A000533 -id:A000533 - cp's OEIS frontend

A199684 a(n) = 4*10^n+1.

5, 41, 401, 4001, 40001, 400001, 4000001, 40000001, 400000001, 4000000001, 40000000001, 400000000001, 4000000000001, 40000000000001, 400000000000001, 4000000000000001, 40000000000000001, 400000000000000001, 4000000000000000001

Offset: 0

Vincenzo Librandi, Nov 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533, A011557.

Magma
```
[4*10^n+1: n in [0..30]];
```

a(n) = 10*a(n-1)-9.
a(n) = 11*a(n-1)-10*a(n-2).
G.f.: (5-14*x)/((1-x)*(1-10*x)).
E.g.f.: exp(x) + 4*exp(10*x). - Stefano Spezia, Oct 20 2022

A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

Original entry on oeis.org

0, 3, 4, 8, 36, 8, 5, 72, 28, 6, 79, 212, 23, 6, 73, 24, 52, 62, 3, 28, 220, 53, 75, 58, 228, 9, 265, 89, 214, 86, 215, 4, 7, 39, 295, 40, 87, 216, 97, 6, 264, 53, 287, 223, 4, 239, 259, 25, 57, 364, 49, 38, 93, 86, 27, 30, 80, 24, 6, 356, 50, 645, 395, 206

Offset: 1

Felix Fröhlich, May 28 2015

n = 1 is the only case where a(n) = 0, since for any n > 1, A138148(n) is divisible by A002275(n).
No n exists such that a(n) = 2, since any number of the form A100706(n)+A011557(n) is of the form A000533(n)*A002275(n+1) (see comment by Robert Israel in A107123).
a(n) = 3 iff n is in A107123.
a(n) = 4 iff n is in A107124.
If k has an even number of digits and is a multiple of 11, then k is not a term. If k = (10^r+1)(10^m-1)/9 for some m > 0, r >= 0, then k is not a term. If A272232(k) = 0, then k is not a term. - Chai Wah Wu, Nov 08 2019

			a(1) = 0, because 101 is prime.
a(5) = 36, because the smallest x >= 0 such that 11111_x_11111 (where '_' denotes concatenation) is prime is 36. The decimal expansion of that prime is 111113611111.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A088281, A090287, A272232.

Table[k = 0; s = Table[1, {n}]; While[Or[!PrimeQ[FromDigits[s ~Join~ IntegerDigits[k] ~Join~ s]], Or[First@ IntegerDigits@ k == 1, Last@ IntegerDigits@ k == 1]], k++]; k, {n, 64}] (* Michael De Vlieger, May 28 2015 *)

a000042(n) = (10^n-1)/9
a(n) = my(k=0); while(k==10 || k%10==1 || k\(10^(#Str(k)-1))==1 || !ispseudoprime(eval(Str(a000042(n), k, a000042(n)))), k++); k

A261544 a(n) = Sum_{k=0..n} 1000^k.

Original entry on oeis.org

1, 1001, 1001001, 1001001001, 1001001001001, 1001001001001001, 1001001001001001001, 1001001001001001001001, 1001001001001001001001001, 1001001001001001001001001001, 1001001001001001001001001001001, 1001001001001001001001001001001001

Offset: 0

Ilya Gutkovskiy, Aug 24 2015

A sequence of palindromic numbers.

			From _Bruno Berselli_, Aug 25 2015: (Start)
a(n)   is the binary representation of    A023001
-------------------------------------------------
1  ...........................................  1
1001  ........................................  9
1001001 .....................................  73
1001001001  ................................  585
1001001001001  ............................  4681
1001001001001001  ........................  37449
1001001001001001001  ....................  299593
1001001001001001001001  ................  2396745
1001001001001001001001001  ............  19173961, etc.
(End)

Colin Barker, Table of n, a(n) for n = 0..333 (corrected by Michel Marcus, Jan 19 2019)
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
Eric Weisstein's World of Mathematics, Palindromic Number.
Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Cf. A000533, A002113, A023001.
Subsequence of A033146.
Sums of 100^k: A094028; sums of 10^k: A000042.
Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

[(1000^(n+1)-1)/999: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* Vincenzo Librandi, Aug 24 2015 *)

Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ Colin Barker, Aug 24 2015

a(n) = (1000^(n + 1) - 1)/999.
a(n) = 1001*a(n-1) - 1000*a(n-2). - Colin Barker, Aug 24 2015
G.f.: 1 / ((x-1)*(1000*x-1)). - Colin Barker, Aug 24 2015
E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - G. C. Greubel, Aug 29 2015

A348487 Positive numbers whose square starts and ends with exactly one 1.

