A002277 -id:A002277 - cp's OEIS frontend

A332134 a(n) = (10^(2n+1)-1)/3 + 10^n.

4, 343, 33433, 3334333, 333343333, 33333433333, 3333334333333, 333333343333333, 33333333433333333, 3333333334333333333, 333333333343333333333, 33333333333433333333333, 3333333333334333333333333, 333333333333343333333333333, 33333333333333433333333333333, 3333333333333334333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

There are no primes in this sequence because a(n) = round(n*2/3)*(5*10^n-1).

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)4(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33433...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Maple
```
A332134 := n -> (10^(2*n+1)-1)/3+10^n;
```

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

apply( {A332134(n)=10^(n*2+1)\3+10^n}, [0..15])

def A332134(n): return 10**(n*2+1)//3+10**n

a(n) = 3*A138148(n) + 4*10^n = A002277(2n+1) + 10^n.
G.f.: (4 - 101*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.

A332138 a(n) = (10^(2n+1)-1)/3 + 510^n.

Original entry on oeis.org

8, 383, 33833, 3338333, 333383333, 33333833333, 3333338333333, 333333383333333, 33333333833333333, 3333333338333333333, 333333333383333333333, 33333333333833333333333, 3333333333338333333333333, 333333333333383333333333333, 33333333333333833333333333333, 3333333333333338333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183177 = {1, 7, 85, 94, 273, 356, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)8(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33833...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077792-1)/2 = A183177: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332138 := n -> (10^(2*n+1)-1)/3+5*10^n;

Array[ (10^(2 # + 1)-1)/3 + 5*10^# &, 15, 0]

apply( {A332138(n)=10^(n*2+1)\3+5*10^n}, [0..15])

def A332138(n): return 10**(n*2+1)//3+5*10**n

a(n) = 3*A138148(n) + 8*10^n = A002277(2n+1) + 5*10^n.
G.f.: (8 - 505*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.

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0

Offset: 0

Stefano Spezia, Sep 14 2023

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A271528 a(n) = 2*(10^n - 1)^2/27.

Original entry on oeis.org

0, 6, 726, 73926, 7405926, 740725926, 74073925926, 7407405925926, 740740725925926, 74074073925925926, 7407407405925925926, 740740740725925925926, 74074074073925925925926, 7407407407405925925925926, 740740740740725925925925926, 74074074074073925925925925926

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

All terms are multiple of 6.
Converges in a 10-adic sense to ...925925925926.
A transformation of the Wonderful Demlo numbers (A002477).
More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

			n=1:                  6 = 2 * 3;
n=2:                726 = 22 * 33;
n=3:              73926 = 222 * 333;
n=4:            7405926 = 2222 * 3333;
n=5:          740725926 = 22222 * 33333;
n=6:        74073925926 = 222222 * 333333;
n=7:      7407405925926 = 2222222 * 3333333;
n=8:    740740725925926 = 22222222 * 33333333;
n=9:  74074073925925926 = 222222222 * 333333333, etc.

Ilya Gutkovskiy, Transformation of the Wonderful Demlo numbers
Eric Weisstein's World of Mathematics, Demlo Number
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275, A002276, A002277, A002477.

Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).

Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]

x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ Altug Alkan, Apr 09 2016

for n in range(0,10**1):print((int)((2*(10**n-1)**2)/27))
# Soumil Mandal, Apr 10 2016

O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).
E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).
Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...
Lim_{n -> infinity} a(n + 1)/a(n) = 100.

A274986 Numbers k such that (10^k + 23)/3 is prime.

Original entry on oeis.org

1, 2, 6, 146, 326, 380, 1116, 1866, 4808, 5528, 5730, 21836, 24804, 38724

Offset: 1

Vincenzo Librandi, Sep 25 2016

Also numbers k for which A093137(k) + 7 or A002277(k) + 8 is prime.

Cf. numbers k such that (10^k+m)/3 is prime: A099411 (m=11), this sequence (m=23).

Cf. A002277, A093137.

