A091022 -id:A091022 - cp's OEIS frontend

A106349 Primes indexed by semiprimes.

7, 13, 23, 29, 43, 47, 73, 79, 97, 101, 137, 139, 149, 163, 167, 199, 227, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 661, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947

Offset: 1

Jonathan Vos Post, Apr 29 2005

This is the sequence of the k-th prime for k = {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,...}. Not to be confused with A106350: semiprimes indexed by primes.

			a(1) = 7 because semiprime(1) = 4, so prime(semiprime(1)) = prime(4) = 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Paul Kinlaw, Megan Triplett, and William Tripp, Almost Primes of Almost Prime Index, INTEGERS, Vol 24 (2024), Article #A99.

Cf. A000040, A000720, A001358, A007097, A091022, A105997, A105998, A106350.

[NthPrime(n): n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Vincenzo Librandi, Nov 28 2015

Prime@ Select[Range@ 161, PrimeOmega@ # == 2 &] (* or *) Select[Prime@ Range@ 161, PrimeOmega@ PrimePi@ # == 2 &] (* Michael De Vlieger, Nov 28 2015 *)

lista(nn) = select(x->(bigomega(primepi(x))==2), primes(nn)); \\ Michel Marcus, Nov 29 2015

a(n) = prime(semiprime(n)).
a(n) = A000040(A001358(n)).
pi(a(n)) = p*q for some primes p and q.
Sum_{n>=1} 1/a(n) is in the interval (0.9910, 0.9915) (Kinlaw et al., 2024, Theorem 6, p. 11). - Amiram Eldar, Nov 09 2024

A106350 Semiprimes indexed by primes.

Original entry on oeis.org

6, 9, 14, 21, 33, 35, 49, 55, 65, 86, 91, 115, 122, 129, 142, 159, 183, 187, 206, 215, 218, 247, 259, 287, 303, 319, 323, 334, 339, 358, 403, 415, 446, 451, 482, 489, 511, 527, 537, 553, 573, 581, 626, 633, 655, 667, 698, 737, 753, 758, 771, 791, 794, 835, 851

Offset: 1

Jonathan Vos Post, Apr 30 2005

This is the sequence of the n-th semiprime for n = {2,3,5,7,11,13,17,19,23,29...}. Not to be confused with A106349: Primes indexed by semiprimes. We seek to know what this sequence is asymptotically, as J. B. Rosser's result, subsequently modified, is that prime(n) ~ n*(log n + log log n - 1). hence semiprime(prime(n)) ~ semiprime(n)*(log semiprime(n) + log log semiprime(n) - 1). But what is, asymptotically, semiprime(n)?

Semiprime(n) ~ n log n / log log n, hence a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011

			a(1) = semiprime(prime(1)) = semiprime(2) = 6.
a(2) = semiprime(prime(2)) = semiprime(3) = 9.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. B. Rosser, The n-th Prime is Greater than n log(n), Proc. London Math. Soc. 45, 21-44, 1939.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A000040, A001358, A007097, A091022, A105997, A105998, A106349.

A001358 := proc(n) if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a ; end if; end do ; end if ; end proc: A106350 := proc(n) A001358(ithprime(n)) ; end proc: seq(A106350(n),n=1..80) ; # R. J. Mathar, Dec 14 2009

terms = 55;
semiPrimes = Select[Range[16 terms], PrimeOmega[#] == 2&];
(* NB If the index Prime[terms] exceeds the size of the table semiPrimes, then the coefficient 16 has to be increased according to the number of terms desired: for instance, for 1000 terms, replace 16 with 32. *)
a[n_] := semiPrimes[[Prime[n]]];
Array[a, terms] (* Jean-François Alcover, Apr 13 2020 *)

a(n) = semiprime(prime(n)). a(n) = A001358(A000040(n)).

a(n) ~ n log^2 n / log log n. - Charles R Greathouse IV, Dec 28 2011

All values after a(32) corrected by R. J. Mathar, Dec 14 2009

A105997 Semiprime function n -> A001358(n) applied three times to n.

