A030430 -id:A030430 - cp's OEIS frontend

A132230 Primes congruent to 1 (mod 30).

31, 61, 151, 181, 211, 241, 271, 331, 421, 541, 571, 601, 631, 661, 691, 751, 811, 991, 1021, 1051, 1171, 1201, 1231, 1291, 1321, 1381, 1471, 1531, 1621, 1741, 1801, 1831, 1861, 1951, 2011, 2131, 2161, 2221, 2251, 2281, 2311, 2341, 2371, 2521, 2551, 2671

Offset: 1

Omar E. Pol, Aug 15 2007

Also primes congruent to 1 (mod 15). - N. J. A. Sloane, Jul 11 2008

Primes ending in 1 with (SOD-1)/3 integer where SOD is sum of digits. - Ki Punches, Feb 04 2009

			From _Muniru A Asiru_, Nov 01 2017: (Start)
31 is a prime and 31 = 30*1 + 1;
61 is a prime and 61 = 30*2 + 1;
151 is a prime and 151 = 30*5 + 1;
211 is a prime and 211 = 30*7 + 1;
241 is a prime and 241 = 30*8 + 1;
271 is a prime and 271 = 30*9 + 1.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A057204, A068228, A129805, A039949, A132231-A132236.

A132230 := Filtered(Filtered([1..10^6], n -> n mod 30 = 1), IsPrime); # Muniru A Asiru, Nov 01 2017

select(isprime, [seq(i,i=1..1000,30)]); # Robert Israel, Jan 19 2016

Select[Range[1, 3000, 30], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)
Select[Prime[Range[400]],Mod[#,30]==1&] (* Harvey P. Dale, May 21 2021 *)

is(n)=isprime(n) && n%30==1 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A111175(n)*30 + 1. - Ray Chandler, Apr 07 2009

Intersection of A030430 and A002476. - Ray Chandler, Apr 07 2009

Edited by Ray Chandler, Apr 07 2009

A024912 Numbers k such that 10*k + 1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 7, 10, 13, 15, 18, 19, 21, 24, 25, 27, 28, 31, 33, 40, 42, 43, 46, 49, 52, 54, 57, 60, 63, 64, 66, 69, 70, 75, 76, 81, 82, 88, 91, 94, 97, 99, 102, 103, 105, 106, 109, 115, 117, 118, 120, 123, 129, 130, 132, 136, 138, 145, 147, 148, 151, 153, 157, 160, 162

Offset: 1

Clark Kimberling

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A030430, A049511, A081759.

[n: n in [1..1000] |IsPrime(10*n+1)]; // Vincenzo Librandi, Nov 19 2010

Select[Range[170], PrimeQ[10# + 1] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(10*n+1) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = (A081759(n)+1)/2.

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

Original entry on oeis.org

211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841

Offset: 1

N. J. A. Sloane

From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.

Cf. A047650, A000045, A024894, subsequence of A030430.

See also A270798 (a subsequence), A270800.

a001583 n = a001583_list !! (n-1)
a001583_list = filter
   (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 06 2017

From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)

More terms from James Sellers, Jan 25 2000

Edited by N. J. A. Sloane, Apr 01 2016

A106291 Period of the Lucas sequence A000032 mod n.

Original entry on oeis.org

1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, 16, 30, 48, 24, 20, 84, 72, 48, 14, 24, 30, 48, 40, 36, 16, 24, 76, 18, 56, 12, 40, 48, 88, 30, 24, 48, 32, 24, 112, 60, 72, 84, 108, 72, 20, 48, 72, 42, 58, 24, 60, 30, 48, 96, 28, 120, 136, 36, 48, 48

Offset: 1

T. D. Noe, May 02 2005

nonn

This sequence differs from the Fibonacci periods (A001175) only when n is a multiple of 5, which can be traced to 5 being the discriminant of the characteristic polynomial x^2-x-1.

This sequence coincides with the Fibonacci periods (A001175) if n is a multiple of 5^j and the following conditions apply: n contains at least one prime factor of the form p = 10*k+1 (A030430) which occurs in Fibonacci(m) or Lucas(m) as prime factor, where m must be the smallest possible index containing p and a factor 5^i and j <= i. If n contains several prime factors from A030430 that satisfy the above conditions, the largest applicable i is decisive. - Klaus Purath, Apr 26 2019

			From _Klaus Purath_, Jul 10 2019: (Start)
n = 3*5*31 = 465, j = 1; L(15) is the smallest Lucas number with prime factor 31; 15 = 3*5, i = 1 = j. Hence Lucas period (mod 465) = Fibonacci period (mod 465) = 120, but if n = 3*5^2*31 = 2325, j = 2 > i. Hence Lucas period (mod 2325) = 120 < Fibonacci period (mod 2325) = 600.
n = 5*701 = 3505, j = 1; F(175) is the smallest Fibonacci number with prime factor 701; 175 = 7*5^2, i = 2 > j. Therefore Lucas period (mod 3505) = Fibonacci period (mod 3505) = 700, but if n = 5^3*701 = 87625, j = 3 > i. Therefore Lucas period (mod 87625) = 700 < Fibonacci period (mod 87625) = 3500.
n = 5^2*11*101 = 27775, j =2; L(5) is the smallest Lucas number with prime factor 11, i = 1; L(25) = is the smallest Lucas number with prime factor 101; 25 = 5^2, i = 2 ( decisive); j = i. Hence Lucas period (mod 27775) = Fibonacci period (mod 27775) = 100, but if n = 5^3*11*101 = 138875, j = 3 > i. Hence Lucas period (mod 138875) = 100 < Fibonacci period (mod 138875) = 500. (End)

