A000533 -id:A000533 - cp's OEIS frontend

A101849 Indices of primes in sequence defined by A(0) = 37, A(n) = 10*A(n-1) + 27 for n > 0.

0, 1, 13, 19, 29, 43, 65, 259, 871, 8845, 26743, 57505, 98471, 106891

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 20 2004

Numbers n such that (360*10^n - 27)/9 is prime.
Numbers n such that digit 3 followed by n >= 0 occurrences of digit 9 followed by digit 7 is prime.
Numbers corresponding to terms <= 871 are certified primes.
a(14) > 10^5. - Robert Price, Mar 17 2015.
a(15) > 2*10^5. - Robert Price, Oct 02 2015

			397 is prime, hence 1 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 399...997.
Index entries for primes involving repunits.

Cf. A000533, A002275, A101155, A169830, A101398.

Select[Range[0, 1000], PrimeQ[(360*10^# - 27)/9] &] (* Robert Price, Mar 17 2015 *)

a=37;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+27)

for(n=0,1500,if(isprime((360*10^n-27)/9),print1(n,",")))

a(n) = A101398(n) - 1.

8845 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
a(11)-a(13) derived from A101398 by Robert Price, Mar 17 2015
a(14) from Robert Price, Oct 02 2015

A101951 Indices of primes in sequence defined by A(0) = 23, A(n) = 10*A(n-1) - 27 for n > 0.

Original entry on oeis.org

0, 2, 4, 5, 6, 11, 15, 16, 21, 23, 34, 114, 119, 357, 1487, 1818, 4678, 9820, 27216, 27692, 194412

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 23 2004

Numbers n such that 20*10^n + 3 is prime.
Numbers n such that digit 2 followed by n >= 0 occurrences of digit 0 followed by digit 3 is prime.
Numbers corresponding to terms <= 357 are certified primes.
a(21) > 10^5. - Robert Price, Nov 16 2014
a(22) > 2*10^5. - Robert Price, Jul 11 2015

			2003 is prime, hence 2 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 200...003.

Cf. A000533, A002275, A081677.

[n: n in [0..500] | IsPrime(20*10^n+3)]; // Vincenzo Librandi, Nov 17 2014

Select[Range[0, 1000], PrimeQ[(20 10^# + 3)] &] (* Vincenzo Librandi, Nov 17 2014 *)

a=23;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-27)

for(n=0,1500,if(isprime(20*10^n+3),print1(n,",")))

a(n) = A081677(n+1) - 1. - Robert Price, Nov 16 2014

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
a(19)-a(20) derived from A081677 by Robert Price, Nov 16 2014
a(21) from Robert Price, Jul 11 2015

A256537 First differences of corner sequence A256536 associated with A151723.

Original entry on oeis.org

1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65, 9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129

Offset: 1

Omar E. Pol, Apr 02 2015

Number of cells turned ON at n-th stage in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.

For an animation see "The movie version" in Links section.

			Written as an irregular triangle in which the row lengths are the absolute values of the terms of A141531, the sequence begins:
  1;
  3;
  5;
  9, 9;
  9, 17, 25, 17;
  9, 17, 29, 37, 33, 41, 57, 33;
  9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65;
  9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129;
  ...
It appears that the right border gives A083318, whose representation in base 2 gives A000533.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000533, A083318, A141531, A151723, A151724, A169779, A170898, A256536.

a(1) = 1; a(2) = 3.
It appears that a(n) = 1 + (A151724(n) + A151724(n-1))/3, n >= 3.
It appears that a(n) = 1 + (A151723(n) - A151723(n-2))/3, n >= 3.
It appears that a(n) = 1 + 2*(A170898(n-2) + A170898(n-3)), n >= 3.
a(3) = 5.
It appears that a(n) = 1 + 2*(A169779(n-2) - A169779(n-4)), n >= 4.

A330135 a(n) = ((10^(n+1))^4 - 1)/9999 for n >= 0.