Original entry on oeis.org

1, 11, 39, 41, 101, 111, 119, 121, 129, 131, 139, 141, 319, 321, 329, 331, 349, 351, 359, 361, 369, 371, 379, 381, 389, 391, 399, 401, 409, 411, 419, 421, 429, 431, 439, 441, 1001, 1009, 1011, 1019, 1021, 1029, 1031, 1039, 1041, 1099, 1101, 1109, 1111, 1119, 1121, 1129, 1131, 1139

Offset: 1

Bernard Schott, Oct 21 2021

When a square ends with 1, this square ends with exactly one 1.

Sequences A000533 and A253213 show that there are an infinity of terms. The square of their terms, for n >= 3, starts and ends with exactly one 1. Also, the numbers 119, 1119, 11119, ..., ((10^k + 71) / 9)^2, (k >= 3) are terms. The squares ((10^k + 71) / 9)^2, have the last digit 1 and because 12*10^(2*k - 3) < ((10^k + 71) / 9)^2 <13*10^(2*k - 3), for k >= 3, the squares ((10^k + 71) / 9)^2, k >= 4, start with 12. - Marius A. Burtea, Oct 21 2021

			39 is a term since 39^2 = 1521.
109 is not a term since 109^2 = 11881.
119 is a term since 119^2 = 14161.

Cf. A045855, A090771, A253213, A273372 (squares ending with 1), A017281, A017377.
Cf. A000533, A253213 for n >= 2 (subsequences).
Subsequence of A305719.

[1] cat [n:n in [2..1200]|Intseq(n*n)[1] eq 1 and Intseq(n*n)[#Intseq(n*n)] eq 1 and Intseq(n*n)[-1+#Intseq(n*n)] ne 1]; // Marius A. Burtea, Oct 21 2021

Join[{1}, Select[Range[11, 1200], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 1 && d[[2]] != 1 &]] (* Amiram Eldar, Oct 21 2021 *)

isok(k) = my(d=digits(sqr(k))); (d[1]==1) && (d[#d]==1) && if (#d>2, (d[2]!=1) && (d[#d-1]!=1), 1); \\ Michel Marcus, Oct 21 2021

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("1")) == len(s.lstrip("1")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [1, 9]))
  return [k for k in r if ok(k)]
print(aupto(1140)) # Michael S. Branicky, Oct 21 2021

A052035 Palindromic primes whose sum of squared digits is also prime.

Original entry on oeis.org

11, 101, 131, 191, 313, 353, 373, 797, 919, 10301, 11311, 12721, 13331, 13931, 14341, 14741, 16361, 17971, 18181, 19391, 30103, 30703, 33533, 71317, 71917, 74747, 75557, 76367, 77977, 79397, 90709, 93139, 93739, 95959, 96769, 97379

Offset: 1

Patrick De Geest, Dec 15 1999

From Bernard Schott, Oct 20 2021: (Start)
Except for 11, all terms have an odd number of digits.
Except for terms of the form 10^k+1, k >= 2, the middle digit is always odd; the unique known term of the form 10^k+1 for 2 <= k <= 100000 is 101 (see comment in A000533). (End)

			373 -> 3^2 + 7^2 + 3^2 = 67, which is prime.

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Michel Marcus, Table of n, a(n) for n = 1..1215
Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Cf. A000533, A002385, A052034, A003132.

Select[Prime@ Range[2, 10^4], And[PalindromeQ@ #, PrimeQ@ Total[IntegerDigits[#]^2]] &] (* Michael De Vlieger, Oct 20 2021 *)

isok(p) = my(d=digits(p)); isprime(p) && (d==Vecrev(d)) && isprime(sum(k=1, #d, d[k]^2)); \\ Michel Marcus, Oct 17 2021

from sympy import isprime
def ok(n):
    s = str(n)
    return s==s[::-1] and isprime(n) and isprime(sum(int(d)**2 for d in s))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 23 2021

# second version for going to large terms
from sympy import isprime
from itertools import product
def ok(pal):
    return isprime(pal) and isprime(sum(int(d)**2 for d in str(pal)))
def agentod(maxdigs):
    yield 11
    for d in range(3, maxdigs+1, 2):
        pal = 10**(d-1) + 1
        if ok(pal): yield pal
        for first in "1379":
            for left in product("0123456789", repeat=(d-3)//2):
                left = "".join(left)
                for mid in "13579":
                    pal = int(first + left + mid + left[::-1] + first)
                    if ok(pal): yield pal
print([an for an in agentod(5)]) # Michael S. Branicky, Nov 23 2021

A056251 Indices of primes in sequence defined by A(0) = 33, A(n) = 10*A(n-1) - 17 for n > 0.

