A106350 -id:A106350 - 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

A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

Original entry on oeis.org

1, 4, 9, 8, 10, 12, 24, 24, 32, 15, 46, 24, 27, 34, 25, 40, 44, 46, 51, 54, 53, 46, 54, 60, 70, 70, 98, 105, 104, 91, 64, 72, 45, 48, 95, 118, 120, 120, 116, 108, 100, 96, 101, 118, 102, 144, 123, 86, 76, 81, 136, 138, 143, 112, 132, 131, 153, 160, 171, 169

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(semiprime(1)) - semiprime(prime(1)) = prime(4) - semiprime(2) = 7 - 6 = 1.
a(2) = prime(semiprime(2)) - semiprime(prime(2)) = prime(6) - semiprime(3) = 13 - 9 = 4.
a(3) = prime(semiprime(3)) - semiprime(prime(3)) = prime(9) - semiprime(5) = 23 - 14 = 9.
a(4) = prime(semiprime(4)) - semiprime(prime(4)) = prime(10) - semiprime(7) = 29 - 21 = 8.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A106349, A106350.

sp = Select[Range[1000], PrimeOmega[#] == 2 &]; Table[ Prime[ sp[[i]]] - sp[[Prime[i]]], {i, PrimePi@ Length@ sp}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = A106349(n) - A106350(n).

a(33)-a(54) corrected by and a(55)-a(60) from Giovanni Resta, Jun 13 2016

A124268 Primes indexed by 3-almost primes.

Original entry on oeis.org

19, 37, 61, 71, 103, 107, 113, 181, 193, 197, 229, 239, 307, 317, 337, 349, 379, 383, 397, 479, 521, 523, 557, 571, 601, 619, 641, 643, 683, 691, 733, 787, 853, 857, 883, 887, 971, 977, 1013, 1019, 1021, 1033, 1039, 1091, 1109, 1123, 1151, 1187, 1279

Offset: 1

Jonathan Vos Post, Oct 23 2006

3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

			a(1) = prime(3almostprime(1)) = prime(8) = 19.
a(2) = prime(3almostprime(2)) = prime(12) = 37.
a(3) = prime(3almostprime(3)) = prime(18) = 61.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A122824, A124269, A124270.

Prime[#]&/@Select[Range[400],PrimeOmega[#]==3&] (* Harvey P. Dale, Mar 19 2020 *)

a(n) = prime(3almostprime(n)) = A000040(A014612(n)). {p such that p is prime and omega(primepi(p)) = 3} = {p such that p is in A000040 and A001222(A000720(p)) = 3}.

A124269 3-almost primes indexed by primes.

Original entry on oeis.org

12, 18, 27, 30, 50, 63, 75, 78, 102, 124, 130, 164, 172, 175, 190, 231, 246, 258, 279, 286, 292, 332, 345, 369, 404, 418, 425, 430, 435, 452, 524, 539, 574, 578, 606, 610, 638, 652, 663, 692, 722, 725, 775, 782, 795, 801, 854, 906, 916, 927, 938, 963, 969

Offset: 1

Jonathan Vos Post, Oct 23 2006

Primes indexed by 3-almostprimes = A124268. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

			a(1) = 3almostprime(prime(1)) = 3almostprime(2) = 12 = 2^2 * 3.
a(2) = 3almostprime(prime(2)) = 3almostprime(3) = 18 = 2 * 3^2.
a(3) = 3almostprime(prime(3)) = 3almostprime(5) = 27 = 3^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A122824, A124268, A124270.

From R. J. Mathar, Oct 15 2010: (Start)
A014612 := proc(n) option remember; if n = 1 then 8; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then return a; end if; end do; end if; end proc:
A124269 := proc(n) A014612(ithprime(n)) ; end proc: seq(A124269(n),n=1..80) ; (End)

p3 = Select[Range[1000], PrimeOmega[#] == 3 &]; p3[[Prime@ Range@ PrimePi@ Length@ p3]] (* Giovanni Resta, Jun 13 2016 *)

a(n) = 3almostprime(prime(n)) = A014612(A000040(n)).

