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 1-6 of 6 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

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

A100467 Semiprimes of special form: sum of a semiprime k and the k-th semiprime.

Original entry on oeis.org

14, 21, 34, 129, 141, 158, 187, 194, 206, 221, 361, 382, 391, 393, 674, 893, 922, 934, 1067, 1094, 1133, 1293, 1415, 1441, 1473, 1569, 1589, 1681, 1703, 1739, 1769, 1982, 2119, 2206, 2362, 2395, 2433, 2481, 2507, 2602, 2614, 2627, 2642, 2823, 2839, 2983
Offset: 1

Views

Author

Jonathan Vos Post, Nov 20 2004

Keywords

Examples

			34 is in the sequence because both 9 and 9+semiprime(9) = 9+25 = 34 are semiprimes.

Links

  • Eric Weisstein, World of Mathematics, Semiprime.

Crossrefs

Cf. A001358, A100493, A100466, A100915, A100916.

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)).

A381792 Numbers k such that k + prime(k) is prime and k + semiprime(k) is semiprime.

Original entry on oeis.org

4, 6, 18, 24, 34, 72, 96, 98, 116, 130, 150, 172, 200, 206, 270, 290, 350, 356, 362, 386, 410, 420, 450, 504, 508, 554, 576, 618, 666, 682, 720, 738, 754, 782, 784, 808, 820, 832, 858, 892, 960, 962, 984, 1016, 1050, 1102, 1110, 1154, 1162, 1168, 1176, 1184, 1206, 1256, 1284, 1296, 1302, 1360
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Mar 07 2025

Keywords

Comments

All terms are even.

Examples

			a(3) = 18 is a term because the 18-th prime and 18-th semiprime are 61 and 51 respectively, 18 + 61 = 79 is prime and 18 + 51 = 69 = 3 * 23 is semiprime.

Links

Crossrefs

Intersection of A064402 and A100915.
Cf. A000040, A001222, A001358,

Programs

  • Maple
    N:= 100: # for a(1) .. a(N)
with(priqueue):
initialize(pq);
insert([-4,2,2],pq);
p:= 1:
R:= NULL: count:= 0:
for n from 1 while count < N do
  p:= nextprime(p);
  t:= extract(pq);
  if n::even and isprime(n + p) and numtheory:-bigomega(n - t[1])=2 then R:= R, n; count:= count+1 fi;
  q:= nextprime(t[3]);
  if t[2] = t[3] then insert([-q^2,q,q],pq) fi;
  insert([-t[2]*q,t[2],q],pq);
od:
R;
  • Mathematica
    lim=1360;i=1;Do[Until[PrimeOmega[i]==2,i++];Sp[n]=i,{n,lim}];Select[Range[lim],PrimeQ[#+Prime[#]]&&PrimeOmega[#+Sp[#]]==2&] (* James C. McMahon, Mar 09 2025 *)

Formula

A001222(a(n) + A000040(a(n))) = 1 and A001222(a(n) + A001358(a(n))) = 2.

A381630 a(n) is the least k such that the sum of k and the k-th number with n prime factors (counted with multiplicity) has n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 4, 8, 14, 16, 16, 96, 80, 304, 448, 640, 1984, 544, 2048, 3584, 20480, 9216, 49152, 65536, 524288, 1245184, 3309568, 204800, 1179648, 28311552, 2426880, 29360128, 6291456, 27787264, 125829120, 67108864, 327155712, 1073741824
Offset: 1

Views

Author

Robert Israel, Mar 07 2025

Keywords

Examples

			a(3) = 8 because the 8th number with 3 prime factors (the 8th triprime) is 42 = 2*3*7, 8 + 42 = 50 = 2 * 5^2 also has 3 prime factors, and 8 is the smallest number that works.

Crossrefs

Cf. A001222, A064402, A100915.

Programs

  • Maple
    f:= proc(n) uses priqueue; local pq,k,t,i,q;
    initialize(pq);
    insert([-2^n,2$n],pq);
    for k from 1 do
       t:= extract(pq);
       if numtheory:-bigomega(k-t[1])=n then return k fi;
       q:= nextprime(t[-1]);
       for i from 1 to n while t[-i] = t[-1] do
         insert([t[1]*(q/t[-1])^i,op(t[2..n+1-i]),q$i],pq);
       od
    od
end proc:
map(f, [$1..30]); # Robert Israel, Mar 07 2025

Extensions

a(32) from Jinyuan Wang, Mar 09 2025
a(33) from Jinyuan Wang, Mar 21 2025
Showing 1-6 of 6 results.

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