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989. See p. 89. - From N. J. A. Sloane, Feb 20 2013

G. C. Greubel and D. Turner, Table of n, a(n) for n = 1..10000
Brennan Benfield and Oliver Lippard, Fixed points of K-Fibonacci sequences, arXiv:2404.08194 [math.NT], 2024. See p. 11.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1).

Cf. A000032, A001175, A030430.

n=2; Table[p=i; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 70}]

from math import lcm
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A106291(n):
    if n < 3:
        return (n<<1)-1
    f = factorint(n).items()
    if len(f) > 1:
        return lcm(*(A106291(a**b) for a,b in f))
    else:
        k,x = 1, (1,3)
        while x != (2,1):
            k += 1
            x = (x[1], (x[0]+x[1]) % n)
        return k # Chai Wah Wu, Apr 25 2025

def a(n): return BinaryRecurrenceSequence(1, 1, 2, 1).period(n)
[a(n) for n in (1..100)] # G. C. Greubel, Apr 27 2019

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

A168147 Primes of the form 10*n^3 + 1.

Original entry on oeis.org

11, 271, 641, 2161, 33751, 40961, 58321, 138241, 196831, 270001, 297911, 466561, 506531, 795071, 1326511, 1406081, 1851931, 2160001, 3890171, 4218751, 5314411, 5513681, 6585031, 7290001, 8043571, 11910161, 12597121, 12950291, 14815441

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 19 2009

(1) These primes all with end digit 1=1^3 are concatenations of two CUBIC numbers: "n^3 1".
(2) It is conjectured that the sequence is infinite.
(3) It is an open problem if 3 consecutive naturals n exist which give such a prime.
No three such integers exist, as every n = 2 (mod 3) yields 10n^3 + 1 = 0 (mod 3). - Charles R Greathouse IV, Apr 24 2010

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005

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

Cf. A030430 (primes of the form 10*n+1).
Cf. A167535 (concatenation of two square numbers which give a prime).
See A168219 for the numbers n.

[ a: n in [1..150] | IsPrime(a) where a is 10*n^3+1 ]; // Vincenzo Librandi, Jul 25 2011

Select[Table[10*n^3+1,{n,1000}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

for(n=1,2e2, isprime(n^3*10+1) && print1(n^3*10+1", "))  \\ M. F. Hasler, Jul 24 2011

a(n) = 10*A168219(n)^3 + 1. \\ M. F. Hasler, Jul 24 2011

A081759 Numbers n such that 5n+6 is prime.

Original entry on oeis.org

1, 5, 7, 11, 13, 19, 25, 29, 35, 37, 41, 47, 49, 53, 55, 61, 65, 79, 83, 85, 91, 97, 103, 107, 113, 119, 125, 127, 131, 137, 139, 149, 151, 161, 163, 175, 181, 187, 193, 197, 203, 205, 209, 211, 217, 229, 233, 235, 239, 245, 257, 259, 263, 271, 275, 289

Offset: 1

Giovanni Teofilatto, Nov 21 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A024912, A030430, A049511, A067076, A087505, A089192, A089253.

[n: n in [0..300]| IsPrime(5*n + 6)]; // Vincenzo Librandi, Oct 16 2012

A081759 := proc(n) option remember: local k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(isprime(5*k+6))then return k: fi: od: end: seq(A081759(n),n=1..100); # Nathaniel Johnston, May 28 2011

Select[Range[300], PrimeQ[5# + 6] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(5*n+6) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2*A024912(n) - 1.

Corrected by Ray Chandler, Nov 22 2003

A132232 Primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 491, 521, 641, 701, 761, 821, 881, 911, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1871, 1901, 1931, 2081, 2111, 2141, 2351, 2381, 2411, 2441, 2531

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A068231, A039949.

Cf. A132230, A132231, A132233, A132234, A132235, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 11 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{11},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[11,2600,30],PrimeQ] (* Harvey P. Dale, Mar 07 2020 *)

is(n)=isprime(n) && n%30==11 \\ Charles R Greathouse IV, Jul 01 2016

From Ray Chandler, Apr 07 2009: (Start)
a(n) = A158614(n)*30 + 11.
Intersection of A030430 and A007528. (End)

Extended by Ray Chandler, Apr 07 2009

A004615 Divisible only by primes congruent to 1 mod 5.

