A164979 -id:A164979 - cp's OEIS frontend

A116656 Slowest growing sequence of semiprimes having the semiprime-pairwise-sum property: for any i

4, 6, 51, 115, 511, 5263, 116623, 204091, 823363, 1144363, 78325123, 883337023, 6860264683, 19613836423, 167589841663

Offset: 1

Zak Seidov, Feb 21 2006

			Triangle of resulting semiprimes begins:
      10
      55,     57
     119,    121,    166
     515,    517,    562,    626
    5267,   5269,   5314,   5378,   5774
  116627, 116629, 116674, 116738, 117134, 121886

Cf. A001358, A164979, A181620.

spQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; L = {0, 4}; Do[n = L[[-1]] + 1; While[! AllTrue[n + L, spQ], n++]; AppendTo[L, n], {9}]; Rest@ L (* Giovanni Resta, Jun 13 2018 *)

lista(nn) = my(m, r, s, t, u, v=vector(nn=max(2, nn))); print1(v[1]=4, ", ", v[2]=6); for(n=3, nn, m=4; r=List([3]); forprime(p=2, oo, if(m*p>v[n-1], break); u=List([]); forprime(q=2, p-1, s=Set(v%(t=p*q)); for(i=1, #s, listput(u, Mod(t-s[i], t)))); s=List([]); for(i=1, #r, forstep(k=r[i], m*p, m, t=1; for(j=1, #u, if(k==u[j], t=0; break)); if(t, listput(s, k)))); r=s; m*=p); listsort(r); forstep(i=0, oo, m, for(j=1, #r, t=i+r[j]; if(t>v[n-1]&&bigomega(t)==2&&bigomega(t+4)==2&&bigomega(t+6)==2, for(k=3, n-1, if(!isprime((t+v[k])\2), t=0; break)); if(t, print1(", ", v[n]=t); break(2)))))); \\ Jinyuan Wang, May 29 2025

a(8)-a(10) from R. J. Mathar, Jan 23 2008
a(11)-a(12) from Donovan Johnson, Nov 11 2008
a(13) from Donovan Johnson, Jul 22 2011
a(14) from Giovanni Resta, Jun 13 2018
a(15) from Giovanni Resta, Jun 14 2018

A114845 Slowest growing sequence of semiprimes having the semiprime-pairwise-average property: for any i,j, (a(i)+a(j))/2 is semiprime.

Original entry on oeis.org

4, 14, 38, 134, 254, 13238, 252254, 691958, 952814, 3316238, 30364918838, 210339665174, 575167942574

Offset: 1

Jonathan Vos Post, Feb 20 2006

Semiprime analog of A113875.

			The pairwise average of the semiprimes {4 = 2^2, 14 = 2*7} is {9 = 3^2}.
The pairwise averages of the semiprimes {4, 14, 38} are {9, 21, 26}.
The pairwise averages of the semiprimes {4, 14, 38, 134} are {9, 21, 26, 69, 74, 86}.
The pairwise averages of the semiprimes {4, 14, 38, 134, 254} are {9, 21, 26, 69, 74, 86, 129, 134, 146, 194}.

Cf. A001358, A113832, A113875, A115760, A164979.

a(n) = 2*A164979(n).

More terms from Zak Seidov, Feb 21 2006
Corrected and extended by Zak Seidov, Sep 03 2009
a(11)-a(12) from Amiram Eldar, Jun 27 2024
a(13) from Jinyuan Wang, May 29 2025

A305887 The least increasing sequence of numbers where all pairwise sums are semiprimes, with a(1)=4.

Original entry on oeis.org

4, 5, 10, 29, 173, 249, 19073, 71489, 1166789, 3800333, 7021253, 15920129, 84551600693, 224223772673

Offset: 1

Zak Seidov, Jun 14 2018

All terms > 10 are congruent to {3, 9} mod 10.
Triangle of resulting semiprimes begins:
    9
   14,  15
   33,  34,  39
  177, 178, 183, 202

Cf. A001358, A116656, A164979, A181620.

Nest[Append[#, Block[{k = Last[#] + 1}, While[! AllTrue[#, PrimeOmega[k + #] == 2 &], k++]; k]] &, {4}, 7] (* Michael De Vlieger, Jun 14 2018 *)

a(13) from Giovanni Resta, Jun 14 2018

a(14) from Jinyuan Wang, May 29 2025

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A116656 Slowest growing sequence of semiprimes having the semiprime-pairwise-sum property: for any i

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A114845 Slowest growing sequence of semiprimes having the semiprime-pairwise-average property: for any i,j, (a(i)+a(j))/2 is semiprime.

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A305887 The least increasing sequence of numbers where all pairwise sums are semiprimes, with a(1)=4.

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Mathematica

Extensions