A159352 -id:A159352 - cp's OEIS frontend

A049054 Numbers k such that 10^k + 3 is prime.

1, 2, 5, 6, 11, 17, 18, 39, 56, 101, 105, 107, 123, 413, 426, 2607, 7668, 10470, 11021, 17753, 26927, 60776, 98288, 300476, 509546

Offset: 1

G. L. Honaker, Jr.

A102006 is another version of the same sequence. - N. J. A. Sloane, Jan 28 2010

Verified existing 21 terms. If another term exists, it is > 39456. - Robert Price, Aug 16 2010

			5 is a term since 10^5 + 3 = 100003 is a prime.
6 is a term since 10^6 + 3 = 1000003 is a prime.

Makoto Kamada, Prime numbers of the form 100...003.
Henri & Renaud Lifchitz, PRP Records.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A102006, A159352.

Do[ If[ PrimeQ[ 10^n + 3], Print[n]], {n, 0, 18000}] (* Robert G. Wilson v Jun 15 2002 *)

is(n)=isprime(10^n + 3) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A102006(n) + 1.

More terms from Robert G. Wilson v, Jun 15 2002
a(16) from Ray Chandler, Oct 09 2003
a(17)-a(20) from Robert G. Wilson v, Jan 18 2005
a(21) from Jason Earls, Jan 01 2008
a(22) from Robert Price, Jan 09 2011
a(23) from Robert Price, Mar 03 2011
a(24) from Edward A. Trice, Oct 21 2012
a(25) from Paul Bourdelais, Jan 28 2021

A062339 Primes whose sum of digits is 4.

Original entry on oeis.org

13, 31, 103, 211, 1021, 1201, 2011, 3001, 10111, 20011, 20101, 21001, 100003, 102001, 1000003, 1011001, 1020001, 1100101, 2100001, 10010101, 10100011, 20001001, 30000001, 101001001, 200001001, 1000000021, 1000001011, 1000010101, 1000020001, 1000200001, 1002000001, 1010000011

Offset: 1

Amarnath Murthy, Jun 21 2001

Is this sequence (and its brothers A062337, A062341 and A062343) infinite?

10^A049054(m)+3 and 3*10^A056807(m)+1 are subsequences. A107715 (primes containing only digits from set {0,1,2,3}) is a supersequence. Terms not containing the digit 3 are either terms of A020449 (primes that contain digits 0 and 1 only) or of A106100 (primes with maximal digit 2) - and thus terms of these sequences' union A036953 (primes containing only digits from set {0,1,2}). - Rick L. Shepherd, May 23 2005

			3001 is a prime with sum of digits = 4, hence belongs to the sequence.

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A062338, A107288, and A107715 (primes with digits <= 3).
A159352 is a subsequence.
Cf. A000040 (primes), A052218 (digit sum = 4), A061239 (primes == 4 (mod 9)).
Cf. Primes p with digital sum equal to k: {2, 11 and 101} for k=2; this sequence (k=4), A062341 (k=5), A062337 (k=7), A062343 (k=8), A107579 (k=10), A106754 (k=11), A106755 (k=13), A106756 (k=14), A106757 (k=16), A106758 (k=17), A106759 (k=19), A106760 (k=20), A106761 (k=22), A106762 (k=23), A106763 (k=25), A106764 (k=26), A048517 (k=28), A106766 (k=29), A106767 (k=31), A106768 (k=32), A106769 (k=34), A106770 (k=35), A106771 (k=37), A106772 (k=38), A106773 (k=40), A106774 (k=41), A106775 (k=43), A106776 (k=44), A106777 (k=46), A106778 (k=47), A106779 (k=49), A106780 (k=50), A106781 (k=52), A106782 (k=53), A106783 (k=55), A106784 (k=56), A106785 (k=58), A106786 (k=59), A106787 (k=61), A107617 (k=62), A107618 (k=64), A107619 (k=65), A106807 (k=67), A244918 (k=68), A181321 (k=70).
Cf. A049054 (10^k+3 is prime), A159352 (these primes).
Cf. A056807 (3*10^k+1 is prime), A259866 (these primes).
Cf. A020449 (primes with digits 0 and 1), A036953 (primes with digits <= 2), A106100 (primes with largest digit = 2), A069663, A069664 (smallest resp. largest n-digit prime with minimum digit sum).

[p: p in PrimesUpTo(800000000) | &+Intseq(p) eq 4]; // Vincenzo Librandi, Jul 08 2014

N:= 20: # to get all terms < 10^N
B[1]:= {1}:
B[2]:= {2}:
B[3]:= {3}:
A:= {}:
for d from 2 to N do
   B[4]:= map(t -> 10*t+1,B[3]) union  map(t -> 10*t+3,B[1]);
   B[3]:= map(t -> 10*t, B[3]) union map(t -> 10*t+1,B[2]) union map(t -> 10*t+2,B[1]);
   B[2]:= map(t -> 10*t, B[2]) union map(t -> 10*t+1,B[1]);
   B[1]:= map(t -> 10*t, B[1]);
   A:= A union select(isprime,B[4]);
od:
sort(convert(A,list)); # Robert Israel, Dec 28 2015

Union[FromDigits/@Select[Flatten[Table[Tuples[{0,1,2,3},k],{k,9}],1],PrimeQ[FromDigits[#]]&&Total[#]==4&]] (* Jayanta Basu, May 19 2013 *)
FromDigits/@Select[Tuples[{0,1,2,3},10],Total[#]==4&&PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Jul 23 2025 *)

for(a=1,20,for(b=0,a,for(c=0,b,if(isprime(k=10^a+10^b+10^c+1),print1(k", "))))) \\ Charles R Greathouse IV, Jul 26 2011

select( {is_A062339(p,s=4)=sumdigits(p)==s&&isprime(p)}, primes([1,10^7])) \\ 2nd optional parameter for similar sequences with other digit sums. M. F. Hasler, Mar 09 2022

A062339_upto_length(L,s=4,a=List(),u=[10^(L-k)|k<-[1..L]])=forvec(d=[[1,L]|i<-[1..s]], isprime(p=vecsum(vecextract(u,d))) && listput(a,p),1); Vecrev(a) \\ M. F. Hasler, Mar 09 2022

Intersection of A052218 (digit sum 4) and A000040 (primes). - M. F. Hasler, Mar 09 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Rick L. Shepherd, May 23 2005
More terms from Lekraj Beedassy, Dec 19 2007

A228034 Primes of the form 9^n + 2.

Original entry on oeis.org

3, 11, 83, 6563, 59051, 4782971, 282429536483, 2541865828331, 150094635296999123, 57264168970223481226273458862846808078011946891, 30432527221704537086371993251530170531786747066637051

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..16
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).

Cf. A004051 (primes of the form 2^a+3^b), A057735 (primes of the form 3^n+2), A090649 (associated n), A104070 (primes of the form 2^n+9), A159352 (primes of the form 10^n+3), A176495 (primes of the form 27^n+2), A182330 (primes of the form 5^n+2).

[a: n in [0..300] | IsPrime(a) where a is 9^n+2];

Select[Table[9^n + 2, {n, 0, 300}], PrimeQ]

A104638 Number of odd digits in n-th prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Zak Seidov, Mar 18 2005

The only zero is the first term. Sequence is unbounded. - Zak Seidov, Jan 12 2016
From Robert Israel, Jan 12 2016: (Start)
For any N, the asymptotic density of terms >= N is 1.
On the other hand, a(n) = 2 if prime(n) is in A159352, which is conjectured to be infinite.
Record values: a(2) = 1, a(5) = 2, a(30) = 3, a(187) = 4, a(1346) = 5, a(10545) = 6, a(86538) = 7, a(733410) = 8.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A104637, A159352.