Original entry on oeis.org

1, 11, 31, 41, 61, 71, 101, 121, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 341, 401, 421, 431, 451, 461, 491, 521, 541, 571, 601, 631, 641, 661, 671, 691, 701, 751, 761, 781, 811, 821, 881, 911, 941

Offset: 1

N. J. A. Sloane

nonn

Also numbers with all divisors ending with digit 1.
Union of number 1, A030430 and A068872. - Jaroslav Krizek, Feb 12 2012
Also numbers with all divisors ending with the same digit; as 1 divides all the integers, this digit is necessarily 1 (see first comment); hence, for these numbers m: A330348(m) = A000005(m). - Bernard Schott, Nov 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027748, A030430 (primes), A068872 (composites).

Cf. A010879, A027750, A002808, A330348, A338784 (subsequence).

a004615 n = a004615_list !! (n-1)
a004615_list = filter (all (== 1) . (map (`mod` 5) . a027748_row)) [1..]
-- Reinhard Zumkeller, Apr 16 2012

[n: n in [1..1500] | forall{d: d in PrimeDivisors(n) | d mod 5 eq 1}]; // Vincenzo Librandi, Aug 21 2012

ok[1]=True;ok[n_]:=And@@(Mod[#,5]==1&)/@FactorInteger[n][[All,1]];Select[Range[2000],ok] (* Vincenzo Librandi, Aug 21 2012 *)
Select[Range[1000],Union[Mod[#,5]&/@FactorInteger[#][[All,1]]]=={1}&] (* Harvey P. Dale, Apr 19 2019 *)

is(n)=#select(p->p%5!=1, factor(n)[,1])==0 \\ Charles R Greathouse IV, Mar 11 2014

A206291 merged in by Franklin T. Adams-Watters, Sep 21 2012

A141849 Primes congruent to 1 mod 11.

Original entry on oeis.org

23, 67, 89, 199, 331, 353, 397, 419, 463, 617, 661, 683, 727, 859, 881, 947, 991, 1013, 1123, 1277, 1321, 1409, 1453, 1607, 1783, 1871, 2003, 2069, 2113, 2179, 2267, 2311, 2333, 2377, 2399, 2531, 2663, 2707, 2729, 2861, 2927, 2971, 3037, 3169, 3191, 3257

Offset: 1

N. J. A. Sloane, Jul 11 2008

Conjecture: Also primes p such that ((x+1)^11-1)/x has 10 distinct irreducible factors of degree 1 over GF(p). - Federico Provvedi, Apr 17 2018

Primes congruent to 1 mod 22. - Chai Wah Wu, Apr 28 2025

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

Cf. A000040, A090187, A102656.
Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), A007528 (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), this sequence (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).

Filtered([1..4000],n->n mod 11=1 and IsPrime(n)); # Muniru A Asiru, Apr 19 2018

[ p: p in PrimesUpTo(5000) | p mod 11 eq 1 ]; // Vincenzo Librandi, Apr 19 2011

a:=select(n->isprime(n) and modp(n,11)=1,[$1..4000]); # Muniru A Asiru, Apr 19 2018

Select[Range[1,10000,11],PrimeQ] (* Vladimir Joseph Stephan Orlovsky, May 18 2011 *)

is(n)=isprime(n) && n%11==1 \\ Charles R Greathouse IV, Jul 01 2016

forstep(n=2, 1e3, 2, if(isprime(p=11*n+1), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018

a(n) ~ 10n log n. - Charles R Greathouse IV, Jul 02 2016

A208270 Primes containing a digit 1.

Original entry on oeis.org

11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661

Offset: 1

Jaroslav Krizek, Mar 04 2012

Subsequence of A011531, A062634, A092911 and A092912.
Supersequence of A106101, A045707 and A030430.
Complement of A208271 with respect to A011531.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A208271 (nonprimes containing a digit 1), A011531 (numbers containing a digit 1).

Complement of A038603 in A000040. - M. F. Hasler, Mar 05 2012

[p: p in PrimesUpTo(400) | 1 in Intseq(p)]; // Vincenzo Librandi, Apr 29 2019

Select[Prime[Range[124]], MemberQ[IntegerDigits[#], 1] &](* Jayanta Basu, Apr 01 2013 *)
Select[Prime[Range[200]],DigitCount[#,10,1]>0&] (* Harvey P. Dale, Dec 15 2020 *)

forprime(p=2,1e3,s=vecsort(eval(Vec(Str(p))),,8);if(s[1]==1||(s[1]==0&&s[2]==1),print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012

is_A208270(n)=isprime(n)&setsearch(Set(Vec(Str(n))),1) \\ M. F. Hasler, Mar 05 2012

a(n) ~ n log n since the sequence contains almost all primes. - Charles R Greathouse IV, Mar 04 2012

cp's OEIS Frontend

A132230 Primes congruent to 1 (mod 30).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

Extensions

A024912 Numbers k such that 10*k + 1 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A106291 Period of the Lucas sequence A000032 mod n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Sage

Formula

A168147 Primes of the form 10*n^3 + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A081759 Numbers n such that 5n+6 is prime.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI