A035095 -id:A035095 - cp's OEIS frontend

A234372 Second smallest prime congruent to 1 modulo n-th prime.

5, 13, 31, 43, 67, 79, 137, 229, 139, 233, 373, 223, 739, 431, 659, 743, 827, 733, 1609, 853, 439, 1423, 499, 1069, 971, 809, 1031, 857, 2399, 1583, 2287, 787, 1097, 1669, 1789, 1511, 3769, 2609, 5011, 1039, 1433, 1811, 2293, 1931, 4729, 2389, 4643, 6691, 5903, 5039, 1399, 1913, 2411, 4519, 9767, 5261, 3229, 2711

Offset: 1

Zak Seidov, Dec 25 2013

nonn

a(10^k) for k=1..10: 233, 11903, 142543, 837833, 36391853, 247773809, 3947342807, 146741381497, 684052904671, 16638454841119.

			a(3) = 31 because prime(3) = 5, first prime == 1 (mod 5) is 11 and 2nd one is 31.
a(4) = 43 because prime(4) = 7, first prime == 1 (mod 7) is 29 and 2nd one is 43.

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

Cf. A035095.

Table[If[n<2,5,p=Prime[n];r=1+2*p;While[!PrimeQ[r],r+=2*p];r+=2*p;While[!PrimeQ[r],r+=2*p];r],  {n,100}]

A247216 Maximum of the smallest prime factors of (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)].

Original entry on oeis.org

3, 13, 31, 55987, 12207031, 16148168401, 50544702849929377, 109912203092239643840221, 11111111111111111111111, 7369130657357778596659, 568972471024107865287021434301977158534824481, 388230138454493

Offset: 1

Vladimir Shevelev, Nov 26 2014

nonn

If in the definition "maximum" is replaced with "minimum," then we obtain A035095 (see comment there).

Cf. A020639, A035095.

with(numtheory):
a:= n-> max(seq(min(factorset(
        (i^ithprime(n)-1)/(i-1))[]), i=2..ithprime(n))):
seq(a(n), n=1..10);  # Alois P. Heinz, Dec 06 2014

a[n_] := Max[Table[Min[FactorInteger[(i^Prime[n]-1)/(i-1)][[All, 1]]], {i, 2, Prime[n]}]];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Mar 25 2017, after Alois P. Heinz *)

a(n) == 1 (mod prime(n)).

More terms from Peter J. C. Moses, Nov 26 2014

A061638 Primes p such that the greatest prime divisor of p-1 is 7.

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 281, 337, 379, 421, 449, 491, 631, 673, 701, 757, 883, 1009, 1051, 1373, 1471, 2017, 2269, 2521, 2647, 2689, 2801, 3137, 3361, 3529, 4201, 4481, 5881, 6301, 7001, 7057, 7351, 7561, 7841, 8233, 8821, 10501, 10753, 12097

Offset: 1

Labos Elemer, Jun 13 2001

nonn

Prime numbers n for which cos(2*Pi/n) is an algebraic number of 7th degree. - Artur Jasinski, Dec 13 2006

			For n = {4, 8, 9, 12}, a(n)-1 = {70, 210, 280, 420} = 7*{10, 30, 40, 60}.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 500 terms from Smith)

The 4th in a family of sequences after A019434(=Fermat-primes), A058383, A061599.
Cf. A019434, A058383, A023503, A034694, A006530, A006093, A035095, A061599.
Cf. A004729, A058383, A125867-A125875, A024899.

Select[Prime[Range[2000]],FactorInteger[#-1][[-1,1]] ==7&]  (* Harvey P. Dale, Mar 12 2011 *)

default(primelimit, 108864001); n=0; forprime (p=3, 108864001, f=factor(p - 1)~; if (f[1, length(f)]==7, write("b061638.txt", n++, " ", p))) \\ Harry J. Smith, Jul 25 2009

list(lim)=my(v=List(), t, t5, t7); lim\=1; lim--; for(a=1, logint(lim\2, 7), t7=2*7^a; for(b=0, logint(lim\t7, 5), t5=5^b*t7; for(c=0, logint(lim\t5, 3), t=3^c*t5; while(t<=lim, if(isprime(t+1), listput(v, t+1)); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Oct 29 2018

Primes of form 2^a*3^b*5^c*7^d + 1 with a and d > 1.

