A030430 -id:A030430 - cp's OEIS frontend

A319099 Number of solutions to x^5 == 1 (mod n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1

Offset: 1

Jianing Song, Sep 10 2018

All terms are powers of 5. Those n such that a(n) > 1 are in A066500.

			Solutions to x^5 == 1 (mod 11): x == 1, 3, 4, 5, 9 (mod 11).
Solutions to x^5 == 1 (mod 25): x == 1, 6, 11, 16, 21 (mod 25) (x == 1 (mod 5)).
Solutions to x^5 == 1 (mod 31): x == 1, 2, 4, 8, 16 (mod 31).

Jianing Song, Table of n, a(n) for n = 1..10000

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), this sequence (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).
Cf. A030430, A066500, A293482, A000010.
Mobius transform gives A307380.

f[p_, e_] := If[Mod[p, 5] == 1, 5, 1]; f[5, 1] = 1; f[5, e_] := 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

a(n)=my(Z=znstar(n)[2]); prod(i=1,#Z,gcd(5,Z[i]));

Multiplicative with a(5) = 1, a(5^e) = 5 if e >= 2; for other primes p, a(p^e) = 5 if p == 1 (mod 5), a(p^e) = 1 otherwise.
If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(5, k_i).
a(n) = A000010(n)/A293482(n). - Jianing Song, Nov 10 2019

A024894 Numbers k such that 5*k + 1 is prime.

Original entry on oeis.org

2, 6, 8, 12, 14, 20, 26, 30, 36, 38, 42, 48, 50, 54, 56, 62, 66, 80, 84, 86, 92, 98, 104, 108, 114, 120, 126, 128, 132, 138, 140, 150, 152, 162, 164, 176, 182, 188, 194, 198, 204, 206, 210, 212, 218, 230, 234, 236, 240, 246, 258, 260, 264, 272, 276, 290, 294, 296, 302, 306, 314

Offset: 1

Clark Kimberling

a(n) = (A030430(n) - 1) / 5. - Reinhard Zumkeller, Aug 15 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A030430.

a024894 = flip div 5 . subtract 1 . a030430
-- Reinhard Zumkeller, Aug 15 2013

[n: n in [0..1000] | IsPrime(5*n+1)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[350], PrimeQ[5 # + 1] &] (* Vincenzo Librandi, May 20 2014 *)

is(n)=isprime(5*n+1) \\ Charles R Greathouse IV, Apr 16 2012

a(n) ~ 0.8n log n. - Charles R Greathouse IV, Apr 16 2012

A049511 Numbers k such that prime(k) == 1 (mod 10).

Original entry on oeis.org

5, 11, 13, 18, 20, 26, 32, 36, 42, 43, 47, 53, 54, 58, 60, 64, 67, 79, 82, 83, 89, 94, 98, 100, 105, 110, 115, 116, 121, 125, 126, 133, 135, 141, 142, 152, 156, 160, 164, 167, 172, 173, 177, 178, 182, 190, 193, 194, 197, 202, 210, 212, 216, 218, 221, 230, 233

Offset: 1

N. J. A. Sloane

nonn

Also k for which prime(k) == 1 (mod 5). - Bruno Berselli, Mar 04 2016

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000720, A024912, A030430, A081759.

Cf. A049508, A049509, A049510.

Select[Range[210], Mod[Prime[ # ], 10] == 1 &] (* Ray Chandler, Nov 07 2006 *)

isok(n) = !((prime(n)-1) % 10); \\ Michel Marcus, Mar 04 2016

[n for n in (1..300) if Mod(nth_prime(n), 10) == 1] # Bruno Berselli, Mar 04 2016

a(n) = A000720(A030430(n)). - Ray Chandler, Nov 07 2006

Extended by Ray Chandler, Nov 28 2003

Formula corrected by Zak Seidov, Sep 20 2011

A062332 Primes starting and ending with 1.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291, 1301, 1321, 1361, 1381, 1451, 1471, 1481, 1511, 1531, 1571, 1601, 1621, 1721, 1741, 1801, 1811, 1831, 1861, 1871, 1901, 1931, 1951, 10061, 10091, 10111, 10141

Offset: 1

Amarnath Murthy, Jun 21 2001

Complement of A208261 (nonprime numbers with all divisors starting and ending with digit 1) with respect to A208262 (numbers with all divisors starting and ending with digit 1). - Jaroslav Krizek, Mar 04 2012

Intersection of A030430 and A045707. - Michel Marcus, Jun 08 2013

			102701 is a member as it is a prime and the first and the last digits are both 1.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A208259 (Numbers starting and ending with digit 1).

