A062800 -id:A062800 - cp's OEIS frontend

A121172 Smallest integer k>0 such that k*10^n + 1 is a prime.

1, 1, 3, 7, 7, 22, 3, 6, 6, 3, 19, 18, 4, 39, 6, 13, 37, 15, 15, 6, 16, 9, 96, 61, 19, 6, 9, 3, 33, 63, 57, 117, 55, 49, 30, 3, 28, 6, 42, 24, 36, 72, 21, 60, 6, 24, 33, 61, 21, 85, 31, 49, 13, 93, 18, 90, 9, 16, 135, 19, 55, 9, 135, 60, 6, 30, 3, 16, 115, 114, 19, 99, 15, 147, 171, 42

Offset: 1

Alexander Adamchuk, Aug 14 2006

nonn

			a(1) = 1 because A030430[1] = 11 is a smallest prime of form 10*k + 1.
a(2) = 1 because A062800[1] = 101 is a smallest prime of form 100*k + 1.

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A030430, A062800, see A070854 for resulting primes.

s={};Do[k=0;Until[PrimeQ[k*10^n+1],k++];AppendTo[s,k],{n,76}];s (* James C. McMahon, Oct 13 2024 *)

a(n) = (A070854(n)-1)/10^n. - Ray Chandler, Feb 10 2009

Minor edits by Ray Chandler, Feb 10 2009

A327347 The 54 prime dates of each year of the form concatenate(day,month) with leading zero for months 1, 3, 7, 9 (no leading zero for days).

Original entry on oeis.org

101, 401, 601, 701, 1201, 1301, 1601, 1801, 1901, 2801, 3001, 103, 503, 1103, 1303, 2003, 2203, 2503, 2803, 2903, 107, 307, 607, 907, 1307, 1607, 1907, 2207, 2707, 109, 409, 509, 709, 809, 1009, 1109, 1409, 1609, 1709, 2309, 2609, 2909, 211, 311, 811, 911, 1511, 1811, 2011, 2111, 2311, 2411, 2711, 3011

Offset: 1

Wolfdieter Lang, Sep 30 2019

All these dates come from January, March, July, September and November, sorted this d.m way, with 11, 9, 9, 13 and 12 dates, respectively, summing to 54. Note that all September dates without leading zero of month m = 9 from A327346 survive after inserting the 0. The November dates coincide, of course.

Cf. A062800 (first 11 members), A101780 (9 members, starting with n = 2), A166547 (9 members, starting with n = 2), A166560 (first 13 members), A167442 (12 members, starting with n = 2), respectively.

Cf. A327346 (74 prime dates d.m without leading 0 for month), A327348 (66 prime dates m.d for non-leap years), A327349 (67 prime dates, like A327348 but for leap years), A327914 (58 prime dates m.d for non-leap years, with leading 0 for d = 1..9), A327915 (59 prime dates, like A327914, but for leap years).

Select[Flatten@ Map[Function[{m, d},  Array[FromDigits[IntegerDigits[#]~Join~m] &, d]] @@ {PadLeft[IntegerDigits@ #, 2], Which[MemberQ[{4, 6, 9, 11}, #], 30, # == 2, 28, True, 31]} &, Select[Range[1, 12, 2], CoprimeQ[#, 10] &]], PrimeQ] (* Michael De Vlieger, Oct 03 2019 *)

A095995 Primes of the form 100n - 1.

Original entry on oeis.org

199, 499, 599, 1399, 1499, 1699, 1999, 2099, 2399, 2699, 2999, 3299, 3499, 4099, 4799, 4999, 5099, 5399, 6199, 6299, 6599, 6899, 7499, 7699, 8599, 8699, 8999, 9199, 10099, 10399, 10499, 10799, 11299, 11399, 11699, 12799, 12899, 13099, 13399

Offset: 1

Alonso del Arte, Jul 19 2004

If n is of the form 3x + 1 then 100n - 1 will be of the form 100*3x + 99, that is, a multiple of 3. The factorizations of other nonprime 100n - 1 has a much more complicated pattern.

			a(2)=499 because 499 = 100 * 5 - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062800.

[a: n in [0..135] | IsPrime(a) where a is 100*n-1]; // Vincenzo Librandi, Jul 17 2012

Select[ 100Range[134] - 1, PrimeQ[ # ] &]
Select[Table[100n-1,{n,0,800}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)

Edited by Robert G. Wilson v, Jul 23 2004

A120124 Smallest prime p such that p*10^n + 1 is a prime.

Original entry on oeis.org

3, 7, 3, 7, 7, 61, 3, 7, 7, 3, 19, 37, 109, 79, 97, 13, 37, 19, 73, 103, 97, 283, 157, 61, 19, 61, 1213, 3, 163, 691, 367, 163, 181, 157, 241, 3, 103, 733, 151, 283, 337, 193, 211, 163, 7, 73, 307, 61, 223, 1549, 31, 127, 13, 547, 103, 151, 193, 811, 337, 19, 1021, 151

Offset: 1

Alexander Adamchuk, Aug 15 2006

nonn

All terms belong to A007645. All terms also belong to A055664. Also many terms including the first 14 smallest primes from 3 to 139 {3,7,13,19,31,37,43,61,73,79,97,103,127,139} belong tpA023203. The smallest term that differs from A023203 is 151.

			a(1) = 3 because 31 = 3*10 + 1 is the smallest prime of form p*10 + 1, where p is a prime.
a(2) = 7 because 701 = 7*100 + 1 is the smallest prime of form p*100 + 1.

