A068231 -id:A068231 - cp's OEIS frontend

A263769 Smallest prime q such that q == -1 (mod prime(n)-1).

2, 3, 3, 5, 19, 11, 31, 17, 43, 83, 29, 71, 79, 41, 137, 103, 173, 59, 131, 139, 71, 233, 163, 263, 191, 199, 101, 211, 107, 223, 251, 389, 271, 137, 443, 149, 311, 647, 331, 859, 1423, 179, 379, 191, 587, 197, 419, 443

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

a(n): A000040(1), A065091(1), A002145(1), A007528(1), A030433(1), A068231(1), A127576(1), A061242(1), A141857(1), A141976(1), A132236(1), A142111(1), A142198(1), A141898(1), A141926(1), A142531(1), A142004(1), A142799(1), A142068(1), A142099(1), ...

Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

			a(4) = 5 because 5 == -1 (mod prime(4)-1) and is prime.

Robert Israel, Table of n, a(n) for n = 1..6000

Cf. A263730, A263770.

Cf. A000040, A065091, A002145, A007528, A030433, A068231, A127576, A061242, A141857, A141976, A132236, A142111, A142198, A141898, A141926, A142531, A142004, A142799, A142068, A142099.

for n from 1 to 100 do
  k:= ithprime(n)-1;
  q:= 2;
  while (1 + q) mod k <> 0 do
    q:= nextprime(q)
  od;
  A[n]:= q;
od:
seq(A[i],i=1..1000); # Robert Israel, Oct 26 2015

Table[q = 2; z = Prime@ n - 1; While[Mod[q, z] != z - 1, q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

Corrected and edited by Robert Israel, Oct 26 2015,

A359387 Primes p such that the smallest prime factor of (2^(p-1)-1)/(3*p) is greater than p.

Original entry on oeis.org

11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1847, 1907, 2027, 2039, 2063, 2099, 2207, 2243, 2447, 2459, 2579, 2687, 2699

Offset: 1

Alain Rocchelli, Dec 29 2022

nonn

This sequence corresponds to the values of p (>7) in A358527 for which p appears in second position in the factorization of 2^(p-1)-1.
All terms are congruent to 11 mod 12, cf. A068231.
It is conjectured that there are infinitely many terms in this sequence, and their estimated asymptotic density n/a(n) ~ C/(log(a(n)))^2 where C is a constant between 0.7 and 0.9.

			7 is not a term since for p=7, (2^(p-1)-1)/(3*p) = (2^6-1)/(3*7) = 3 and 3 is not greater than 7.
11 is a term since for p=11, (2^(p-1)-1)/(3*p) = (2^10-1)/(3*11) = 31, which is greater than 11.
23 is a term since (2^22-1)/(3*23) = 60787 = 89*683 and 89 is greater than 23.

Cf. A068231, A096060, A358527.

q[p_] := AllTrue[Range[p], ! PrimeQ[#] || PowerMod[2, p - 1, 3*p*#] > 1 &]; Select[Prime[Range[4, 400]], q] (* Amiram Eldar, Dec 31 2022 *)

isok(p) = (p%12==11 && isprime(p)) || return(0); forprime(div=5, p-1, if(Mod(2,div)^(p-1)==1, return(0))); 1;

A119657 Denominator of BernoulliB[10p] divided by 66, where p=Prime[n].

Original entry on oeis.org

5, 217, 1, 71, 23, 131, 1, 191, 47, 59, 311, 1, 83, 431, 1, 107, 1, 1, 1, 1, 1, 1, 167, 179, 971, 1, 1031, 1, 1091, 227, 1, 263, 1, 1, 1, 1511, 1571, 1, 1, 347, 359, 1811, 383, 1931, 1, 1, 2111

Offset: 1

Alexander Adamchuk, Jul 28 2006

nonn

The only composite in this sequence is a(2) = 217 = 7*31. All other a(n) are equal to 1 (for n=3,7,12,15,17,18,19,20,21,22,26,28,31,33,34,35,38,39,45,46..) or prime: a(1) = 5, all other primes in a(n) belong to A068231[n]: Primes congruent to 11 (mod 12). It appears that every prime from A068231[n] except 11 shows up in a(n) just once.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A068231, A119456, A051230, A051229, A002445.