Original entry on oeis.org

26, 39, 74, 77, 118, 119, 178, 194, 219, 235, 299, 301, 329, 377, 381, 454, 471, 502, 535, 565, 566, 634, 679, 703, 721, 779, 842, 886, 893, 914, 973, 995, 998, 1006, 1126, 1174, 1227, 1282, 1294, 1317, 1337, 1343, 1389, 1418, 1457, 1563, 1577, 1623, 1642

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(1))) = semiprime(semiprime(4)) = semiprime(10) = 26.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105998, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@3)(n):
seq(a(n), n=1..49);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 1700], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 3], {n, 50}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105997(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return A001358(A001358(A001358(n))) # Chai Wah Wu, Aug 16 2024

a(n) = A001358(A001358(A001358(n))).

Corrected and extended by Robert G. Wilson v, Apr 30 2005

A105998 Semiprime function n -> A001358(n) applied four times to n.

Original entry on oeis.org

77, 119, 219, 235, 377, 381, 566, 634, 721, 779, 998, 1006, 1126, 1282, 1294, 1563, 1642, 1745, 1853, 1959, 1961, 2209, 2402, 2483, 2554, 2785, 3005, 3149, 3173, 3242, 3481, 3574, 3587, 3622, 4101, 4282, 4471, 4681, 4714, 4798, 4859, 4882, 5095, 5201

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(semiprime(1)))) = semiprime(semiprime(semiprime(4))) = semiprime(semiprime(10)) = semiprime(26) = 77.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105997, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@4)(n):
seq(a(n), n=1..44);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 5210], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 4], {n, 45}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105998(n):
    def f(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n)+n
        while m != k:
            m, k = k, f(k)+n
        return m
    return A001358(A001358(A001358(A001358(n)))) # Chai Wah Wu, Aug 16 2024

a(n) = A001358(A001358(A001358(A001358(n)))).

More terms from Robert G. Wilson v, Apr 30 2005

A105999 Semiprimeth recurrence: a(0) = 1, a(n+1) = semiprime(a(n)) = A001358(a(n)).

Original entry on oeis.org

1, 4, 10, 26, 77, 235, 779, 2785, 10643, 43697, 192893, 915218, 4657929, 25380749, 147721169, 916036271, 6037442989, 42191467826, 311911160465, 2434014941905, 20007995450483, 172911791611798, 1568190042677867

Offset: 0

Jonathan Vos Post, Apr 29 2005

nonn

Semiprime equivalent of R. G. Wilson's primeth recurrence: A007097.

			a(1) = A001358(1) = 4,
a(2) = A001358(a(1)) = A001358(4) = 10,
a(3) = A001358(a(2)) = A001358(10) = 26.

Cf. A001358, A007097, A091022, A105997, A105998.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; NestList[SemiPrime@# &, 1, 18] (* Robert G. Wilson v, May 31 2006 *)

a(5)-a(15) from Robert G. Wilson v, Apr 30 2005
a(16)-a(20) from Robert G. Wilson v, May 31 2006
a(21)-a(22) from Donovan Johnson, Sep 24 2010

A105346 3-almost primes whose indices are 3-almost primes.

Original entry on oeis.org

42, 52, 76, 92, 116, 117, 125, 174, 182, 186, 212, 230, 266, 275, 282, 285, 316, 318, 325, 385, 406, 410, 423, 428, 436, 455, 470, 474, 507, 508, 534, 575, 604, 605, 618, 627, 654, 657, 670, 678, 682, 705, 710, 730, 754, 762, 772, 788, 834, 861, 903, 931

Offset: 1

Jonathan Vos Post, Apr 30 2005

The n-th 3-almost prime function applied to itself. This is the 3-almost prime equivalent of A091022, the latter being the n-th 2-almost prime function applied to itself. Note that this new iterated 3-almost prime sequence begins with the meaning of "Life, the Universe and Everything" and then generalizes to include the number of playing cards in a deck and the boiling point of water on the Fahrenheit scale.

			a(1) = 3-almost-prime(3-almost-prime(1)) = 3-almost-prime(8) = 42.
a(2) = 3-almost-prime(3-almost-prime(2)) = 3-almost-prime(12) = 52.
a(3) = 3-almost-prime(3-almost-prime(3)) = 3-almost-prime(18) = 76.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A014612, A000040, A001358, A007097, A091022, A105997, A105998, A101349, A106350.

isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: A014612 := proc(n) option remember ; if n =1 then 8; else for a from procname(n-1)+1 do if isA014612(a) then RETURN(a) ; fi; od; fi; end: for n from 1 to 100 do q := A014612(A014612(n)) ; printf("%d,",q) ; od: # R. J. Mathar, Jan 27 2009

With[{tap=Select[Range[2000],PrimeOmega[#]==3&]},Table[tap[[tap[[n]]]],{n,100}]] (* Harvey P. Dale, May 20 2019 *)

do(lim)=my(v=List(), t); forprime(p=2, lim\4, forprime(q=2, min(lim\(2*p), p), t=p*q; forprime(r=2, min(lim\t, q), listput(v, t*r)))); v=Set(v); t=setsearch(v,#v); if(!t, t=setsearch(v,#v,1)-1); vector(t,i,v[v[i]]) \\ Charles R Greathouse IV, Feb 05 2017

a(n) = A014612(A014612(n)).