[n: n in [0..400] | IsPrime((10^n+23) div 3)];

Select[Range[1000], PrimeQ[(10^# + 23) / 3] &]

is(n)=ispseudoprime((10^n+23)/3) \\ Charles R Greathouse IV, Jun 13 2017

a(9)-a(11) from Michael S. Branicky, Aug 16 2021
a(12)-a(13) from Michael S. Branicky, May 14 2023
a(14) from Kamada data by Tyler Busby, May 05 2024

A332135 a(n) = (10^(2n+1)-1)/3 + 2*10^n.

Original entry on oeis.org

5, 353, 33533, 3335333, 333353333, 33333533333, 3333335333333, 333333353333333, 33333333533333333, 3333333335333333333, 333333333353333333333, 33333333333533333333333, 3333333333335333333333333, 333333333333353333333333333, 33333333333333533333333333333, 3333333333333335333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183175 = {1, 2, 17, 79, 118, 162, 177, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)5(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33533...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077784-1)/2 = A183175: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332125 .. A332195 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332135 := n -> (10^(2*n+1)-1)/3+2*10^n;

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

apply( {A332135(n)=10^(n*2+1)\3+2*10^n}, [0..15])

def A332135(n): return 10**(n*2+1)//3+2*10**n

a(n) = 3*A138148(n) + 5*10^n = A002277(2n+1) + 2*10^n.
G.f.: (5 - 202*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.
E.g.f.: exp(x)*(10*exp(99*x) + 6*exp(9*x) - 1)/3. - Stefano Spezia, Sep 24 2024

A332136 a(n) = 3(10^(2n+1)-1)/9 + 310^n.

Original entry on oeis.org

6, 363, 33633, 3336333, 333363333, 33333633333, 3333336333333, 333333363333333, 33333333633333333, 3333333336333333333, 333333333363333333333, 33333333333633333333333, 3333333333336333333333333, 333333333333363333333333333, 33333333333333633333333333333, 3333333333333336333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

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

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332136 := n -> (10^(2*n+1)-1)/3+3*10^n);

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

apply( {A332136(n)=10^(n*2+1)\3+3*10^n}, [0..15])

def A332136(n): return 10**(n*2+1)//3+3*10**n

a(n) = 3*A138148(n) + 6*10^n = A002277(2n+1) + 3*10^n = 3*A332112(n).
G.f.: (6 - 303*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.

A332137 a(n) = (10^(2n+1)-1)/3 + 4*10^n.

Original entry on oeis.org

7, 373, 33733, 3337333, 333373333, 33333733333, 3333337333333, 333333373333333, 33333333733333333, 3333333337333333333, 333333333373333333333, 33333333333733333333333, 3333333333337333333333333, 333333333333373333333333333, 33333333333333733333333333333, 3333333333333337333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183176 = {1, 3, 7, 11, 13, 17, 29, 31, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)7(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33733...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077790-1)/2 = A183176: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332127 .. A332197 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332137 := n -> (10^(2*n+1)-1)/3+4*10^n;

Array[ (10^(2 # + 1)-1)/3 + 4*10^# &, 15, 0]

apply( {A332137(n)=10^(n*2+1)\3+4*10^n}, [0..15])

def A332137(n): return 10**(n*2+1)//3+4*10**n

a(n) = 3*A138148(n) + 7*10^n = A002277(2n+1) + 4*10^n.
G.f.: (7 - 404*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.

A361820 Palindromes in A329150.

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 22, 33, 55, 77, 202, 222, 232, 252, 272, 303, 313, 323, 333, 353, 373, 505, 525, 535, 555, 575, 707, 717, 727, 737, 757, 777, 1111, 2002, 2112, 2222, 2332, 2552, 2772, 3003, 3113, 3223, 3333, 3553, 3773, 5005, 5115, 5225, 5335, 5555, 5775, 7007, 7117

Offset: 1

Bernard Schott, Mar 25 2023

If m is a palindrome with no digit greater than 5 in A118597, then A329147(m) is a term, but there exist terms that are not of this form as 313, 717, ...