A215609 Smallest prime p congruent to 1 modulo prime(10^n).

Original entry on oeis.org

3, 59, 9739, 63353, 209459, 15596509, 154858631, 1794246731, 4076149487, 45603526979, 11092303227413, 93864728285579, 1319833868136653, 11656098322067917, 27803086068196217, 1781976386163140977, 32382366940106296979, 100447117955224696057, 707388643757314709297

Offset: 0

Zak Seidov, Aug 17 2012

nonn

			a(0) = 3 because p = prime(1) = 2, 3 = 1+p.
a(1) = 59 because p = prime(10) = 29, 59 = 1+2*p.
a(2) = 9739 because p = prime(10) = 29, 59 = 1+18*p.

Amiram Eldar, Table of n, a(n) for n = 0..24

Cf. A035095, A006988.

a[n_]:=(If[n<2,3,p=Prime[n];r=2p+1;While[!PrimeQ[r],r=r+2p];r]);Table[a[10^n],{n,0,12}]

a(n) = A035095(10^n).

a(13)-a(18) from Amiram Eldar, Apr 30 2024

A234387 a(n) = n-th smallest prime congruent to 1 modulo prime(n).

Original entry on oeis.org

3, 13, 41, 113, 331, 443, 613, 1103, 1013, 1741, 2543, 3257, 3691, 4129, 4889, 6997, 6491, 8053, 8443, 12071, 11681, 12799, 15439, 18869, 20759, 21211, 20807, 27179, 33791, 28703, 37339, 39301, 37813, 53377, 51853, 54059, 62801, 60637, 74149, 72661, 77687, 62989, 81749, 79903, 79589, 109849, 102547

Offset: 1

Zak Seidov, Dec 25 2013

nonn

			a(3) = 41 because prime(3) = 5 and primes == 1 mod 5 are 11, 31, 41;
a(4) = 113 because prime(4) = 7 and primes == 1 mod 7 are 29, 43, 71, 113.

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seidov)

Cf. A035095, A234372.

Reap[Sow[3];Do[c=0;q=Prime[n];p=1;While[c

a(n)=if(n<2,return(3)); my(p=prime(n),q=2*p+1); while(n, if(isprime(q), n--); q+= 2*p); q-2*p \\ Charles R Greathouse IV, Dec 26 2013

A247229 Minimum of the greatest prime factors of (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)].

Original entry on oeis.org

3, 7, 11, 127, 89, 8191, 30839, 363889, 178481, 2089, 4404047, 616318177, 164511353, 2099863, 13264529, 20394401

Offset: 1

Vladimir Shevelev, Nov 27 2014

If in the definition "greatest" is replaced with "smallest", then we obtain A035095 (see comment there).

Cf. A035095, A247216.

a(n) = {mini = 0; p = prime(n); for (i=2, p, f = factor((i^p-1)/(i-1)); gpf = f[#f~, 1]; if (! mini, mini = gpf, mini = min(mini, gpf));); mini;} \\ Michel Marcus, Dec 09 2014

a(n) == 1 (mod prime(n)).

More terms from Peter J. C. Moses, Nov 27 2014
a(13)-a(14) from Michel Marcus, Dec 09 2014
a(15)-a(16) from Jinyuan Wang, Mar 03 2020

A307434 a(n) is the smallest k such that the p-rank of (Z/kZ)* is 2, where p = prime(n) and (Z/kZ)* is the multiplicative group of integers modulo n.