Cf. A154764 (1 odd digit), A155071 (2 odd digits), A030096 (all digits odd).

seq(nops(select(type, convert(ithprime(i),base,10),odd)),i=1..100); # Robert Israel, Jan 12 2016
# alternative
A104638 := proc(n)
    local a,dgs,d ;
    ithprime(n) ;
    dgs := convert(%,base,10) ;
    a := 0 ;
    for d in dgs do
        a := a+modp(d,2) ;
    end do:
    a ;
end proc:
seq(A104638(n),n=1..40) ; # R. J. Mathar, Jul 13 2025

Table[Count[IntegerDigits[Prime[n]],?OddQ],{n,100}] (* _Harvey P. Dale, Jan 22 2012 *)
Table[Total[Mod[IntegerDigits[Prime[n]], 2]], {n, 100}] (* Vincenzo Librandi, Jan 13 2016 *)

a(n)=vecsum(digits(prime(n)%2)) \\ Zak Seidov, Jan 12 2016

a(n) = A196564(A000040(n)). - Michel Marcus, Oct 05 2013

A102006 Indices of primes in sequence defined by A(0) = 13, A(n) = 10*A(n-1) - 27 for n > 0.

Original entry on oeis.org

0, 1, 4, 5, 10, 16, 17, 38, 55, 100, 104, 106, 122, 412, 425, 2606, 7667, 10469, 11020, 17752, 26926, 60775, 98287, 300475

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 28 2004

Numbers n such that 10*10^n + 3 is prime.
Numbers n such that digit 1 followed by n >= 0 occurrences of digit 0 followed by digit 3 is prime.
Numbers corresponding to terms <= 425 are certified primes.
No other terms <99,999.

			100003 is prime, hence 4 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 100...003.

Cf. A000533, A002275, A049054, A159352.

a=13;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-27)

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

a(n) = A049054(n) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
a(22)=60775, a(23)=98287 from Robert Price, Mar 03 2011
a(24) from A049054 by Ray Chandler, May 01 2015

A228026 Primes of the form 4^k + 3.

Original entry on oeis.org

7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819

Offset: 1

Vincenzo Librandi, Aug 11 2013

			67 is a term because 4^3 + 3 = 67 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..16

Cf. A089437 (associated k).

Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), this sequence (r=4, h=3), A228027 (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  4^n+3];

Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]

a(n) = 4^A089437(n) + 3. - Elmo R. Oliveira, Nov 14 2023

Cross-references corrected by Robert Price, Aug 01 2017

A228032 Primes of the form 8^n + 3.

Original entry on oeis.org

11, 67, 4099, 32771, 262147, 1073741827, 19342813113834066795298819

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..14

Cf. A217354 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=3), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  8^n+3];

Select[Table[8^n + 3, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A228029 Primes of the form 5^n + 6.

Original entry on oeis.org

7, 11, 31, 131, 631, 1220703131

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..15

Cf. A089142 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), this sequence (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+6];

Select[Table[5^n + 6, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A228030 Primes of the form 7^n + 6.

Original entry on oeis.org

7, 13, 349, 33232930569607, 2651730845859653471779023381607

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..7

Cf. A217130 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=7, h=6), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  7^n+6];

Select[Table[7^n + 6, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A228031 Primes of the form 7^n + 10.

Original entry on oeis.org

11, 17, 59, 353, 2411, 117659, 823553, 1977326753, 9387480337647754305659, 3219905755813179726837617, 44567640326363195900190045974568017, 616873509628062366290756156815389726793178417, 30226801971775055948247051683954096612865741953

Offset: 1

Vincenzo Librandi, Aug 11 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..21

Cf. A217132 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=7, h=10), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..300] | IsPrime(a) where a is  7^n+10];

Select[Table[7^n + 10, {n, 0, 300}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

cp's OEIS Frontend

A049054 Numbers k such that 10^k + 3 is prime.

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A062339 Primes whose sum of digits is 4.

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A228034 Primes of the form 9^n + 2.

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A104638 Number of odd digits in n-th prime.

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A102006 Indices of primes in sequence defined by A(0) = 13, A(n) = 10*A(n-1) - 27 for n > 0.

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A228026 Primes of the form 4^k + 3.

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A228032 Primes of the form 8^n + 3.

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