			232 is a term which has two preimages since A329147(91) = A329147(121) = 232.
313 = A329147(26) is a term whose preimage is not in A118597.
2002 is a term since A329147(1001) = 2002.
2112 is a term since A329147(151) = 2112.
27172 = A329147(1471) is a term whose preimage is not in A118597.

Intersection of A002113 and A329150.
Cf. A118597, A329147, A361750, A361821.
Subsequences: A002276, A002277, A002279, A002281, A099814.

p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten@ IntegerDigits@ (p /@ IntegerDigits[n])], {n, 0, 10^ndigmax - 1}]}, Union@ Select[t, # < 10^ndigmax && PalindromeQ[#] &]]; seq[4] (* Amiram Eldar, Mar 26 2023 *)

ispal(n) = my(d=digits(n)); d==Vecrev(d);
f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if ((m <= nn) && ispal(m), listput(list, m));); vecsort(Set(list)); \\ Michel Marcus, Mar 26 2023

A385515 Repdigit numbers whose square does not contain the repeated digit.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 22, 33, 44, 77, 88, 333, 444, 3333, 33333, 44444, 88888, 333333, 3333333, 33333333, 333333333, 3333333333, 33333333333, 333333333333, 3333333333333, 33333333333333, 333333333333333, 3333333333333333, 33333333333333333, 333333333333333333

Offset: 1

Gonzalo Martínez, Jul 01 2025

For n >= 18, all terms are of the form 33...3; that is, elements of A002277.
A002277(m) is a term, for m > 0. Proof: 3 is in a(n) because 3^2 = 9. If 33...3 is composed of k 3's, with k > 1, it is satisfied that 33...3^2 = 11...1088...89; i.e., (k - 1) 1's followed by a 0, then (k - 1) 8's and a 9, so that 3 is not among the digits of its square.
Let's see that there are no other terms of the form 33...3 besides 2, 4, 7, 8, 9, 22, 44, 77, 88, 444, 44444, 88888. In this sequence there are no repdigits of the form 11...1, 55...5, 66...6, since their squares end in 1, 5 and 6 respectively. On the other hand, 9 is the only number of the form 999...9, since if it has 2 or more 9's its square starts with 9. Suppose that dd...d contains 6 or more digits. We already saw that the cases d = 1, 5, 6 and 9 are discarded. Let us analyze what happens for d = 2, 4, 7 and 8:
For d = 2, we have that 22...2^2 == 284 (mod 10^3).
For d = 4, we have that 44...4^2 == 469136 (mod 10^6).
For d = 7, we have that 77...7^2 == 729 (mod 10^3).
For d = 8, we have that 88...8^2 == 876544 (mod 10^6).
Thus, we conclude that a(n) only consists of digits 3 for n >= 18. And, in fact, a(n) consists of (n - 12) 3's.

			22 is a term since 22^2 = 484 does not contain the digit 2.

Cf. A002277, A010785, A029783.

Intersection of A010785 and A029783.

Select[Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 18}] ,ContainsNone[IntegerDigits[#^2],IntegerDigits[#]]&] (* James C. McMahon, Jul 07 2025 *)

a(n) = A002277(n - 12), for n >= 18.

cp's OEIS Frontend

A332134 a(n) = (10^(2n+1)-1)/3 + 10^n.

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A332138 a(n) = (10^(2*n+1)-1)/3 + 5*10^n.

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A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

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A271528 a(n) = 2*(10^n - 1)^2/27.

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A274986 Numbers k such that (10^k + 23)/3 is prime.

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A332135 a(n) = (10^(2n+1)-1)/3 + 2*10^n.

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A332136 a(n) = 3*(10^(2n+1)-1)/9 + 3*10^n.

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A332138 a(n) = (10^(2n+1)-1)/3 + 510^n.

A332136 a(n) = 3(10^(2n+1)-1)/9 + 310^n.