A056804 -id:A056804 - cp's OEIS frontend

A274676 Numbers k such that 7*10^k + 13 is prime.

1, 3, 9, 12, 18, 19, 36, 37, 49, 67, 337, 893, 1924, 8044, 11610, 13560, 18777, 35376, 53601, 56022, 66488, 89801, 190210

Offset: 1

Vincenzo Librandi, Jul 03 2016

a(15) > 10000. - Felix Fröhlich, Jul 03 2016

			3 is in this sequence because 7*10^3 + 13 = 7013 is prime.
4 is not in the sequence because 7*10^4 + 13 = 70013 = 53 * 1321.
Initial terms and associated primes:
a(1) =  1: 83;
a(2) =  3: 7013;
a(3) =  9: 7000000013;
a(4) = 12: 7000000000013, etc.

Makoto Kamada, Search for 70w13.

Cf. numbers k such that 7*10^k + m is prime: A056804 (m=1), A097970 (m=3), A097954 (m=9), this sequence (m=13), A274677 (m=19), A274678 (m=27), A111021 (m=31), A274679 (m=33), A274700 (m=37), A274692 (m=43), A270974 (m=57).

[n: n in [1..800] | IsPrime(7*10^n+13)];

select(t -> isprime(7*10^t+13), [$1..2000]); # Robert Israel, Jul 03 2016

Select[Range[0, 3000], PrimeQ[7 * 10^# + 13] &]

is(n) = ispseudoprime(7*10^n+13) \\ Felix Fröhlich, Jul 03 2016

lista(nn) = for(n=1, nn, if(ispseudoprime(7*10^n+13), print1(n, ", "))); \\ Altug Alkan, Jul 03 2016

a(15) from Michael S. Branicky, Jan 22 2023
a(16)-a(17) from Michael S. Branicky, Apr 10 2023
a(18)-a(23) from Kamada data by Tyler Busby, Apr 15 2024

A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

Original entry on oeis.org

11, 13, 17, 41, 43, 97, 101, 131, 157, 181, 233, 239, 271, 311, 353, 401, 421, 491, 521, 541, 599, 617, 631, 647, 673, 743, 811, 859, 953, 1021, 1031, 1051, 1093, 1171, 1201, 1249, 1259, 1301, 1303, 1327, 1373, 1531, 1601, 1621, 1801, 1871, 2029, 2111, 2129, 2161

Offset: 1

Burak Muslu, Sep 10 2021

			97 is a term since its sum of digits is 9+7 = 16, and 97 mod 16 = 1.

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

Cf. A000040, A007605, A136251.
Subsequence of A209871.
A259866 \ {31}, and the primes associated with A056804 \ {1, 2} and A056797 are subsequences.

select(t -> isprime(t) and t mod convert(convert(t,base,10),`+`) = 1, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 05 2024

Select[Range[2000], PrimeQ[#] && Mod[#, Plus @@ IntegerDigits[#]] == 1 &] (* Amiram Eldar, Sep 10 2021 *)

isok(p) = isprime(p) && ((p % sumdigits(p)) == 1); \\ Michel Marcus, Sep 10 2021

from sympy import primerange
def ok(p): return p%sum(map(int, str(p))) == 1
print(list(filter(ok, primerange(1, 2130)))) # Michael S. Branicky, Sep 10 2021

A101128 Indices of primes in sequence defined by A(0) = 71, A(n) = 10*A(n-1) - 9 for n > 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 44, 135, 141, 157, 242, 922, 1234, 2195, 4649, 6118, 7323, 9542, 13493, 20309, 20359, 232919, 830864, 902707

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 03 2004

Numbers n such that 70*10^n + 1 is prime.
Numbers n such that digit 7 followed by n >= 0 occurrences of digit 0 followed by digit 1 is prime.
Numbers corresponding to terms <= 922 are certified primes.

			700001 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 700...001.
Index entries for primes involving repunits.

Cf. A000533, A002275, A056804.

a=71;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-9)

for(n=0,1500,if(isprime(70*10^n+1),print1(n,",")))

a(n) = A056804(n) - 1.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

a(21)-a(25) from Kamada data by Ray Chandler, Apr 29 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

A109749 Numbers n such that 88 * 10^n + 1 is prime.

Original entry on oeis.org

1, 3, 4, 7, 18, 30, 82, 99, 105, 106, 147, 334, 1092, 1221, 3705, 5524, 30355, 35962

Offset: 1

Jason Earls, Aug 11 2005

All terms have been certified. Primality proof for the largest: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing 88*10^30355+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3 Calling Brillhart-Lehmer-Selfridge with factored part 69.89% 88*10^30355+1 is prime! (491.3713s+0.8690s)

Makoto Kamada, Search for 880w1.

Cf. A056804, A056797.

is(n)=ispseudoprime(88*10^n+1) \\ Charles R Greathouse IV, Jun 13 2017

a(18) from Kamada data by Tyler Busby, Apr 16 2024

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

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A274676 Numbers k such that 7*10^k + 13 is prime.

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A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

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A101128 Indices of primes in sequence defined by A(0) = 71, 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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A109749 Numbers n such that 88 * 10^n + 1 is prime.

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

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

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