Extended by R. J. Mathar, Jan 27 2009

A176654 Numbers k such that both semiprime(k)/p and semiprime(semiprime(k))/p are prime for some prime p.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 14, 20, 21, 22, 24, 27, 28, 42, 43, 47, 52, 58, 62, 64, 65, 66, 70, 73, 75, 82, 87, 92, 97, 105, 109, 111, 116, 129, 130, 133, 135, 147, 149, 150, 161, 170, 171, 172, 189, 191, 195, 208, 220, 222, 224, 227, 241, 246, 267, 274, 276, 277, 281, 287

Offset: 1

Juri-Stepan Gerasimov, Apr 22 2010

nonn

Indices n such that A001358(n) and A091022(n) share at least one prime factor. - R. J. Mathar, Apr 26 2010

			1 is a term because both semiprime(1)/2 = 4/2 = 2 and semiprime(semiprime(1))/2 = 10/2 = 5 are prime;
2 is a term because both semiprime(2)/3 = 6/3 = 2 and semiprime(semiprime(2))/3 = 15/3 = 5 are prime;
4 is a term because both semiprime(4)/2 = 10/2 = 5 and semiprime(semiprime(4))/2 = 26/2 = 13 are prime.

Cf. A001358, A091022.

A091022 := proc(n) A001358(A001358(n)) ; end proc: seq(A091022(n),n=1..20) ; isA176654 := proc(n) pfsn := convert(numtheory[factorset]( A001358(n) ),list) ; pfsn1 := convert(numtheory[factorset]( A091022(n) ),list) ; op(1,pfsn) = op(1,pfsn1) or op(1,pfsn) = op(-1,pfsn1) or op(-1,pfsn) = op(1,pfsn1) or op(-1,pfsn) = op(-1,pfsn1) ; end proc: for n from 1 to 1600 do if isA176654(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 26 2010

Most values after a(6) replaced by R. J. Mathar, Apr 26 2010

A176658 Numbers k such that semiprime(semiprime(k)) + 1 = semiprime(semiprime(k+1)).

Original entry on oeis.org

3, 5, 11, 14, 20, 32, 52, 57, 70, 72, 81, 95, 114, 124, 231, 240, 273, 276, 287, 291, 371, 380, 441, 507, 528, 544, 573, 607, 629, 647, 672, 695, 716, 739, 828, 830, 832, 873, 1002, 1035, 1037, 1044, 1100, 1104, 1182, 1208, 1236, 1278, 1321, 1340, 1367, 1522

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2010

nonn

Numbers k such that A001358(A001358(k)) + 1 = A001358(A001358(k+1)).

Numbers k such that A091022(k) + 1 = A091022(k+1).

			3 is a term because semiprime(semiprime(3)) + 1 = 25 + 1 = semiprime(semiprime(3+1)).

Cf. A001358, A091022.

A091022 := proc(n) A001358(A001358(n)) ; end proc: isA176658 := proc(n) A091022(n)+1 = A091022(n+1) ; end proc: for n from 1 to 1600 do if isA176658(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 26 2010

Corrected (72 inserted, 85 replaced by 95, 124 inserted) and extended by R. J. Mathar, Apr 26 2010

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A106349 Primes indexed by semiprimes.

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A106350 Semiprimes indexed by primes.

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A105997 Semiprime function n -> A001358(n) applied three times to n.

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A105998 Semiprime function n -> A001358(n) applied four times to n.

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A105999 Semiprimeth recurrence: a(0) = 1, a(n+1) = semiprime(a(n)) = A001358(a(n)).

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A105346 3-almost primes whose indices are 3-almost primes.

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A176654 Numbers k such that both semiprime(k)/p and semiprime(semiprime(k))/p are prime for some prime p.

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