) = 7(10^(2n+1)-1)/9 - 710^n.

Original entry on oeis.org

0, 707, 77077, 7770777, 777707777, 77777077777, 7777770777777, 777777707777777, 77777777077777777, 7777777770777777777, 777777777707777777777, 77777777777077777777777, 7777777777770777777777777, 777777777777707777777777777, 77777777777777077777777777777, 7777777777777770777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332170 := n -> 7*(10^(2*n+1)-1)/9-7*10^n;

Array[7 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]

apply( {A332170(n)=(10^(n*2+1)\9-10^n)*7}, [0..15])

def A332170(n): return (10**(n*2+1)//9-10^n)*7

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.

A332172 a(n) = 7(10^(2n+1)-1)/9 - 510^n.

Original entry on oeis.org

2, 727, 77277, 7772777, 777727777, 77777277777, 7777772777777, 777777727777777, 77777777277777777, 7777777772777777777, 777777777727777777777, 77777777777277777777777, 7777777777772777777777777, 777777777777727777777777777, 77777777777777277777777777777, 7777777777777772777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

Indices of prime terms: {0, 1, 3, 7, 10, 12, 480, 949, ...} = A183178.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332172 := n -> 7*(10^(n*2+1)-1)/9 -5*10^n;

Array[7 (10^(2 # +1)-1)/9 -5*10^# &, 15, 0]

apply( {A332172(n)=10^(n*2+1)\9*7-5*10^n}, [0..25])

def A332172(n): return 10**(n*2+1)//9*7-5*10^n

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.

A332173 a(n) = 7(10^(2n+1)-1)/9 - 410^n.

Original entry on oeis.org

3, 737, 77377, 7773777, 777737777, 77777377777, 7777773777777, 777777737777777, 77777777377777777, 7777777773777777777, 777777777737777777777, 77777777777377777777777, 7777777777773777777777777, 777777777777737777777777777, 77777777777777377777777777777, 7777777777777773777777777777777

Offset: 0

M. F. Hasler, Feb 06 2020

According to M. Kamada, n = 0 and n = 2 are the only indices of a prime up to n = 2*10^4.

Makoto Kamada, Factorization of 77...77377...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332173 := n -> 7*(10^(n*2+1)-1)/9 - 4*10^n;

Array[7 (10^(2 # + 1) - 1)/9 - 4*10^# &, 15, 0]

apply( {A332173(n)=10^(n*2+1)\9*7-4*10^n}, [0..15])

def A332173(n): return 10**(n*2+1)//9*7-4*10^n

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.
E.g.f.: exp(x)*(70*exp(99*x) - 36*exp(9*x) - 7)/9. - Stefano Spezia, Feb 19 2025

A332174 a(n) = 7(10^(2n+1)-1)/9 - 310^n.

Original entry on oeis.org

4, 747, 77477, 7774777, 777747777, 77777477777, 7777774777777, 777777747777777, 77777777477777777, 7777777774777777777, 777777777747777777777, 77777777777477777777777, 7777777777774777777777777, 777777777777747777777777777, 77777777777777477777777777777, 7777777777777774777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183179 = {2, 3, 6, 23, 36, 69, 561, ...} for the indices of primes.

Makoto Kamada, Factorization of 77...77477...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332174 := n -> 7*(10^(n*2+1)-1)/9 - 3*10^n;

Array[7 (10^(2 # + 1) - 1)/9 - 3*10^# &, 15, 0]

apply( {A332174(n)=10^(n*2+1)\9*7-3*10^n}, [0..15])

def A332174(n): return 10**(n*2+1)//9*7-3*10^n

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

A332175 a(n) = 7(10^(2n+1)-1)/9 - 210^n.

Original entry on oeis.org

5, 757, 77577, 7775777, 777757777, 77777577777, 7777775777777, 777777757777777, 77777777577777777, 7777777775777777777, 777777777757777777777, 77777777777577777777777, 7777777777775777777777777, 777777777777757777777777777, 77777777777777577777777777777, 7777777777777775777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183180 = {0, 1, 7, 13, 58, 129, 253, ...} for the indices of primes.

Makoto Kamada, Factorization of 77...77577...77, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332175 := n -> 7*(10^(n*2+1)-1)/9 - 2*10^n;

Array[7 (10^(2 # + 1) - 1)/9 - 2*10^# &, 15, 0]

apply( {A332175(n)=10^(n*2+1)\9*7-2*10^n}, [0..15])

def A332175(n): return 10**(n*2+1)//9*7-2*10^n

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

A332176 a(n) = 7*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

6, 767, 77677, 7776777, 777767777, 77777677777, 7777776777777, 777777767777777, 77777777677777777, 7777777776777777777, 777777777767777777777, 77777777777677777777777, 7777777777776777777777777, 777777777777767777777777777, 77777777777777677777777777777, 7777777777777776777777777777777

Offset: 0

M. F. Hasler, Feb 08 2020

See A183181 = {4, 5, 8, 11, 1244, 1685, ...} for the indices of primes.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

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).

A332176 := n -> 7*(10^(n*2+1)-1)/9 - 10^n;

Array[7 (10^(2 # + 1) - 1)/9 - 10^# &, 15, 0]

apply( {A332176(n)=10^(n*2+1)\9*7-10^n}, [0..15])

def A332176(n): return 10**(n*2+1)//9*7-10^n

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.

cp's OEIS Frontend

A332171 a(n) = 7*(10^(2n+1)-1)/9 - 6*10^n.

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A332178 a(n) = 7*(10^(2n+1)-1)/9 + 10^n.

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A332170 a(n) = 7*(10^(2n+1)-1)/9 - 7*10^n.

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A332172 a(n) = 7*(10^(2n+1)-1)/9 - 5*10^n.

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A332173 a(n) = 7*(10^(2n+1)-1)/9 - 4*10^n.

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A332174 a(n) = 7*(10^(2n+1)-1)/9 - 3*10^n.

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A332175 a(n) = 7*(10^(2n+1)-1)/9 - 2*10^n.

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A332171 a(n) = 7(10^(2n+1)-1)/9 - 610^n.

A332170 a(n) = 7(10^(2n+1)-1)/9 - 710^n.

A332172 a(n) = 7(10^(2n+1)-1)/9 - 510^n.

A332173 a(n) = 7(10^(2n+1)-1)/9 - 410^n.

A332174 a(n) = 7(10^(2n+1)-1)/9 - 310^n.

A332175 a(n) = 7(10^(2n+1)-1)/9 - 210^n.