Table[Denominator[BernoulliB[10Prime[n]]]/66,{n,1,47}]

a(n) = Denominator[BernoulliB[10Prime[n]]]/66.

A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

Original entry on oeis.org

4, 3, 3, 4, 5, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 5, 2, 4, 5, 2, 4, 5, 3, 2, 3, 4, 5, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 5, 3, 5, 2, 5, 2, 3, 4, 4, 4, 5, 2, 3, 5, 2, 5, 3, 5, 3, 4, 2, 3, 4, 3, 4, 5, 2, 3, 4, 2, 5, 2, 3, 5, 4, 2, 4, 5, 3, 2, 3, 2, 5, 2, 5, 2, 4, 5, 3, 2, 3, 4, 5, 5, 4, 5, 4, 5, 3, 3, 4, 2, 4, 3

Offset: 1

Zak Seidov, Jan 16 2011

nonn

a(n) = 2 iff prime(n) == 1 (mod 12); a(n) = 2 for prime(n) = 13, 37, 61, 73, 97, 109, ... (A068228).
a(n) = 5 iff prime(n) == 11 (mod 12); a(n) = 5 for prime(n) = 11, 23, 47, 59, 71, 83, ... (A068231).
For n > 2, a(n) = 3 iff prime(n) == 5 (mod 12); a(n) = 3 for prime(n) = 5, 17, 29, 41, 53, 89, ... (A040117).
For n > 2, a(n) = 4 iff prime(n) == 7 (mod 12); a(n) = 4 for prime(n) = 7, 19, 31, 43, 67, 79, ... (A068229).

Equals A039701 + A039702.

Cf. A068228, A068231, A040117, A068229, A039710.

A180217:=func< n | p mod 3 + p mod 4 where p is NthPrime(n) >; [ A180217(n): n in [1..105] ]; // Klaus Brockhaus, Jan 18 2011

Mod[#,3]+Mod[#,4]&/@Prime[Range[110]] (* Harvey P. Dale, Nov 09 2011 *)

A232040 Primes p congruent to 11 mod 12 such that (p - 1)/2 does not divide the numerator of the Bernoulli number B(p-1).

Original entry on oeis.org

1871, 2531, 3191, 3851, 5171, 6491, 7151, 9791, 13751, 14411, 15731, 18371, 19031, 20747, 21011, 24851, 24971, 26951, 27611, 30911, 34031, 34211, 34871, 35531, 36191, 37511, 37643, 40151, 41999, 42131, 43451, 44111, 44771, 46091, 46751, 48731, 49391

Offset: 1

Arkadiusz Wesolowski, Nov 17 2013

nonn

A prime p is in the sequence if p is of the form 660*n + 551.

Index entries for sequences related to Bernoulli numbers

Cf. A000367, A000928, A068231, A232039.

forstep(p=11, 49391, 12, if(isprime(p)&&!Mod(numerator(bernfrac(p-1)), (p-1)/2)==0, print1(p, ", ")));

A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 8, 4, 3, 0, 3, 6, 0, 0, 5, 0, 7, 4, 0, 0, 0, 18, 0, 0, 0, 0, 9, 4, 12, 4, 0, 14, 0, 0, 5, 8, 11, 0, 0, 6, 0, 12, 9, 0, 5, 0, 13, 6, 5, 10, 7, 14, 0, 0, 5, 0, 31, 0, 5, 0, 7, 30, 0, 12, 0, 0, 17, 6, 0, 0, 13, 18, 9, 8

Offset: 1

Omar E. Pol and Robert G. Wilson v, Nov 04 2017

a(n) = 0 iff n is in A138620.

First occurrence of k > -1: 1, 3, 8, 13, 18, 31, 28, 33, 23, 43, 66, 53, 45, 63, 48, 101, 166, etc.

			a(13) = 3 since d(12*13-1)/12 - 13 = 192/12 - 13 = 16 - 13 = 3.

Inspired by A291900.

Cf. A000203, A017653, A068231, A086463, A138620.

a[n_] := DivisorSigma[1, 12 n - 1]/12 - n; Array[a, 90]

PARI
```
a(n) = sigma(12*n-1)/12 - n;
```

a(n) = sigma(12*n-1)/12 - n = A000203(A017653(n-1))/12 - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A319535 Primes of the form 2*6^k - 1.