Original entry on oeis.org

8, 63, 275, 1247, 1541, 4187, 14111, 43739, 6533, 13747, 116003, 33227, 61337, 74563, 186497, 79501, 586343, 269011, 432821, 485357, 128627, 451091, 83333, 191351, 377719, 491063, 638189, 551051, 2617309, 359341, 1164083, 206981, 902831, 929633, 2134277

Offset: 1

Jianing Song, Apr 08 2019

nonn

The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. By definition, a(n) is the smallest k such that x^p == 1 (mod k) has exactly p^2 solutions, p = prime(n).
a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there exists a primes q congruent to 1 modulo p, in which case C_p X C_p is a subgroup of (Z/(q*p^2)Z)*, p = prime(n).
a(n) is the smallest k such that C_p X C_p is a subgroup of (Z/kZ)*, p = prime(n).
Also, a(n) is the smallest k such that there exists some x, y such that ord(x,k) = ord(y,k) = prime(n) and the set of powers of x and the set of powers of y modulo k have trivial intersection {1}, where ord(x,k) is the multiplicative order of x modulo k.
Let p, q be odd primes, then (Z/(q^e)Z)* has p-rank 1 if q == 1 (mod p), or q = p and e >= 2; 0 otherwise. As a result, for n >= 2, a(n) is of the form (q_1)*(q_2) or (q_1)*p^2 where q_1 and q_2 are the smallest two primes congruent to 1 modulo prime(n).
It seems that for n >= 4, a(n) is not divisible by prime(n)^2, that is, there exists at least two primes that are smaller than prime(n)^2 and congruent to 1 modulo prime(n).

			(Z/8Z)* = C_2 X C_2, in which the solutions to x^2 == 1 (mod 8) are x == 1, 3, 5, 7 (mod 8) (4 solutions);
(Z/63Z)* = C_6 X C_6, in which the solutions to x^3 == 1 (mod 63) are x == 1, 4, 16, 22, 25, 37, 43, 46, 58 (mod 63) (9 solutions);
(Z/275Z)* = C_10 X C_20, in which the solutions to x^5 == 1 (mod 275) are x == 1, 16, 26, 31, 36, 56, 71, 81, 86, 91, 111, 126, 136, 141, 146, 166, 181, 191, 196, 201, 221, 236, 246, 251, 256 (mod 275) (25 solutions).

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Factorization of a(n) for n = 4..10000
Wikipedia, Multiplicative group of integers modulo n

Cf. A307436, A035095.

a(n) = if(n==1, 8, my(p=prime(n), i=0, q=0); for(k=1, +oo, if(isprime(2*k*p+1), i++; if(i==1, q=2*k*p+1)); if(i==2, return(q*min(p^2, 2*k*p+1)))))

a(1) = 8; a(n) = A307436(prime(n)) for n >= 2.

A330775 Irregular triangle read by rows: row n gives the primes of the form m*prime(n)+1 where m is an even number <= prime(n) and prime(n) is the n-th prime, or 0 if no such prime exists for any n.

Original entry on oeis.org

5, 7, 11, 29, 43, 23, 67, 89, 53, 79, 131, 157, 103, 137, 239, 191, 229, 47, 139, 277, 461, 59, 233, 349, 523, 311, 373, 683, 149, 223, 593, 1259, 83, 739, 821, 1231, 1559, 173, 431, 947, 1033, 1291, 1549, 1721, 283, 659, 941, 1129, 1223, 1693, 1787, 2069, 107, 743, 1061, 1697, 2333

Offset: 1

Metin Sariyar, Dec 30 2019

All safe primes are in this sequence.
Conjecture: For every prime p, there is at least one even m <= p such that m*p+1 is prime; this implies that no row is empty and there is no "0" in the sequence.
Conjecture: For every prime p, there is always a positive integer k <= p such that k*p+m is prime for any odd integer m, 0 < m < p. For example, for p = 11, k*11+m is prime for pairs {k,m}: {2,1}, {4,3}, {6,5}, {2,7}, {2,9}. - Metin Sariyar, Jan 26 2021

			For n = 4, m = {4, 6}, prime(4) = 7, and 4*7+1 = 29, 6*7+1 = 43 are primes.
Rows of the triangle:
n=1 => {5}
n=2 => {7}
n=3 => {11}
n=4 => {29, 43}
n=5 => {23, 67, 89}
n=6 => {53, 79, 131, 157}
n=7 => {103, 137, 239}
n=8 => {191, 229}
n=9 => {47, 139, 277, 461}
...