Cf. A010051, A017281, A131835, A000030.

a062332 n = a062332_list !! (n-1)
a062332_list = filter ((== 1) . a010051') a208259_list
-- Reinhard Zumkeller, Jul 16 2014

fl1Q[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Last[idn]==1]; Select[ Prime[Range[1300]],fl1Q] (* Harvey P. Dale, Apr 30 2012 *)

{ n=-1; t=log(10); forprime (p=2, 5*10^5, if ((p-10*(p\10)) == 1 && (p\10^(log(p)\t)) == 1, write("b062332.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

A010051(a(n)) * A000030(a(n)) * (a(n) mod 10) = 1. - Reinhard Zumkeller, Jul 16 2014

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jun 29 2001

Missing term a(36)=1901 added by Harry J. Smith, Aug 05 2009

A385782 Primes having only {1, 6, 8} as digits.

Original entry on oeis.org

11, 61, 181, 661, 811, 881, 1181, 1811, 1861, 6661, 8111, 8161, 8681, 8861, 11161, 11681, 16111, 16661, 16811, 18181, 18661, 61681, 61861, 66161, 68111, 68161, 68611, 68881, 81181, 81611, 86111, 86161, 86861, 88661, 88681, 88811, 88861, 111611, 116681, 116881

Offset: 1

Jason Bard, Jul 13 2025

Subsequence of A030430. Supersequence of A020454, A020456.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 6, 8]];

Flatten[Table[Select[FromDigits /@ Tuples[{1, 6, 8}, n], PrimeQ], {n, 7}]]
Select[10Flatten[Table[FromDigits/@Tuples[{1,6,8},n],{n,5}]]+1,PrimeQ] (* Harvey P. Dale, Aug 27 2025 *)

primes_with(, 1, [1, 6, 8]) \\ uses function in A385776

print(list(islice(primes_with("168"), 41))) # uses function/imports in A385776

A042991 Primes congruent to {0, 2, 3, 4} (mod 5).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 103, 107, 109, 113, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 223, 227, 229, 233, 239, 257, 263, 269, 277, 283, 293, 307

Offset: 1

N. J. A. Sloane

Complement of A030430 relative to A000040. [Bruno Berselli, Jan 26 2016]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A030430.

Has same beginning as A040159 but is strictly different.

[p: p in PrimesUpTo(500) | p mod 5 in [0,2,3,4]]; // Vincenzo Librandi, Aug 09 2012

Select[Prime[Range[200]],MemberQ[{0,2,3,4},Mod[#,5]]&] (* Vincenzo Librandi, Aug 09 2012 *)

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x)    (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x)   (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x)   (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018

[p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018

Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)

is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Simpler definition from N. J. A. Sloane, Jul 11 2008

A340808 Decimal expansion of Product_{primes p == 1 (mod 5)} 1/(1-p^(-4)).

Original entry on oeis.org

1, 0, 0, 0, 0, 6, 9, 8, 7, 2, 8, 3, 2, 1, 8, 4, 2, 6, 1, 4, 1, 4, 1, 9, 6, 3, 5, 2, 6, 4, 6, 0, 0, 6, 2, 5, 1, 5, 3, 2, 3, 6, 8, 1, 4, 6, 7, 9, 6, 1, 5, 3, 4, 0, 6, 2, 7, 2, 4, 3, 4, 4, 3, 2, 6, 2, 7, 1, 4, 9, 4, 0, 1, 4, 0, 6, 7, 9, 6, 8, 5, 8, 7, 0, 9, 5, 2, 1, 5, 1, 1, 7, 9, 4, 1, 7, 3, 0, 2, 0, 1, 8, 9, 4, 0

Offset: 1

R. J. Mathar, Jan 22 2021

Equals also the same product over the primes p == 1 (mod 10).

			1.0000698728321842614141963526460062515  = (14641/14640) * (923521/923520) * (2825761/2825760) *...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions... arXiv:1008.2547, Zeta_{m=5,n=1}(s=4).