Original entry on oeis.org

11, 71, 431, 2591, 15551, 4353564671, 5642219814911, 341163456359156416511, 2046980738154938499071, 20628849596981071092343898111, 26734989077687468135677691953151, 207891275068097752223029732627709951, 269427092488254686881046533485512097791

Offset: 1

Jianing Song, Sep 22 2018

nonn

Primes in A164559.

Companion sequence of A057472. There are 49 terms known in this sequence.

			2*6^1 - 1 = 11, 2*6^2 - 1 = 71, 2*6^3 - 1 = 431, 2*6^4 - 1 = 2591 and 2*6^5 - 1 = 15551 are primes, but 2*6^6 - 1 = 93311 = 23*4057 is not.

K. D. Bajpai, Table of n, a(n) for n = 1..26

Integers k such that 2*b^k - 1 is prime: A090748 (b=2), A003307 (b=3), A120375 (b=5), A057472 (b=6), A002959 (b=7), A002957 (b=10), A120378 (b=11).
Primes of the form 2*b^k - 1: A000668 (b=2), A079363 (b=3), A120376 (b=5), this sequence (b=6), A158795 (b=7), A055558 (b=10), A120377 (b=11).
Cf. also A000043, A002958, A068231, A164559.

[k: n in [1..100] | IsPrime(k) where k is 2*6^n-1];  // K. D. Bajpai, Nov 15 2019

A319535:= n-> (2*6^n-1): select(isprime, [seq((A319535(n), n=1..200))]);  # K. D. Bajpai, Nov 15 2019

Select[Table[2*6^k-1,{k,1600}], PrimeQ[#]&]  (* K. D. Bajpai, Nov 15 2019 *)

for(n=1, 99, my(t); if(ispseudoprime(t=2*6^n-1), print1(t", ")))

a(n) = 2*6^A057472(n) - 1.

A132244 Twin primes congruent to {11, 13, 17, 19} mod 30.

Original entry on oeis.org

11, 13, 17, 19, 41, 43, 71, 73, 101, 103, 107, 109, 137, 139, 191, 193, 197, 199, 227, 229, 281, 283, 311, 313, 347, 349, 431, 433, 461, 463, 521, 523, 617, 619, 641, 643, 821, 823, 827, 829, 857, 859, 881, 883, 1031, 1033, 1061, 1063

Offset: 1

Omar E. Pol, Aug 17 2007

C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001097, A001359, A006512, A039949, A068231.

Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]]==2&&MemberQ[ {11,13,17,19},Mod[ #[[1]],30]]&&MemberQ[{11,13,17,19},Mod[#[[2]],30]]&]//Flatten (* Harvey P. Dale, Jul 04 2022 *)

A132245 Twin primes congruent to {1, 11, 13, 29} mod 30.

Original entry on oeis.org

11, 13, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 149, 151, 179, 181, 191, 193, 239, 241, 269, 271, 281, 283, 311, 313, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 641, 643, 659, 661, 809, 811, 821, 823, 881, 883, 1019, 1021

Offset: 1

Omar E. Pol, Aug 17 2007

C. K. Caldwell, The Prime Pages.
C. K. Caldwell, Twin Primes.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001097, A001359, A006512, A039949, A068231, A129805.

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1, q=2(prime), a=3, b=4, c=5, s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150, A155186, A068231, A129517, A054574, A132281.

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[q],AppendTo[lst,q]]],{n,8!}];lst

cp's OEIS Frontend

A263769 Smallest prime q such that q == -1 (mod prime(n)-1).

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A359387 Primes p such that the smallest prime factor of (2^(p-1)-1)/(3*p) is greater than p.

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A119657 Denominator of BernoulliB[10p] divided by 66, where p=Prime[n].

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A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

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A232040 Primes p congruent to 11 mod 12 such that (p - 1)/2 does not divide the numerator of the Bernoulli number B(p-1).

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A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

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A319535 Primes of the form 2*6^k - 1.

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A132244 Twin primes congruent to {11, 13, 17, 19} mod 30.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.