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Numbers k such that 9*10^k+1 is prime.
+10
11
3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, 380734, 583696, 719055, 823037, 862868
EXAMPLE
For k=9 we have (9*(10^9))+1 = 9000000001, which is prime.
MATHEMATICA
Do[ If[ PrimeQ[9*10^n + 1], Print[n]], {n, 0, 10000}]
Numbers k such that 4*10^k+3 is prime.
+10
2
0, 1, 3, 7, 10, 40, 419, 449, 1737, 2245, 3131, 3813, 5345, 5659, 5681, 8410, 9097, 11293, 21061
COMMENTS
See Kamada link for search limit and prime vs. PRP status.
EXAMPLE
n = 1, 3, 7, 10 are members since 43, 4003, 40000003 and 40000000003 are prime numbers.
MATHEMATICA
Do[ If[ PrimeQ[4*10^n + 3], Print[n]], {n, 0, 10000}] (* Robert G. Wilson v, Jan 18 2005 *)
AUTHOR
Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005
Primes of the form 4*10^n+1.
+10
2
5, 41, 401, 4001, 40000000000001
PROG
(Magma) [a: n in [0..250] | IsPrime(a) where a is 4*10^n+1];
Numbers k such that 4*10^k+9 is prime.
+10
1
0, 2, 4, 5, 8, 9, 28, 191, 196, 2038, 34414, 39266, 50579, 94286, 108412, 130480, 178091, 185355
EXAMPLE
n = 2, 4, 5, 8, 9 are members since 409, 40009, 400009, 400000009 and 4000000009 are all prime.
MATHEMATICA
Do[ If[ PrimeQ[4*10^n + 9], Print[n]], {n, 0, 10000}]
AUTHOR
Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005
EXTENSIONS
a(10)=2038 from Joao da Silva (zxawyh66(AT)yahoo.com), Sep 30 2005
Numbers k such that 4*10^k+7 is prime.
+10
1
COMMENTS
No further terms < 50000.
EXAMPLE
n = 1, 3, 9 are members since 47, 4007 and 4000000007 are primes.
MATHEMATICA
Do[ If[ PrimeQ[4*10^n + 7], Print[n]], {n, 0, 10000}]
AUTHOR
Julien Peter Benney (jpbenney(AT)ftml.net), Jan 15 2005
Indices of primes in sequence defined by A(0) = 41, A(n) = 10*A(n-1) - 9 for n > 0.
+10
1
0, 1, 2, 12, 228, 241, 308, 956, 1472, 1493, 3181, 3726, 4176, 23209, 25718, 32834, 36989, 103957, 789954, 1038889
COMMENTS
Numbers k such that 40*10^k + 1 is prime.
Numbers k such that digit 4 followed by k >= 0 occurrences of digit 0 followed by digit 1 is prime.
REFERENCES
Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.
EXAMPLE
4001 is prime, hence 2 is a term.
MATHEMATICA
Select[Range[0, 1500], PrimeQ[4*10^# + 1] &] (* Robert Price, Mar 19 2015 *)
PROG
(PARI) a=41; for(n=0, 1500, if(isprime(a), print1(n, ", ")); a=10*a-9)
(PARI) for(n=0, 1500, if(isprime(40*10^n+1), print1(n, ", ")))
AUTHOR
Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 14 2004
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 02 2008
Numbers n such that 99 * 10^n + 1 is prime.
+10
1
1, 2, 4, 8, 16, 20, 24, 72, 200, 359, 454, 624, 1054, 2060, 6301, 8083, 8407, 13159, 65059, 74957
COMMENTS
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]
EXAMPLE
For n=8 we have 99*10^8+1 = 9900000001, which is prime.
EXTENSIONS
a(19)-a(20) from Kamada data by Tyler Busby, Apr 16 2024
Numbers n such that 55*10^n + 1 is prime.
+10
0
2, 3, 7, 9, 33, 61, 93, 112, 284, 615, 1293, 2558, 2925, 5961, 6454, 7960, 17521, 40838
COMMENTS
All values proved prime. No more up to 25000. Primality proof for the largest: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 55*10^17521+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 55*10^17521+1 is prime! (48.1892s+0.1481s)
Integers n such that (25*10^n)+1 is prime.
+10
0
1, 8, 255, 320, 609, 688, 1436, 3271, 3921, 6520, 19604, 38348, 63531
COMMENTS
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".
EXAMPLE
For n=8 we have (25*10^8)+1 = 25*100000000+1 = 2500000000+1 = 2500000001, which is prime.
AUTHOR
Julien Peter Benney (jpbenney(AT)ftml.net), Dec 13 2009
EXTENSIONS
a(12)-a(13) from Kamada data by Tyler Busby, May 03 2024
Numbers k such that 12*10^k + 1 is prime.
+10
0
0, 2, 38, 80, 9230, 25598, 39500
COMMENTS
k must be even since 12*10^k + 1 is divisible by 11 if k is odd. - Robert G. Wilson v, Feb 12 2018
EXAMPLE
13 and 1201 are prime, so 0 and 2 are the initial values.
MATHEMATICA
ParallelMap[ If[ PrimeQ[12*10^# +1], #, Nothing] &, 2 + 6Range@ 4500] (* Robert G. Wilson v, Feb 13 2018 *)
PROG
(PARI) isok(k) = isprime(12*10^k + 1); \\ Altug Alkan, Mar 04 2018
CROSSREFS
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).
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