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 1-4 of 4 results.

A100493 a(n) = n + n-th semiprime.

Original entry on oeis.org

5, 8, 12, 14, 19, 21, 28, 30, 34, 36, 44, 46, 48, 52, 54, 62, 66, 69, 74, 77, 79, 84, 88, 93, 99, 103, 109, 113, 115, 117, 122, 125, 127, 129, 141, 147, 152, 156, 158, 161, 163, 165, 172, 177, 179, 187, 189, 191, 194, 196, 206, 210, 212, 215, 221, 225, 234, 236
Offset: 1

Views

Author

Jonathan Vos Post, Nov 20 2004

Keywords

Comments

This is the semiprime analog of A014688.

Examples

			a(7) = 7 + semiprime(7) = 7 + 21 = 28.

Links

Crossrefs

Cf. A001358, A014688, A100466, A100467, A100915, A100916.

Programs

  • Magma
    m:=300;
A001222:=[n eq 1 select 0 else (&+[p[2]: p in Factorization(n)]): n in [1..4*m]];
A001358:=[n: n in [1..4*m] | A001222[n] eq 2];
A100493:= func< n | n + A001358[n] >;
[A100493(n): n in [1..m]]; // G. C. Greubel, Apr 04 2023
  • Maple
    N:= 1000: # to use semiprimes <= N
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
Semiprimes:= sort(convert(select(`<=`,{seq(seq(Primes[i]*Primes[j],i=1..j),j=1..nops(Primes))},N),list)):
seq(i+Semiprimes[i],i=1..nops(Semiprimes)); # Robert Israel, Dec 20 2015
  • Mathematica
    m=300;
A001358:= A001358= Select[Range[5*m], PrimeOmega[#]==2 &];
A100493[n_]:= n + A001358[[n]];
Table[A100493[n], {n, m}] (* G. C. Greubel, Apr 04 2023 *)
  • PARI
    lista(n)= my(s=0); vector(n, i, while(2!=bigomega(s++), ); i+s); \\ Ruud H.G. van Tol, Mar 10 2025
  • SageMath
    from sympy import primeomega
b=[n for n in (1..1000) if primeomega(n)==2]
[n+b[n-1] for n in range(1,301)] # G. C. Greubel, Apr 04 2023

Formula

a(n) = n + A001358(n).
a(n) ~ n log n / log log n. [Charles R Greathouse IV, Dec 28 2011]

Extensions

Edited, corrected and extended by Ray Chandler, Nov 26 2004

A100915 Numbers n such that n plus n-th semiprime is semiprime.

Original entry on oeis.org

4, 6, 9, 12, 16, 18, 19, 20, 24, 29, 31, 34, 35, 39, 40, 44, 46, 49, 51, 54, 55, 72, 73, 76, 79, 80, 81, 84, 87, 91, 93, 94, 96, 98, 110, 113, 116, 120, 128, 130, 136, 137, 148, 150, 154, 159, 165, 168, 170, 172, 175, 188, 190, 191, 199, 200, 206, 215, 217, 220, 230
Offset: 1

Views

Author

Ray Chandler, Nov 26 2004

Keywords

Comments

This is the semiprime analog of A064402.

Examples

			a(3) = 9 because 9 + semiprime(9) = 9 + 25 = 34 is semiprime.

Links

Crossrefs

Cf. A001358, A064402, A100493, A100466, A100467, A100916.

Programs

  • Mathematica
    With[{c=Select[Range[1000],PrimeOmega[#]==2&]},Transpose[Select[Thread[ {c,Range[ Length[c]]}], PrimeOmega[Total[#]]==2&]][[2]]] (* Harvey P. Dale, Oct 25 2011 *)

Formula

a(n) = A100466(n) - A100916(n) = A100466(n) - A001358(A100915(n)).

A100466 Semiprimes of special form: sum of an integer k and the k-th semiprime.

Original entry on oeis.org

14, 21, 34, 46, 62, 69, 74, 77, 93, 115, 122, 129, 141, 158, 161, 177, 187, 194, 206, 215, 221, 289, 291, 302, 326, 329, 334, 346, 361, 382, 391, 393, 398, 403, 451, 471, 481, 502, 535, 543, 581, 583, 629, 635, 655, 674, 698, 706, 713, 723, 734, 802, 813
Offset: 1

Views

Author

Jonathan Vos Post, Nov 20 2004

Keywords

Comments

This is the semiprime analog of A061068.

Examples

			a(3) = 34 because 34 is the third semiprime appearing in A100493.

Links

  • Eric Weisstein, World of Mathematics, Semiprime.

Crossrefs

Cf. A001358, A061068, A100493, A100467, A100915, A100916.

Formula

a(n) = A100915(n) + A100916(n) = A100915(n) + A001358(A100915(n)).

Extensions

Edited, corrected and extended by Ray Chandler, Nov 26 2004

A100916 Sum of a semiprime and its semiprime index is a new semiprime.

Original entry on oeis.org

10, 15, 25, 34, 46, 51, 55, 57, 69, 86, 91, 95, 106, 119, 121, 133, 141, 145, 155, 161, 166, 217, 218, 226, 247, 249, 253, 262, 274, 291, 298, 299, 302, 305, 341, 358, 365, 382, 407, 413, 445, 446, 481, 485, 501, 515, 533, 538, 543, 551, 559, 614, 623, 626
Offset: 1

Views

Author

Ray Chandler, Nov 26 2004

Keywords

Comments

This is the semiprime analog of A061067.

Examples

			a(1) = 10 because 10 = semiprime(4) and semiprime(4) + 4 = 14 is
semiprime.
a(2) = 15 because 15 = semiprime(6) and semiprime(6) + 6 = 21 is
semiprime.

Links

  • Eric Weisstein, World of Mathematics, Semiprime.

Crossrefs

Cf. A001358, A061067, A100493, A100466, A100467, A100915.

Programs

  • Mathematica
    Module[{sp=Select[Range[1000],PrimeOmega[#]==2&],len},len=Length[sp];Select[ Thread[{sp,Range[len]}],PrimeOmega[Total[#]]==2&]][[All,1]] (* Harvey P. Dale, Jan 13 2019 *)

Formula

a(n) = A100466(n) - A100915(n) = A001358(A100915(n)).
Showing 1-4 of 4 results.

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