A056797 -id:A056797 - cp's OEIS frontend

A056806 Numbers n such that 4*10^n + 1 is prime.

0, 1, 2, 3, 13, 229, 242, 309, 957, 1473, 1494, 3182, 3727, 4177, 23210, 25719, 32835, 36990, 103958, 789955, 1038890

Offset: 1

Robert G. Wilson v, Aug 22 2000

Some of the results were computed using the PrimeFormGW (PFGW) primality-testing program. - Hugo Pfoertner, Nov 14 2019

Makoto Kamada, Prime numbers of the form 400...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A056797 (9*10^n+1 is prime), A101712.

Do[ If[ PrimeQ[ 4*10^n + 1], Print[ n ]], {n, 0, 30000}]

is(n)=isprime(4*10^n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A101712(n-1) + 1.

a(12)-a(16) using PrimeForm from Hugo Pfoertner, Jul 08 2004
32835 from Ray Chandler, Aug 30 2010
36990 from Peter Benson, Aug 23 2003 confirmed as next term by Ray Chandler, Sep 07 2010
103958 from Peter Benson, Dec 31 2004 confirmed as next term by Ray Chandler, Feb 18 2012
a(20)-a(21) from Kamada data by Tyler Busby, May 03 2024

A056807 Numbers k such that 3*10^k + 1 is prime.

Original entry on oeis.org

1, 3, 7, 10, 28, 36, 67, 81, 147, 483, 643, 1020, 1900, 2620, 10453, 27720, 52824, 105589, 111988, 618853, 665829

Offset: 1

Robert G. Wilson v, Aug 22 2000

			k = 3 gives (3*10^3+1) = 3000+1 = 3001, which is prime.

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]
Makoto Kamada, Prime numbers of the form 300...001.
Sabin Tabirca and Kieran Reynolds, Lacunary Prime Numbers.

Cf. A056797, A062339, A101823, A199683, A259866.

Do[ If[ PrimeQ[ 3*10^k + 1], Print[ k ]], {k, 0, 20000}]

is(k)=isprime(3*10^k+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A101823(n) + 1.

a(13)-a(14) from Julien Peter Benney (jpbenney(AT)ftml.net), Nov 23 2004
a(15) from Hugo Pfoertner, Jan 18 2005
a(16)-a(17) from Robert G. Wilson v, Jan 18 2005
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) from Kamada data by Robert Price, Jan 26 2015

A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

Original entry on oeis.org

3, 5, 7, 17, 19, 43, 101, 157, 163, 257, 487, 1459, 2029, 4423, 6163, 14407, 19183, 22651, 23549, 26407, 37057, 39367, 62501, 65537, 77659, 113233, 121453, 143263, 208393, 292141, 342733, 375157, 412807, 527803, 564899, 590593, 697049, 843643

Offset: 1

T. D. Noe, Aug 15 2003

nonn

It is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A002383 (primes of the form n^2 + n + 1, which is the same as n^2 - n + 1).

Cf. A019434 (Fermat primes), A003306 (2*3^n + 1 is prime), A056799 (8*9^n + 1 is prime), A056797 (9*10^n + 1 is prime), A087139 (least k such that p^k - p^(k-1) + 1 is prime for p = prime(n)).

lst={}; maxNum=10^6; n=1; While[p=Prime[n]; p^2-p+1

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

A100997 Indices of primes in sequence defined by A(0) = 91, A(m) = 10*A(m-1) - 9 for m > 0.

Original entry on oeis.org

2, 3, 4, 8, 21, 26, 35, 56, 61, 77, 200, 536, 695, 789, 904, 1037, 66885, 70499, 91835, 100612, 127239, 380733, 583695

Offset: 1

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Nov 26 2004

Numbers k such that 90*10^k + 1 is prime.
Numbers k such that digit 9 followed by k >= 0 occurrences of digit 0 followed by digit 1 is prime.
Numbers corresponding to terms <= 1037 are certified primes.

			900001 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 900...001.
Index entries for primes involving repunits.

Cf. A000533, A002275, A056797.

a=91;for(n=0,1500,if(isprime(a),print1(n,","));a=10*a-9)

for(n=0,1500,if(isprime(90*10^n+1),print1(n,",")))

a(n) = A056797(n) - 1.

66885 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(18)-a(21) from Kamada data by Ray Chandler, Apr 28 2015
a(22)-a(23) from Kamada data by Mohammed Yaseen, Jul 20 2021

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

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

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

cp's OEIS Frontend

A056806 Numbers n such that 4*10^n + 1 is prime.

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A056807 Numbers k such that 3*10^k + 1 is prime.

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A087126 Primes of the form p^k - p^(k-1) + 1 for some prime p and integer k > 1.

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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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A100997 Indices of primes in sequence defined by A(0) = 91, A(m) = 10*A(m-1) - 9 for m > 0.

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

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A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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PARI