Cf. A004615, A030430, A340004 (s=2), A340629, A340809, A340926, A340927.

A340629 = A340004 ^2 / this.

Equals Sum_{k>=1} 1/A004615(k)^4. - Amiram Eldar, Jan 24 2021

More digits from Vaclav Kotesovec, Jan 22 2021

A068188 Tetradic primes (primes in A006072).

Original entry on oeis.org

11, 101, 181, 18181, 1008001, 1180811, 1880881, 1881881, 100111001, 100888001, 108101801, 110111011, 111010111, 111181111, 118818811, 180101081, 181111181, 181888181, 188010881, 188888881, 10008180001, 10081818001

Offset: 1

Eric W. Weisstein, Feb 18 2002

Primes that are palindromes and use only the digits 0, 1 and 8, so they read the same backwards and upside down.

11 is the only term with an even number of digits. The number of terms for an odd number of digits (3-37) is: 2, 1, 4, 12, 26, 62, 173, 392, 1087, 3197, 8189, 23354, 65128, 181486, 514255, 1447637, 4052813, 11682721. That makes the number of terms less than 10^2n (n to 19): 1, 3, 4, 8, 20, 46, 108, 281, 673, 1760, 4957, 13146, 36500, 101628, 283114, 797369, 2245006, 6297819, 17980540. - Hans Havermann, Dec 16 2017

Hans Havermann, Table of n, a(n) for n = 1..36500
Eric Weisstein's World of Mathematics, Tetradic Number

Cf. A006072, subsequence of A030430.

TetrPrmsUpTo10powerK[k_]:= Select[FromDigits/@ Tuples[{0,1,8}, k],
PrimeQ[#] && IntegerDigits[#] == Reverse[IntegerDigits[#]] &]; TetrPrmsUpTo10powerK[13] (* Mikk Heidemaa, May 20 2017 *)

Edited by Jud McCranie, Jun 02 2003

Offset corrected by Arkadiusz Wesolowski, Oct 17 2011

A112386 Smallest prime obtained by appending one or more 1's to n, -1 if no such prime exists.

Original entry on oeis.org

11, 211, 31, 41, 511111, 61, 71, 811, 911, 101, 1111111111111111111, 121111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111, 131, 14111111111, 151, 16111, 1711111111, 181, 191, 2011, 211, 22111, 2311, 241

Offset: 1

Michel Dauchez (mdzdm(AT)yahoo.fr), Dec 04 2005

a(37) = -1 since there is a covering of the set {371, 3711, 37111, ...} by the prime moduli 3, 7, 13, 37. Hence, there are infinitely many values -1 in the sequence (at 371, 3711, 37111, ...). - Emmanuel Vantieghem, Oct 27 2022

a(38) = -1 because 38 followed by m >= 1 1's is divisible by 3 or 37 or by (7*10^k-1)/3 if m = 3k. - Toshitaka Suzuki, Nov 07 2023

			a(5) = 511111 because 51, 511, 5111 and 51111 are not primes.

Toshitaka Suzuki, Table of n, a(n) for n = 1..55

Cf. A030430, A069568.

f[n_] := Block[{k = 1, e = Floor[Log[10, n] + 1]}, While[ !PrimeQ[n*10^k + (10^k - 1)/9], k++ ]; n*10^k + (10^k - 1)/9]; Array[f, 24] (* Robert G. Wilson v, Dec 05 2005 *)
Table[SelectFirst[Table[FromDigits[PadRight[IntegerDigits[k],n,1]],{n,IntegerLength[k]+1,250}],PrimeQ],{k,25}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 30 2017 *)

Edited, corrected and extended by Robert G. Wilson v, Dec 05 2005

Name edited by Emmanuel Vantieghem, Oct 27 2022

cp's OEIS Frontend

A319099 Number of solutions to x^5 == 1 (mod n).

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A024894 Numbers k such that 5*k + 1 is prime.

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A049511 Numbers k such that prime(k) == 1 (mod 10).

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A062332 Primes starting and ending with 1.

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A385782 Primes having only {1, 6, 8} as digits.

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A042991 Primes congruent to {0, 2, 3, 4} (mod 5).

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A140444 Primes congruent to 1 (mod 14).

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