A358149 -id:A358149 - cp's OEIS frontend

A358155 First of four consecutive primes p,q,r,s such that (2p+q)/5, (q+r)/10 and (r+2s)/5 are prime.

11, 2696717, 3500381, 3989903, 4515113, 8164073, 12451013, 18793013, 23567267, 24057713, 30312409, 36391853, 44569853, 45657881, 53442343, 54721253, 54944761, 56652203, 63993803, 76763081, 90662303, 92889127, 94670143, 105790973, 106339481, 108988223, 117213871, 118802533, 130741007, 145543523

Offset: 1

J. M. Bergot and Robert Israel, Nov 01 2022

nonn

11 is the only term that is in A007530, because if p is in A007530 (so q = p+2, r = p+6 and s = p+8), one of p, q, r, s, 2*p+q, q+r and r+2*s is divisible by 7.

			a(3) = 3500381 is a term because 3500381, 3500383, 3500407, 3500429 are four consecutive primes with (2*3500381 + 3500383)/5 = 2100229, (3500383 + 3500407)/10 = 700079, and (3500407 + 2*3500429)/5 = 2100253 all prime.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A007530, A358149.

Res:= NULL: count:= 0:
q:= 2: r:= 3: s:= 5:
while count < 40 do
  p:= q; q:= r; r:= s; s:= nextprime(s);
  t:= (2*p+q)/5; u:= (q+r)/10; v:= (r+2*s)/5;
  if (t::integer and u::integer and v::integer and isprime(t) and isprime(u) and isprime(v)) then
    count:= count+1; Res:= Res, p;
  fi
od:
Res;

Select[Partition[Prime[Range[8.3*10^6]], 4, 1], PrimeQ[(2*#[[1]] + #[[2]])/5] && PrimeQ[(#[[2]] + #[[3]])/10] && PrimeQ[(#[[3]] + 2*#[[4]])/5] &][[;; , 1]] (* Amiram Eldar, Nov 01 2022 *)

A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2p+q)/5 is a prime and (r+2s)/5^n is a prime.

Original entry on oeis.org

11, 101, 243701, 6758951, 3257480201, 5493848951, 58634348951, 218007942701, 21840280598951, 213065296223951, 186522444661451, 383378987630201, 7794174397786451, 110420241292317701, 67327687581380201, 91455128987630201, 3987035878499348951, 80659241994222005201, 4289429982503255201

Offset: 1

J. M. Bergot and Robert Israel, Nov 02 2022

nonn

			a(3) = 243701 because p = 247301, q = p+2 = 247303, r = p+6 = 243707, s = p+8 = 243709, (2*p+q)/5 = 146221 and (r+2*s)/5^3 = 5849 are primes, and p is the least prime that works.

Cf. A007530, A358149.

f:= proc(n) local t,p,m;
        m:= 5^n;
        t:= 3;
        do
          t:= nextprime(t);
          if t*m mod 3 <> 1 then next fi;
          p:= (t*m-22)/3;
          if isprime(p) and isprime(p+2) and isprime(p+6) and isprime(p+8) and isprime((3*p+2)/5) then return p fi;
        od;
end proc;
map(f, [$1..20]);

a[n_] := a[n] = Module[{t = 3, p, m = 5^n}, While[True, t = NextPrime[t]; If[Mod[t*m, 3] != 1, Continue[]]; p = (t*m - 22)/3; If[AllTrue[{p, p+2, p+6, p+8, (3p+2)/5}, PrimeQ], Return[p]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* Jean-François Alcover, Jan 31 2023, after Maple program *)

cp's OEIS Frontend

A358155 First of four consecutive primes p,q,r,s such that (2p+q)/5, (q+r)/10 and (r+2s)/5 are prime.

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A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2p+q)/5 is a prime and (r+2s)/5^n is a prime.

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Maple

Mathematica

cp's OEIS Frontend

A358155 First of four consecutive primes p,q,r,s such that (2*p+q)/5, (q+r)/10 and (r+2*s)/5 are prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2*p+q)/5 is a prime and (r+2*s)/5^n is a prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A358155 First of four consecutive primes p,q,r,s such that (2p+q)/5, (q+r)/10 and (r+2s)/5 are prime.

A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2p+q)/5 is a prime and (r+2s)/5^n is a prime.