Original entry on oeis.org

1, 11, 13, 29, 103, 125, 341, 599, 9823

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (280*10^n + 17)/9 is prime.
Numbers n such that digit 3 followed by n >= 0 occurrences of digit 1 followed by digit 3 is prime.
Numbers corresponding to terms <= 599 are certified primes.
All terms are odd since 11 is the only palindromic prime with an even number of digits. - Chai Wah Wu, Nov 05 2019
a(10) > 2*10^5. - Tyler Busby, Feb 01 2023

			313 is prime, hence 1 is a term.

Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Patrick De Geest, PDP Reference Table - 313.
Makoto Kamada, Prime numbers of the form 311...113.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068650, A082704.

Flatten[Position[NestList[10#-17&,33,600],?PrimeQ]-1] (* This generates the first 8 terms of the sequence. Changing the last constant from 600 to 9825 will generate all 9 terms of the sequence but it will take a long time to do so. - _Harvey P. Dale, May 16 2012 *)
Select[Range[0, 2000], PrimeQ[(280 10^# + 17) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=33;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-17)

for(n=0,1500,if(isprime((280*10^n+17)/9),print1(n,",")))

a(n) = A082704(n) - 2.

Additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 20 2004
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Added and updated the links section, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056265 Indices of primes in sequence defined by A(0) = 99, A(n) = 10*A(n-1) - 61 for n > 0.

Original entry on oeis.org

1, 5, 11, 109, 3607, 37783

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (830*10^n + 61)/9 is prime. - Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Nov 27 2004
Numbers n such that digit 9 followed by n >= 0 occurrences of digit 2 followed by digit 9 is prime.
Numbers corresponding to terms <= 3607 are certified primes. For number corresponding to 37783 see P. De Geest, PDP Reference Table.

			9222229 is prime, hence 5 is a term.

Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Patrick De Geest, PDP Reference Table - 929.
Makoto Kamada, Prime numbers of the form 922...229.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068651, A082718.

[n: n in [0..1000] | IsPrime((830*10^n + 61) div 9)]; // Vincenzo Librandi, Nov 02 2014

Do[If[PrimeQ[(9*10^n + 2*(10^n - 1)/9)*10 + 9], Print[n]], {n, 1, 2500}]
Select[Range[2000], PrimeQ[(830 10^# + 61) / 9] &] (* Vincenzo Librandi, Nov 02 2014 *)

a=99;for(n=0,1000,if(isprime(a),print1(n,","));a=10*a-61) \\ Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Nov 27 2004

for(n=0,1000,if(isprime((830*10^n + 61)/9),print1(n,","))) \\ Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Nov 27 2004

a(n) = A082718(n) - 2.

3607 from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Nov 27 2004
37783 from Patrick De Geest, Jun 26 2005
Edited by N. J. A. Sloane, Jan 14 2008
Edited by Ray Chandler, Oct 20 2010
Comments section edited by Patrick De Geest, Nov 02 2014
Editied by Ray Chandler, Nov 05 2014

A056810 Numbers whose fourth power is a palindrome.

Original entry on oeis.org

0, 1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001

Offset: 1

Robert G. Wilson v, Aug 21 2000

Suppose a number is of the form a=10...01 (see A000533) then a^2=10..020..01, so a^2 is always a palindrome. a^3=10..030..030..01, so a^3 is always a palindrome. Similarly we also have a^4=10..040..060..040..01, so a^4 is always a palindrome. However, a^5 is in general not a palindrome, for example 101^5=10510100501. - Dmitry Kamenetsky, Apr 17 2009
The sequence contains no term with digit sum 3. - Vladimir Shevelev, May 23 2011.  Proof: There are four possibilities for n:
1) 1+10^k+10^m, 00, 3) 2+10^s, s>0, 4) 3*10^t, t>=0.
In the last two cases n^4 is trivially not a palindrome.
For r>=2, in the second case we have n^4 = (1 + 2*10^r)^4 = 1 + 8*10^r + 4*10^(2*r) + 2*10^(2*r + 1) + 2*10^(3*r) + 3*10^(3*r + 1) + 6*10^(4*r) + 10^(4*r + 1)
which cannot be a palindrome.
If r=1, we have 1+8*10+...9*10^4+10^5 which also is not a palindrome.
The proof for the first case is similar. QED - Vladimir Shevelev, Oct 24 2015
Does every term have the structure 100...0001? Referring to the Simmons (1972) paper, we can also ask, if n is a number whose cube is a palindrome in base 4, must the base-4 expansion of n have the form 100...0001? - N. J. A. Sloane, Oct 22 2015

