A183180 -id:A183180 - cp's OEIS frontend

A077785 Odd numbers k such that the palindromic wing number (a.k.a. near-repdigit palindrome) 7(10^k - 1)/9 - 210^((k-1)/2) is prime.

3, 15, 27, 117, 259, 507, 3315, 4489, 4875, 15849, 19807, 23799, 36315, 37915, 47331, 211219

Offset: 1

Patrick De Geest, Nov 16 2002

Original name was "Palindromic wing primes (a.k.a. near-repdigit palindromes) of the form 7*(10^a(n)-1)/9-2*10^[ a(n)/2 ]."
Prime versus probable prime status and proofs are given in the author's table.
a(16) > 2*10^5. - Robert Price, Jun 23 2017
1 could be considered part of this sequence since the formula evaluates to 5 which is a degenerate form of the near-repdigit palindrome 777...77577...777 that has zero occurrences of the digit 7. - Robert Price, Jun 23 2017

			15 is in the sequence because 7*(10^15 - 1)/9 - 2*10^7 = 777777757777777 is prime.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 77...77577...77
Index entries for primes involving repunits.

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Do[ If[ PrimeQ[(7*10^n - 18*10^Floor[n/2] - 7)/9], Print[n]], {n, 3, 40000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183180(n) + 1.

a(15) from Robert Price, Jun 23 2017
Example edited by Jon E. Schoenfield, Jun 23 2017
Name edited by Jon E. Schoenfield, Jun 24 2017
a(16) from Robert Price, Oct 12 2023

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

cp's OEIS Frontend

A077785 Odd numbers k such that the palindromic wing number (a.k.a. near-repdigit palindrome) 7(10^k - 1)/9 - 210^((k-1)/2) is prime.

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

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cp's OEIS Frontend

A077785 Odd numbers k such that the palindromic wing number (a.k.a. near-repdigit palindrome) 7*(10^k - 1)/9 - 2*10^((k-1)/2) is prime.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

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Programs

Maple

Mathematica

PARI

Python

Formula

A077785 Odd numbers k such that the palindromic wing number (a.k.a. near-repdigit palindrome) 7(10^k - 1)/9 - 210^((k-1)/2) is prime.

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