A056807 -id:A056807 - cp's OEIS frontend

A259867 Exponent of 2 modulo the prime A056807(n).

5, 1500, 234375, 300000000, 7500000000000000000000000000, 750000000000000000000000000000000000, 1500000000000000000000000000000000000000000000000000000000000000000

Offset: 1

N. J. A. Sloane, Jul 08 2015

nonn

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

Cf. A056807.

A259867 := proc(n)
    p := 3*10^A056807(n)+1 ;
    numtheory[order](2,p) ;
end proc: # R. J. Mathar, Jul 15 2015

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

A199683 a(n) = 3*10^n + 1.

Original entry on oeis.org

4, 31, 301, 3001, 30001, 300001, 3000001, 30000001, 300000001, 3000000001, 30000000001, 300000000001, 3000000000001, 30000000000001, 300000000000001, 3000000000000001, 30000000000000001, 300000000000000001, 3000000000000000001, 30000000000000000001, 300000000000000000001

Offset: 0

Vincenzo Librandi, Nov 09 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Primes: A056807, A259866.

Magma
```
[3*10^n+1: n in [0..30]];
```

3*10^Range[0,20]+1 (* or *) LinearRecurrence[{11,-10},{4,31},20] (* Harvey P. Dale, Dec 12 2016 *)

a(n) = 10*a(n-1) - 9.
a(n) = 11*a(n-1) - 10*a(n-2).
G.f.: (4-13*x)/((1-x)*(1-10*x)).
E.g.f.: exp(x)*(1 + 3*exp(9*x)). - Elmo R. Oliveira, Jun 10 2025

A259866 Primes of the form 3*10^k + 1.

Original entry on oeis.org

31, 3001, 30000001, 30000000001, 30000000000000000000000000001, 3000000000000000000000000000000000001, 30000000000000000000000000000000000000000000000000000000000000000001

Offset: 1

N. J. A. Sloane, Jul 08 2015

nonn

S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
S. W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

Cf. A199683, A056807, A062339.

Select[3*10^Range[70]+1,PrimeQ] (* Harvey P. Dale, Jul 05 2020 *)

a(n) = 3*10^A056807(n)+1. - R. J. Mathar, Jul 15 2015

A101823 Indices of primes in sequence defined by A(0) = 31, A(n) = 10*A(n-1) - 9 for n > 0.

Original entry on oeis.org

0, 2, 6, 9, 27, 35, 66, 80, 146, 482, 642, 1019, 1899, 2619, 10452, 27719, 52823, 105588, 111987, 618852, 665828

Offset: 1

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

Numbers n such that 30*10^n + 1 is prime.

Numbers n such that digit 3 followed by n >= 0 occurrences of digit 0 followed by digit 1 is prime.

			3001 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 300...001.

Cf. A000533, A002275, A056807.

a=31;for(n=0,2000,if(isprime(a),print1(n,","));a=10*a-9)

for(n=0,2000,if(isprime(30*10^n+1),print1(n,",")))

a(n) = A056807(n) - 1.

More terms from Kamada link by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
a(18) from Roman Makarchuk, Dec 05 2008 confirmed as next term by Ray Chandler, Mar 02 2012
a(19) from Alexander Gramolin, Feb 24 2012 confirmed as next term by Ray Chandler, Mar 02 2012
a(20)-a(21) derived from A056807 by Robert Price, Jan 26 2015

A109713 Numbers n such that 99 * 10^n + 1 is prime.

Original entry on oeis.org

1, 2, 4, 8, 16, 20, 24, 72, 200, 359, 454, 624, 1054, 2060, 6301, 8083, 8407, 13159, 65059, 74957

Offset: 1

Jason Earls, Aug 08 2005

Terms < 21000 have been certified. Primality proof for 13159: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 99*10^13159+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 17 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 99*10^13159+1 is prime! (29.5646s+0.0028s) [Comment edited by N. J. A. Sloane, Jan 28 2025]

			For n=8 we have 99*10^8+1 = 9900000001, which is prime.