G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A186080.

palQ[n_] := Block[{}, Reverse[idn = IntegerDigits@ n] == idn]; k = 0; lst = {}; While[k < 1000000002, If[ palQ[k^4], AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Oct 23 2015 *)

def ispal(n): s = str(n); return s == s[::-1]
def afind(limit):
    for k in range(limit+1):
        if ispal(k**4): print(k, end=", ")
afind(10000001) # Michael S. Branicky, Sep 05 2021

a(11) from Robert G. Wilson v, Oct 23 2015

a(12)-a(13) from Michael S. Branicky, Sep 05 2021

A088265 Numbers of the form 10^k + 1, 3, 7, or 9 for k>=1.

Original entry on oeis.org

11, 13, 17, 19, 101, 103, 107, 109, 1001, 1003, 1007, 1009, 10001, 10003, 10007, 10009, 100001, 100003, 100007, 100009, 1000001, 1000003, 1000007, 1000009, 10000001, 10000003, 10000007, 10000009, 100000001, 100000003, 100000007, 100000009, 1000000001, 1000000003, 1000000007, 1000000009

Offset: 1

Amarnath Murthy, Sep 28 2003

A 10-automatic sequence: terms match the regular expression 10*[1379]. - Charles R Greathouse IV, Oct 15 2012

Primes in the sequence are at positions 1 to 8, 12, 15, 16, 18, 22, 31, 35, 36, 42, 66, 70, 72, 88, 95,... R. J. Mathar, Oct 16 2012

Index entries for 10-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,11,0,0,0,-10).

Cf. A088266, A000533.

A088265 := proc(n)
    if n <=8 then
        op(n,[11,13,17,19,101,103,107,109]) ;
    else
        11*procname(n-4)-10*procname(n-8) ;
    end if;
end proc: # R. J. Mathar, Oct 16 2012

Flatten @ Table[10^n + m, {n, 50}, {m, {1, 3, 7, 9}}] (* Mikk Heidemaa, Mar 06 2020 *)

A088265(n)=10^((n+3)\4)+[9,1,3,7][n%4+1]  \\ M. F. Hasler, Oct 15 2012

a(4n+r) = 10^(n+1) + k, (r, k) = (1, 1), (2, 3), (3, 7) or (4, 9).

G.f. -x*(-11-13*x-17*x^2-19*x^3+20*x^4+40*x^5+80*x^6+100*x^7) / ( (x-1)*(1+x)*(1+x^2)*(10*x^4-1) ). a(n)= 11*a(n-4) -10*a(n-8). - R. J. Mathar, Oct 16 2012

More terms from Max Alekseyev, Oct 14 2012

A101155 Indices of primes in sequence defined by A(0) = 73, A(n) = 10*A(n-1) + 63 for n > 0.

Original entry on oeis.org

0, 2, 4, 5, 9, 11, 12, 38, 47, 53, 63, 81, 146, 147, 359, 398, 1637, 1875, 2145, 2193, 15788, 23073, 38465, 68399

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 03 2004

Numbers n such that (720*10^n - 63)/9 is prime.
Numbers n such that digit 7 followed by n >= 0 occurrences of digit 9 followed by digit 3 is prime.
Numbers corresponding to terms <= 398 are certified primes.
a(25) > 2*10^5. - Robert Price, Nov 11 2015

			7999993 is prime, hence 5 is a term.

Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Makoto Kamada, Prime numbers of the form 799...993.
Index entries for primes involving repunits.

Cf. A000533, A002275, A101849, A169830, A099190.

Select[Range[0, 100000], PrimeQ[(720*10^# - 63)/9] &] (* Robert Price, Nov 11 2015 *)

a=73;for(n=0,1000,if(isprime(a),print1(n,","));a=10*a+63)

for(n=0,1000,if(isprime((720*10^n-63)/9),print1(n,",")))

a(n) = A099190(n) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

a(21)-a(24) from Kamada data by Ray Chandler, Apr 30 2015

cp's OEIS Frontend

A199684 a(n) = 4*10^n+1.

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A258372 Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.

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A261544 a(n) = Sum_{k=0..n} 1000^k.

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A348487 Positive numbers whose square starts and ends with exactly one 1.

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A052035 Palindromic primes whose sum of squared digits is also prime.

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A056251 Indices of primes in sequence defined by A(0) = 33, A(n) = 10*A(n-1) - 17 for n > 0.

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A056265 Indices of primes in sequence defined by A(0) = 99, A(n) = 10*A(n-1) - 61 for n > 0.

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