A118818 -id:A118818 - cp's OEIS frontend

A103828 Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1.

1, 3, 9, 27, 69, 429, 1059, 56499, 166839, 5020059, 7681809, 274343589, 8316187179, 2866819175649, 7180244842749, 216549352241349, 22129340663539629, 2504509324460255499

Offset: 1

Walter Kehowski, May 29 2006

Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?
Except for a(1), a(n) == 3 (mod 6). - Robert G. Wilson v, Jun 02 2006.
The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n>2, a(n)+2 and a(n)+4 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - N. J. A. Sloane, Apr 21 2007
From the mod 30 property of A115760 we conclude that a(n) == 9 (mod 15) for n>4. This implies that either a(n) == 9 (mod 30) or == 24 (mod 30), but == 24 (mod 30) is impossible because then == 0 (mod 6). Therefore a(n) == 9 (mod 30) for n>4. - Don Reble, Aug 17 2021

			a(1)=1, a(2)=3, but 5+1+1=7, 5+3+1=9; 7+1+1=9, 7+3+1=11; 9+1+1=11, 9+3+1=13 so a(3)=9.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A093483, A115760, A115782 (primes arising from this sequence), A118818, A128933 (a(n)+1), A291163.

EP:=[]: for w to 1 do for n from 1 to 8*10^6 do s:=2*n-1; Q:=map(z->z+s+1, EP); if andmap(isprime,Q) then EP:=[op(EP),s]; print(nops(EP),s); fi od od; EP;

a[1] = 1; a[2] = 3; a[n_] := a[n] = Block[{k = a[n - 1] + 6, t = Table[ a[i], {i, n - 1}] + 1}, While[ First@ Union@ PrimeQ[k + t] == False, k += 6]; k]; Do[ Print[ a[n]], {n, 15}] (* Robert G. Wilson v, Jun 03 2006 *)

a(n) = (A115760(n) - 1)/2.

a(12) from Robert G. Wilson v, Jun 03 2006
a(13) from Walter Kehowski, Jun 03 2006
Definition corrected by Walter Kehowski, Nov 03 2008
a(14)-a(16) from Don Reble added by N. J. A. Sloane, Sep 18 2012
a(17)-a(18) from Don Reble, Aug 17 2021

A172480 Odd primes p such that there are as many primitive roots (mod p) in the interval [0,p/2] as in the interval [p/2,p].

Original entry on oeis.org

5, 7, 13, 17, 29, 31, 37, 41, 43, 53, 61, 67, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 367, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487, 509, 521, 541, 557, 569

Offset: 1

Emmanuel Vantieghem, Feb 04 2010

nonn

The sequence contains all the primes of the form 4m+1 (A002144).

The sequence also contains some primes of the form 4m+3 (see them in A172490).

Robert Israel, Table of n, a(n) for n = 1..10000

A002144 is a subsequence.

Cf. A118818, A172490

filter:= proc(p) local m; uses NumberTheory;
  if not isprime(p) then return false fi;
  if p mod 4 = 1 then return true fi;
  m:= Totient(Totient(p))/2;
  PrimitiveRoot(p,ith=m+1)=PrimitiveRoot(p,greaterthan=floor(p/2))
end proc:
select(filter, [seq(i,i=5..1000,2)]); # Robert Israel, Nov 23 2019

<< NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, s = {s, p}, q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1,] ]; If[e == 2t, s = {s, p},] ] ]; s = Flatten[s]

A128859 Let p be the n-th odd prime; a(n) is the number of primitive roots of p which are <= (p-1)/2.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 2, 4, 6, 4, 6, 8, 6, 9, 12, 10, 8, 10, 9, 12, 10, 16, 20, 16, 20, 13, 22, 18, 24, 17, 18, 32, 18, 36, 16, 24, 26, 36, 42, 37, 24, 29, 32, 42, 23, 16, 38, 49, 36, 56, 44, 32, 44, 64, 59, 66, 33, 44, 48, 40, 72, 48, 55, 48, 78, 35, 48, 79, 56, 80, 80, 60, 60, 55

Offset: 1

N. J. A. Sloane, Apr 20 2007

Roger Osborn, Tables of All Primitive Roots of Odd Primes Less Than 1000, Univ. Texas Press, Austin, TX, 1961, pp. 69-70.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A118818.

a[n_] := Count[PrimitiveRootList[(p = Prime[n+1])], _?(# <= (p-1)/2 &)]; Array[a, 100]

More terms from Don Reble, Apr 20 2007

A172490 Primes p of the form 4m+3 for which there are exactly as many primitive roots modulo p in the interval [0,p/2] as in the interval [p/2,p].

Original entry on oeis.org

7, 31, 43, 67, 307, 367, 487, 643, 1327, 1663, 2371, 3643, 3847, 4327, 4951, 6091, 6571, 8263, 9151, 9187, 11239, 11383, 11863, 15307, 24007, 24151, 27847, 30091, 30643, 33619, 36871, 42187, 44171, 46279, 46591, 48787, 70843, 71887, 72103, 72379, 73363, 79867, 82003, 92503, 95467, 106243, 110431, 120943, 126031, 130363, 139759, 143827, 162751, 167107, 173191, 174859, 183247

Offset: 1

Emmanuel Vantieghem, Feb 05 2010

Primes 4*k+3 where half of the primitive roots are <= (p-1)/2.
The sequence is probably infinite.
Primes of the form 4m+1 always have as many primitive roots in [0,p/2] as in [p/2,p] (see A172480).

Cf. A118818, A172480.

with(numtheory): p:=3: while p<1000 do if(p mod 4 = 3)then b1:=0: b2:=0: m:=primroot(p): while not m=FAIL do if(mNathaniel Johnston, Jun 26 2011

<< NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, , q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1,] ]; If[e == 2t, s = {s, p},] ] ]; s = Flatten[s]

isA172490(p)=isprime(p)&&p%4==3&&sum(n=0,p\2,gcd(n,p)==1&&znorder(Mod(n,p))==p-1)==sum(n=p-p\2,p,gcd(n,p)==1&&znorder(Mod(n,p))==p-1) \\ Charles R Greathouse IV, Jun 27 2011

More terms from Robert Israel, Nov 23 2019

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A103828 Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1.

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A172480 Odd primes p such that there are as many primitive roots (mod p) in the interval [0,p/2] as in the interval [p/2,p].

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A128859 Let p be the n-th odd prime; a(n) is the number of primitive roots of p which are <= (p-1)/2.

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A172490 Primes p of the form 4m+3 for which there are exactly as many primitive roots modulo p in the interval [0,p/2] as in the interval [p/2,p].

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