Original entry on oeis.org

1, 10001, 100010001, 1000100010001, 10001000100010001, 100010001000100010001, 1000100010001000100010001, 10001000100010001000100010001, 100010001000100010001000100010001

Offset: 0

Bernard Schott, Dec 02 2019

This sequence was the subject of the 6th problem of the 15th British Mathematical Olympiad in 1979 (see the link BMO).
There are no prime numbers in this infinite sequence. Why?
a(0) = 1 and a(1) = 10001 = 73 * 137;
if n even = 2*k, k >= 1, then A094028(n) divides a(n);
if n odd = 2*k+1, k >= 1, then a(k) divides a(n).

			a(2) = ((10^3)^4 - 1)/9999 = 100010001 = 10101 * 9901 where 10101 = A094028(2).
a(3) = ((10^4)^4 - 1)/9999 = 1000100010001 = 10001 * 100000001 where 10001 = a(1).
From _Omar E. Pol_, Dec 04 2019: (Start)
Illustration of initial terms:
                  1;
                10001;
              100010001;
            1000100010001;
          10001000100010001;
        100010001000100010001;
      1000100010001000100010001;
    10001000100010001000100010001;
  100010001000100010001000100010001;
...
(End)

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 6 pp. 68 and 201 (1979).

Colin Barker, Table of n, a(n) for n = 0..200
British Mathematical Olympiad, 1979 - Problem 6.
Index entries for linear recurrences with constant coefficients, signature (10001,-10000).

Cf. A000533 (1000...0001), A094028 (10101...101), A261544 (1001001...1001).

Cf. A131865 (similar, with 2^(n+1)).

Maple
```
A: = seq((10^(4*n+4)-1)/9999, n=1..4);
```

Table[((10^(n+1))^4 - 1)/9999, {n, 0, 8}] (* Amiram Eldar, Dec 04 2019 *)

Vec(1 / ((1 - x)*(1 - 10000*x)) + O(x^11)) \\ Colin Barker, Dec 05 2019

a(n) = (10^(4*n+4) - 1)/9999 for n >= 0.

G.f.: 1 / ((1 - x)*(1 - 10000*x)). - Colin Barker, Dec 05 2019

A056244 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 21 for n > 0.

Original entry on oeis.org

0, 1, 3, 5, 93, 159, 359, 1469, 2897, 3093, 3111, 15697, 17955, 42261, 111031

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (120*10^n - 21)/9 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 3 followed by digit 1 is prime.
Numbers corresponding to terms <= 3111 are certified primes. For larger numbers see P. De Geest, PDP Reference Table.

			131 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 - 131.
Makoto Kamada, Prime numbers of the form 133...331.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068645, A082697.

Do[If[PrimeQ[(1*10^n + 3*(10^n - 1)/9)*10 + 1], Print[n]], {n, 1, 2500}]
Select[Range[0, 2000], PrimeQ[(120 10^# - 21) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=11;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+21)

for(n=0,1500,if(isprime((120*10^n-21)/9),print1(n,",")))

a(n) = A082697(n-1) - 2 for n > 1.

More terms and additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004
Edited by N. J. A. Sloane, Jun 15 2007
Updates from De Geest site by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(15)=111031 from Ray Chandler, Apr 14 2011
Updated comments section and a link, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056246 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 41 for n > 0.

Original entry on oeis.org

0, 1, 3, 19, 31, 399, 561, 7015, 37683

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (140*10^n - 41)/9 is a prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 5 followed by digit 1 is a prime.
Numbers corresponding to terms <= 561 are certified primes.

			151 is a 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 - 151.
Makoto Kamada, Prime numbers of the form 155...551.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068646, A082699.

Select[Range[0, 2000], PrimeQ[(140 10^# - 41) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=11;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+41)

for(n=0,1500,if(isprime((140*10^n-41)/9),print1(n,",")))

a(n) = A082699(n-1) - 2 for n > 1.

Additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004
Edited by N. J. A. Sloane, Jun 15 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Added and updated a link, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056247 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 51 for n > 0.

Original entry on oeis.org

0, 3, 11, 15, 17, 35, 51, 71, 99, 6231, 24027, 40221, 66393

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (150*10^n - 51)/9 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 6 followed by digit 1 is prime.
Numbers corresponding to terms <= 99 are certified primes.