Metin Sariyar, Rows n = 1..220

Cf. A005384 (Sophie Germain primes), A005385 (safe primes), A035095.

row[n_] := Select[2 * Range[Floor[(p = Prime[n])/2]] * p + 1, PrimeQ]; row /@ Range[16] //Flatten (* Amiram Eldar, Jan 02 2020 *)

row(n) = select(x->isprime(x), vector(prime(n)\2, k, 2*k*prime(n)+1)); \\ Michel Marcus, Feb 05 2020

T(n, 1) = A035095(n) for n > 1. - Michel Marcus, Jan 02 2020

A343908 a(n) is the least prime == 4 (mod prime(n)).

Original entry on oeis.org

2, 7, 19, 11, 37, 17, 89, 23, 73, 149, 97, 41, 127, 47, 239, 163, 181, 431, 71, 359, 223, 83, 419, 271, 101, 307, 107, 967, 113, 569, 131, 397, 1237, 421, 2239, 457, 1103, 167, 839, 523, 541, 547, 577, 197, 1777, 601, 1481, 227, 3863, 233, 3499, 2633, 727, 757, 1289, 1319, 811, 1901, 281, 1409

Offset: 1

Zak Seidov, May 03 2021

nonn

			a(3) = 19 because 19 is the least prime == 4 (mod prime(3)).
a(4) = 11 because 11 is the least prime == 4 (mod prime(4)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A023200 (primes p such that p+4 is also prime), A034694, A035095, A279756.

a:= proc(n) local q, p; p:= ithprime(n); q:= p;
      do if irem(q-4, p)=0 then break fi;
         q:= nextprime(q);
      od; q
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 03 2021

s = {}; p = 5; Do[q = p + 2; While[Mod[q, p] != 4, q = NextPrime[q]]; AppendTo[s, q]; p = NextPrime[p], {100}]; s

a(n) = my(p=prime(n)); forprime(q=2,, if (Mod(q, p) == 4, return(q))); \\ Michel Marcus, May 03 2021

A356467 Smallest prime congruent to 1 (mod prime(n)) which is the norm of some principal ideal in the ring of prime(n)-th cyclotomic integers.

Original entry on oeis.org

7, 11, 29, 23, 53, 103, 191, 599, 4931, 5953, 32783, 101107, 178021, 549149

Offset: 2

Paul Vanderveen, Aug 08 2022

For the p-th cyclotomic ring of integers, primes which are norms of principal ideals should have asymptotic density 1/(h(p)*(p-1)) where h(p) is the class number of the p-th cyclotomic field.

			a(3) = 11 since 11 is the smallest prime congruent to 1 mod 5 (prime(3) = 5), which is the norm of some element in the 5th cyclotomic ring of integers. The algebraic integer x^2-x-1 has norm 11 where x is a primitive 5th root of unity.
a(2) - a(8) are the smallest primes congruent to 1 mod prime(n) as those corresponding cyclotomic fields have class number 1.
a(9) = 599. The 23rd cyclotomic ring of integers does not have class number 1. The smallest prime congruent to 1 (mod 23) is 47, and there is no cyclotomic integer with norm 47. The algebraic integer x^3-x-1 has norm 599 where x is a primitive 23rd root of unity.

Cf. A035095.

a(n)={ p=prime(n); t=0; K=bnfinit(polcyclo(p)); q=1; while(t==0, q=nextprime(q+1); if(q%p==1 && #bnfisintnorm(K,q)>0, t=1); ); return(q); }

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A234372 Second smallest prime congruent to 1 modulo n-th prime.

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A247216 Maximum of the smallest prime factors of (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)].

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A061638 Primes p such that the greatest prime divisor of p-1 is 7.

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A215609 Smallest prime p congruent to 1 modulo prime(10^n).

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A234387 a(n) = n-th smallest prime congruent to 1 modulo prime(n).

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A247229 Minimum of the greatest prime factors of (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)].

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A307434 a(n) is the smallest k such that the p-rank of (Z/kZ)* is 2, where p = prime(n) and (Z/kZ)* is the multiplicative group of integers modulo n.

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A330775 Irregular triangle read by rows: row n gives the primes of the form m*prime(n)+1 where m is an even number <= prime(n) and prime(n) is the n-th prime, or 0 if no such prime exists for any n.

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