A161786 -id:A161786 - cp's OEIS frontend

A161796 Primes with at least one digit appearing exactly five times in the decimal expansion.

101111, 111119, 111121, 111191, 111211, 111611, 112111, 113111, 131111, 199999, 311111, 313333, 323333, 331333, 333233, 333323, 333331, 333337, 333383, 333433, 333533, 334333, 343333, 353333, 444443, 444449, 511111, 555557, 599999, 611111

Offset: 1

Ki Punches, Jun 19 2009

The sequence is probably infinite.

Cf. A105992, A161786.

isdgctm := proc(n,d) local dgs,a,i ; dgs := convert(n,base,10) ; a := [seq(0,j=0..9)] ;
for i in dgs do a := subsop(i+1=op(i+1,a)+1,a) ; od: if convert(a,set) intersect {d} <> {} then true; else false; fi; end:
for n from 1 to 100000 do p := ithprime(n) ; if isdgctm(p,5) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jun 21 2009

Edited and corrected by R. J. Mathar, Jun 21 2009

A165508 Numbers k such that 10^k + 111 is prime.

Original entry on oeis.org

2, 4, 184, 460, 784, 3248, 5194, 92386, 156428, 228208

Offset: 1

Rick L. Shepherd, Sep 21 2009

Terms must be congruent to 2 or 4 mod 6. Other than the first term, which produces 10^2 + 111 = 211, these terms produce primes whose decimal representation is 1  111 concatenated. These are only known to be highly probable primes for 184 and beyond. No more terms up to 15000.
a(8) > 55000. - Tyler NeSmith, Jul 10 2021
The corresponding primes have digit sum 4 (A062339). - Jeppe Stig Nielsen, Feb 10 2023
a(9) > 10^5. - Jeppe Stig Nielsen, Feb 11 2023
a(11) > 6.6*10^5. - Boyan Hu, Nov 14 2024

			As 10111 = 10^4 + 111 is a prime, 4 is a term.

Cf. A096912, A157711, A062339, A020449, A036929, A161786.

Select[Range[5200], PrimeQ[10^# + 111] &] (* G. C. Greubel, Oct 21 2018 *)

is(k)=ispseudoprime(10^k+111) \\ Charles R Greathouse IV, Jun 13 2017

a(8) from Jeppe Stig Nielsen, Feb 10 2023

a(9)-a(10) from Boyan Hu, Oct 23 2024

A257937 Primes p such that one digit appears exactly six times together with a single different digit.

Original entry on oeis.org

1111151, 1111181, 1111211, 1111711, 1114111, 1117111, 1171111, 2999999, 3233333, 3331333, 3333133, 3333233, 3333313, 3333331, 3333373, 3333383, 3333433, 3337333, 3353333, 3433333, 3733333, 4999999, 7477777, 7577777, 7727777, 7772777, 7774777, 7777727, 7778777, 7877777, 9899999, 9929999, 9999299, 9999929, 9999991

Offset: 1

K. D. Bajpai, Jul 13 2015

All the terms are congruent to 1 or 2 (mod 3).

In no term does the digit 0, 2, 4, 5, 6, or 8 appear six times.

			a(1) = 1111151 has exactly six 1's together with a single digit 5.
a(8) = 2999999 has exactly six 9's together with a single digit 2.

K. D. Bajpai, Table of n, a(n) for n = 1..35

Cf. A105992, A161786, A161796.

sort(select(isprime, [seq(seq(seq(d*1111111 + (a-d)*10^k, k=0..6), a={$1..9} minus {d}),d=1..9)])); # Robert Israel, Jul 13 2015

kQ[n_]:= Module[{d=Select[DigitCount[n], # > 0 &]},Length[d] == 2 && Min[d] == 1 && Max[d] == 6]; Select[Table[Prime[n], {n, 1000000}], kQ]
Select[Prime[Range[80000, 400000]], MemberQ[DigitCount[#], 6] &] (* Vincenzo Librandi, Jul 14 2015 *)

A161848 Primes with at least one digit appearing exactly three times in the decimal expansion.

Original entry on oeis.org

1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 2333, 2777, 2999, 3313, 3323, 3331, 3343, 3373, 3433, 3533, 3733, 3833, 4111, 4441, 4447, 4999, 5333, 5557, 6661, 7177, 7333, 7477, 7577, 7717, 7727, 7757, 7877, 8111, 8887, 8999, 9199, 9929

Offset: 1

Ki Punches, Jun 20 2009

Sequence is probably infinite.

			2333, 3313, 3833 all repeat some digit 3 three times.

Cf. A161786, A161796.

isdgctm := proc(n,d) local dgs,a,i ; dgs := convert(n,base,10) ; a := [seq(0,j=0..9)] ;
for i in dgs do a := subsop(i+1=op(i+1,a)+1,a) ; od: if convert(a,set) intersect {d} <> {} then true; else false; fi; end:
for n from 1 to 2000 do p := ithprime(n) ; if isdgctm(p,3) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jun 21 2009

Edited and corrected by R. J. Mathar, Jun 21 2009

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A161796 Primes with at least one digit appearing exactly five times in the decimal expansion.

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A165508 Numbers k such that 10^k + 111 is prime.

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A257937 Primes p such that one digit appears exactly six times together with a single different digit.

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A161848 Primes with at least one digit appearing exactly three times in the decimal expansion.

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