Robert Israel, Table of n, a(n) for n = 1..1000 (first 300 terms from _Vincenzo Librandi_)

Cf. A121172, A030430, A062800, A007645, A055664, A023203.

Primes:= select(isprime,[$1..10^5]):
for n from 1 to 1000 do
   for p in Primes do
      if isprime(p*10^n+1) then
        A[n]:= p
      fi
    od
od:
seq(A[n],n=1..1000); # Robert Israel, May 29 2014

prs=Prime[Range[2000]];Table[i=1;While[!PrimeQ[First[Take[prs,{i}]] 10^n+1],i++];Prime[i],{n,200}] (* Harvey P. Dale, May 15 2011 *)

A120642 Smallest integer k>0 such that k*10^n - 1 is a prime.

Original entry on oeis.org

2, 2, 2, 5, 2, 3, 2, 8, 11, 6, 11, 35, 6, 5, 15, 14, 11, 21, 3, 21, 14, 6, 6, 80, 8, 2, 2, 6, 9, 48, 48, 21, 15, 6, 44, 11, 9, 15, 18, 6, 33, 30, 3, 278, 74, 92, 89, 33, 8, 71, 59, 11, 2, 5, 3, 24, 108, 47, 39, 41, 6, 14, 53, 173, 26, 26, 51, 114, 23, 17, 246, 44, 6, 131, 56, 8, 26, 77, 74

Offset: 1

Jonathan Vos Post, Aug 17 2006

			The primes are 19, 199, 1999, 49999, 199999, 2999999,
19999999, 799999999, 10999999999, 59999999999, ...,.

Pierre CAMI, Table of n,a(n) for n=1,...,3000

Cf. A030430, A037071, A062800, A065582, A121172.

f[n_] := Block[{k = 1}, While[ !PrimeQ[k*10^n - 1], k++ ]; k]; Array[f, 79] (* Robert G. Wilson v *)

a(11) onwards from Robert G. Wilson v, Aug 20 2006

A259084 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 0.

Original entry on oeis.org

86, 68, 58, 35, 41, 14, 27, 44, 10, 14, 16, 16, 9, 10, 8, 7, 14, 16, 14, 8, 6, 9, 4, 23, 8, 0, 14, 10, 12, 10, 6, 14, 5, 8, 5, 13, 7, 16, 7, 17, 6, 3, 9, 9, 16, 7, 12, 11, 4, 13, 7, 16, 8, 9, 3, 10, 4, 9, 6, 4, 5, 13, 3, 12, 7, 9, 6, 8, 4, 39, 13, 12, 10, 4

Offset: 1

N. J. A. Sloane, Jun 18 2015

These values are only conjectural.

a(n) = 0 if prime(n) is in A062800. - Robert Israel, Jun 19 2015

			a(1)=86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A062800, A094776, A259081-A259086.

N:= 100: K:= 100:  # to get a(1) to a(N), searching up to k = K
for n from 1 to N do
  p:= ithprime(n);
  A[n]:= 0;
  for k from 1 to K do
    if not has(convert(p^k,base,10),0) then
       A[n]:= k
    fi
  od
od:
seq(A[n],n=1..N); # Robert Israel, Jun 19 2015

a(14)-a(57) from Hiroaki Yamanouchi, Jun 19 2015

A120729 Smallest integer k>0 such that k*10^n + 1 is a semiprime.

Original entry on oeis.org

3, 2, 2, 5, 1, 1, 1, 1, 1, 2, 4, 2, 3, 7, 4, 3, 6, 6, 4, 1, 2, 4, 13, 2, 4, 3, 7, 21, 6, 9, 3, 1, 5, 4, 16, 19, 28, 19, 9, 3

Offset: 0

Jonathan Vos Post, Aug 17 2006

The corresponding semiprimes are 4, 21, 201, 5001, 10001, 100001, 100001, 10000001, 2000000001, 40000000001, ... Semiprime analog of A121172.

			a(0) = 3 because 3*10^0 + 1 = 4 = 2^2 is a semiprime.
a(1) = 2 because 2*10^1 + 1 = 21 = 3*7 is a semiprime.
a(2) = 2 because 2*10^2 + 1 = 201 = 3*67 is a semiprime.
a(3) = 5 because 5*10^3 + 1 = 5001 = 3*1667 is a semiprime.
a(4) = 1 because 1*10^4 + 1 = 10001 = 73*137 is a semiprime.
a(5) = 1 because 1*10^5 + 1 = 100001 = 11*9091 is a semiprime.

Cf. A001358, A030430, A037071, A062800, A065582, A121172.

sik[n_]:=Module[{k=1,c=10^n},While[PrimeOmega[k*c+1]!=2,k++];k]; Array[sik,40,0] (* Harvey P. Dale, Aug 20 2012 *)

Smallest integer k>0 such that k*10^n + 1 is in A001358.

More terms from Harvey P. Dale, Aug 20 2012

cp's OEIS Frontend

A121172 Smallest integer k>0 such that k*10^n + 1 is a prime.

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A327347 The 54 prime dates of each year of the form concatenate(day,month) with leading zero for months 1, 3, 7, 9 (no leading zero for days).

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A095995 Primes of the form 100n - 1.

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A120124 Smallest prime p such that p*10^n + 1 is a prime.

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Maple

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A120642 Smallest integer k>0 such that k*10^n - 1 is a prime.

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A259084 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 0.

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Maple

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A120729 Smallest integer k>0 such that k*10^n + 1 is a semiprime.

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Mathematica

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Extensions