			16661 is prime, hence 3 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 - 161.
Makoto Kamada, Prime numbers of the form 166...661.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068647, A082700.

Select[Range[0, 2000], PrimeQ[(150 10^# - 51) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=11;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+51)

for(n=0,1500,if(isprime((150*10^n-51)/9),print1(n,",")))

a(n) = A082700(n-1) - 2 for n > 1.

Additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004
Edited by N. J. A. Sloane, Jun 15 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Added one more term from the PDP table and a link, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056249 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 71 for n > 0.

Original entry on oeis.org

0, 1, 7, 13, 39, 91, 127, 883, 9423, 14767, 19257, 31233

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (170*10^n - 71)/9 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 8 followed by digit 1 is prime.
Numbers corresponding to terms <= 883 are certified primes.

			181 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 - 181.
Makoto Kamada, Prime numbers of the form 188...881.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068648, A082702.

Select[Range[0, 2000], PrimeQ[(170 10^# - 71) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=11;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+71)

for(n=0,1500,if(isprime((170*10^n-71)/9),print1(n,",")))

a(n) = A082702(n-1) - 2 for n > 1.

Additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004
Edited by N. J. A. Sloane, Jun 15 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Updated and added a link, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056250 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 81 for n > 0.

Original entry on oeis.org

0, 1, 3, 7, 39, 85, 199, 729, 1459, 23671, 28629

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (180*10^n - 81)/9 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 9 followed by digit 1 is prime.
Numbers corresponding to terms <= 1459 are certified primes. For larger numbers see P. De Geest, PDP Reference Table.

			191 is prime, hence 1 is a term.

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

Tony Crilly, A family of discrete spirals, The Mathematical Gazette (2020) Vol. 104, Issue 560, 215-224.
Patrick De Geest, PDP Reference Table - 191.
Makoto Kamada, Prime numbers of the form 199...991.
Index entries for primes involving repunits.

Cf. A000533, A002275, A068649, A082703.

Select[Range[0, 2000], PrimeQ[(180 10^# - 81) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=11;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a+81)

for(n=0,1500,if(isprime((180*10^n-81)/9),print1(n,",")))

a(n) = A082703(n-1) - 2 for n > 1.

Additional comments from Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004
Edited by N. J. A. Sloane, Jun 15 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Updated comments section and a link, by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 04 2014

A056252 Indices of primes in sequence defined by A(0) = 33, A(n) = 10*A(n-1) - 7 for n > 0.

Original entry on oeis.org

5, 7, 893, 1523, 3035, 21155

Offset: 1

Robert G. Wilson v, Aug 18 2000

Numbers n such that (290*10^n + 7)/9 is prime.
Numbers n such that the digit 3 followed by n >= 0 occurrences of the digit 2 followed by the digit 3 is prime.
Numbers corresponding to terms <= 3035 are certified primes.

			3222223 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 - 323.
Makoto Kamada, Prime numbers of the form 322...223.
Index entries for primes involving repunits.

Cf. A000533, A002275, A082705.

Select[Range[0, 2000], PrimeQ[(290 10^# + 7) / 9] &] (* Vincenzo Librandi, Nov 03 2014 *)

a=33;for(n=0,1600,if(isprime(a),print1(n,","));a=10*a-7)

for(n=0,1600,if(isprime((290*10^n+7)/9),print1(n,",")))

a(n) = A082705(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, Apr 17 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Comments section updated and a link added by Patrick De Geest, Nov 02 2014
Edited by Ray Chandler, Nov 05 2014

cp's OEIS Frontend

A101849 Indices of primes in sequence defined by A(0) = 37, A(n) = 10*A(n-1) + 27 for n > 0.

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A101951 Indices of primes in sequence defined by A(0) = 23, A(n) = 10*A(n-1) - 27 for n > 0.

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A256537 First differences of corner sequence A256536 associated with A151723.

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A330135 a(n) = ((10^(n+1))^4 - 1)/9999 for n >= 0.

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Maple

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A056244 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 21 for n > 0.

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Programs

Mathematica

PARI

PARI

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A056246 Indices of primes in sequence defined by A(0) = 11, A(n) = 10*A(n-1) + 41 for n > 0.

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