cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A164979 Slowest growing sequence of primes having the semiprime-pairwise property: for any i,j, a(i)+a(j) is semiprime.

Original entry on oeis.org

2, 7, 19, 67, 127, 6619, 126127, 345979, 476407, 1658119, 15182459419, 105169832587, 287583971287
Offset: 1

Views

Author

Zak Seidov, Sep 03 2009

Keywords

Comments

By Dirichlet's theorem and Linnik's theorem, a(n) exists for all n. - Charles R Greathouse IV, Jun 03 2025

Crossrefs

Subsequence of A045375.
Cf. A001358, A114845, A116656.

Programs

  • PARI
    lista(pmax) = {my(v = [2], ans); print1(v[1], ", "); forprime(p=3, pmax, ans = 1; for(i=1, #v, if(bigomega(p + v[i]) != 2, ans = 0; break)); if(ans, print1(p, ", "); v=concat(v, p)));} \\ Amiram Eldar, Jun 27 2024

Formula

a(n) = A114845(n)/2.
a(n) << A070826(n)^5. - Charles R Greathouse IV, Jun 03 2025

Extensions

a(12) from Amiram Eldar, Jun 27 2024
a(13) from Jinyuan Wang, May 29 2025

A346297 Slowest growing sequence of semiprimes such that any accumulating sum is a semiprime.

Original entry on oeis.org

4, 6, 15, 21, 39, 49, 51, 62, 74, 77, 87, 94, 95, 111, 129, 133, 142, 158, 166, 178, 183, 185, 187, 203, 205, 209, 214, 218, 226, 237, 287, 298, 302, 309, 323, 326, 334, 346, 355, 361, 362, 365, 371, 382, 394, 398, 451, 473, 478, 489, 497, 519, 529, 554, 562, 591, 597, 623, 649, 662, 669, 679, 689, 697, 707, 717, 746
Offset: 1

Views

Author

Zak Seidov, Sep 28 2021

Keywords

Examples

			4 + 6 = 10 = 2*5, 10 + 15 = 25 = 5*5, 25 + 21 = 46 = 2*23.

Links

Crossrefs

Cf. A001222, A001358, A116656.

Programs

  • Mathematica
    s = {4}; t = 4; Do[While[2 != PrimeOmega[n] || 2 != PrimeOmega[t + n] , n++]; AppendTo[s, n]; t = t + n; n++, {50}]; s
  • PARI
    issemi(n)=bigomega(n)==2;
first(n)=my(v=vector(n),s,t); s=v[1]=4; for(k=2,n, t=v[k-1]; while(!issemi(t++) || !issemi(s+t), ); s+=v[k]=t); v; \\ Charles R Greathouse IV, Oct 02 2021

A367855 The slowest increasing sequence of semiprimes such that a(n-1) + a(n) is prime.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 34, 39, 58, 69, 82, 85, 94, 129, 134, 143, 194, 203, 206, 213, 218, 221, 278, 291, 302, 305, 314, 327, 334, 339, 362, 365, 386, 411, 446, 473, 566, 597, 626, 633, 674, 687, 694, 745, 766, 793, 1018, 1081, 1126, 1141, 1198, 1219, 1402, 1417, 1486, 1513, 1654, 1689, 1718, 1731
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Dec 02 2023

Keywords

Comments

a(2*n) is odd and a(2*n-1) is twice a prime where n is a positive integer. - David A. Corneth, Dec 03 2023

Examples

			a(4) = 21 because a(3) = 10, 21 = 3 * 7 is a semiprime > 10, 10 + 21 = 31 is prime, and no smaller semiprime > 10 works.

Links

Crossrefs

Cf. A001358, A116656, A126938.

Programs

  • Maple
    R:= 4: s:= 4:
for count from 2 to 100 do
  for t from s+1 by 2 do
    if isprime(s+t) and numtheory:-bigomega(t) = 2 then
      R:= R,t; s:= t; break
    fi
  od
od:
R;
  • Mathematica
    s = {q = 4}; Do[p = q + 1; While[ PrimeOmega[p] != 2, p = p + 2]; AppendTo[s, q = p], {120}]; s

A280061 a(0)=0; thereafter a(n) is the smallest prime not less than a(n-1) such that a(n - 1) + a(n) is a product of n primes.

Original entry on oeis.org

0, 2, 2, 43, 47, 61, 83, 109, 467, 1453, 2003, 4909, 18131, 24877, 32467, 225581, 603859, 944429, 1267411, 1485101, 2447059, 9349421, 25253587, 53389613, 88168147, 100575533, 151082707, 989767981, 3596703443, 6738061613, 6851483347
Offset: 0

Views

Author

Zak Seidov, Jan 30 2017

Keywords

Comments

6851483347+21334239533=28185722880=2^28*3^1*5^1*7^1 (31-almost prime).
hence a(31) <= 21334239533.

Examples

			a(0)+a(1)=0+2=2 (prime = 1-almost prime)
a(1)+a(2)=2+2=4 (semiprime prime = 2-almost prime)
a(2)+a(3)=2+43=45=2*2*5 (3-almost prime)
a(29)+a(30)=6738061613+6851483347=13589544960=2^25*3^4*5^1 (30-almost prime).

Crossrefs

Cf. A001358, A116656.

Programs

  • Mathematica
    f[n_] := f[n] =  Block[{p = f[n - 1], q = NextPrime[ f[n - 1] - 1]}, While[ PrimeOmega[p + q] != n, q = NextPrime@ q]; q]; f[0] = 0; Array[f, 22] (* Robert G. Wilson v, Jan 30 2017 *)

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

Views

Author

Zak Seidov, Jun 14 2018

Keywords

Comments

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

Crossrefs

Cf. A001358, A116656, A164979, A181620.

Programs

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

Extensions

a(13) from Giovanni Resta, Jun 14 2018
a(14) from Jinyuan Wang, May 29 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.