Makoto Kamada, Search for 990w1.

Cf. A056797, A056804, A056805, A056806, A056807.

is(n)=ispseudoprime(99*10^n+1) \\ Charles R Greathouse IV, Jun 12 2017

Edited by N. J. A. Sloane at the suggestion of Herman Jamke, Jan 13 2008

a(19)-a(20) from Kamada data by Tyler Busby, Apr 16 2024

A109397 Numbers k such that 22 * 10^k + 1 is prime.

Original entry on oeis.org

0, 6, 10, 11, 102, 146, 296, 1415, 2324, 4374, 12590

Offset: 1

Jason Earls, Aug 26 2005

All terms have been certified. Primality proof for the largest: PFGW Version 1.2.0 for Windows [FFT v23.8] Primality testing 22*10^12590+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 22*10^12590+1 is prime! (47.5292s+0.3107s)
No more terms up to 31000.
No more terms up to 50000. - Michael S. Branicky, Oct 11 2024

Cf. A056807.

is(n)=ispseudoprime(22*10^n+1) \\ Charles R Greathouse IV, Jun 13 2017

0 added by Arkadiusz Wesolowski, Oct 27 2011

A171612 Integers n such that (25*10^n)+1 is prime.

Original entry on oeis.org

1, 8, 255, 320, 609, 688, 1436, 3271, 3921, 6520, 19604, 38348, 63531

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Dec 13 2009

No others less than 20000.

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "25001".

			For n=8 we have (25*10^8)+1 = 25*100000000+1 = 2500000000+1 = 2500000001, which is prime.

Makoto Kamada, List of near-repdigit-related prime numbers.
Index entries for primes involving repunits.

Cf. A056797, A056804, A056805, A056806, A056807

Edited by Ray Chandler, Dec 23 2010

a(12)-a(13) from Kamada data by Tyler Busby, May 03 2024

A294396 Numbers k such that 12*10^k + 1 is prime.

Original entry on oeis.org

0, 2, 38, 80, 9230, 25598, 39500

Offset: 1

Patrick A. Thomas, Feb 12 2018

k must be even since 12*10^k + 1 is divisible by 11 if k is odd. - Robert G. Wilson v, Feb 12 2018
a(7) > 27440. - Robert G. Wilson v, Feb 17 2018
a(8) > 10^5. - Jeppe Stig Nielsen, Jan 31 2023

			13 and 1201 are prime, so 0 and 2 are the initial values.

Cf. A001562, A096507, A056797, A056807, A056806, A102940, A056805, A056804, A096508, A102975, A289051, A099017, A295325, A273002, A102945, A282456, A267420.

Cf. A062339 (primes with digit sum 4).

ParallelMap[ If[ PrimeQ[12*10^# +1], #, Nothing] &, 2 + 6Range@ 4500] (* Robert G. Wilson v, Feb 13 2018 *)

isok(k) = isprime(12*10^k + 1); \\ Altug Alkan, Mar 04 2018

a(5) from Robert G. Wilson v, Feb 12 2018
a(6) from Robert G. Wilson v, Feb 13 2018
a(7) from Jeppe Stig Nielsen, Jan 28 2023

cp's OEIS Frontend

A259867 Exponent of 2 modulo the prime A056807(n).

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A062339 Primes whose sum of digits is 4.

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A199683 a(n) = 3*10^n + 1.

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A259866 Primes of the form 3*10^k + 1.

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A101823 Indices of primes in sequence defined by A(0) = 31, A(n) = 10*A(n-1) - 9 for n > 0.

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A109713 Numbers n such that 99 * 10^n + 1 is prime.

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A109397 Numbers k such that 22 * 10^k + 1 is prime.

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A171612 Integers n such that (25*10^n)+1 is prime.