More terms from R. J. Mathar, Oct 15 2010

A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

Original entry on oeis.org

7, 19, 34, 41, 53, 44, 38, 103, 91, 73, 99, 75, 135, 142, 147, 118, 133, 125, 118, 193, 229, 191, 212, 202, 197, 201, 216, 213, 248, 239, 209, 248, 279, 279, 277, 277, 333, 325, 350, 327, 299, 308, 264, 309, 314, 322, 297, 281, 363, 374, 461, 488, 484, 482

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(3almostprime(1)) - 3almostprime(prime(1)) = prime(8) - 3almostprime(2) = 19 - 12 = 7.
a(2) = prime(3almostprime(2)) - 3almostprime(prime(2)) = prime(12) - 3almostprime(3) = 37 - 18 = 19.
a(3) = prime(3almostprime(3)) - 3almostprime(prime(3)) = prime(18) - 3almostprime(5) = 61 - 27 = 34.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040 (primes), A014612 (3-almost primes).
Cf. A124268 (prime(3-almost prime(n))), A124269 (3-almost prime(prime(n))).
Cf. A106349 (prime(semiprime(n))), A106350 (semiprime(prime(n))), A122824 (prime(semiprime(n)) - semiprime(prime(n))).

lista(nn) = {p = primes(nn); pp = select(x->bigomega(x)==3, vector(nn, n, n)); for (n=1, nn, print1(p[pp[n]] - pp[p[n]], ", "););} \\ Michel Marcus, Oct 15 2014

a(n) = A000040(A014612(n)) - A014612(A000040(n)).

a(n) = A124268(n) - A124269(n).

A124282 Primes indexed by 4-almost primes.

Original entry on oeis.org

53, 89, 151, 173, 251, 263, 281, 419, 433, 457, 463, 541, 569, 701, 743, 761, 769, 809, 863, 881, 911, 1097, 1129, 1193, 1213, 1249, 1291, 1373, 1427, 1439, 1459, 1481, 1571, 1583, 1657, 1783, 1931, 1949, 1951, 2017, 2029, 2087, 2203, 2213, 2287, 2297

Offset: 1

Jonathan Vos Post, Oct 24 2006

4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

			a(1) = prime(4almostprime(1)) = prime(16) = 53.
a(2) = prime(4almostprime(2)) = prime(24) = 89.
a(3) = prime(4almostprime(3)) = prime(36) = 151.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124283, A124284.

a(n) = prime(4almostprime(n)) = A000040(A014613(n)). {p such that p is prime and omega(primepi(p)) = 4} = {p such that p is in A000040 and A001222(A000720(p)) = 4}.

A124283 4-almost primes indexed by primes.

Original entry on oeis.org

24, 36, 54, 60, 90, 104, 136, 150, 189, 225, 232, 294, 308, 328, 344, 375, 441, 459, 488, 510, 516, 550, 570, 621, 676, 708, 714, 738, 748, 776, 852, 860, 884, 910, 999, 1014, 1060, 1096, 1112, 1161, 1197, 1206, 1256, 1274, 1284, 1290, 1356, 1432, 1450, 1482

Offset: 1

Jonathan Vos Post, Oct 24 2006

Primes indexed by 4-almost primes = A124282. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

			a(1) = 4almostprime(prime(1)) = 4almostprime(2) = 24.
a(2) = 4almostprime(prime(2)) = 4almostprime(3) = 36.
a(3) = 4almostprime(prime(3)) = 4almostprime(5) = 54.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124284.

from math import isqrt
from sympy import prime, primepi, integer_nthroot, primerange
def A124283(n):
    def f(x): return int(prime(n)+x-sum(primepi(x//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(x, 4)[0]+1)) for b, m in enumerate(primerange(k, integer_nthroot(x//k, 3)[0]+1), a) for c, r in enumerate(primerange(m, isqrt(x//(k*m))+1), b)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

a(n) = 4almostprime(prime(n)) = A014613(A000040(n)).

a(17)-a(50) from Giovanni Resta, Jun 13 2016

A124284 Prime(4almostprime(n))-4almostprime(prime(n)). Commutator [A000040,A014613] at n.

Original entry on oeis.org

29, 53, 97, 113, 161, 159, 145, 269, 244, 232, 231, 247, 261, 373, 399, 386, 328, 350, 375, 371, 395, 547, 559, 572, 537, 541, 577, 635, 679, 663, 607, 621, 687, 673, 658, 769, 871, 853, 839, 856, 832, 881, 947, 939, 1003, 1007, 955, 915, 907, 889, 941, 989

Offset: 1

Jonathan Vos Post, Oct 24 2006

			a(1) = prime(4almostprime(1)) - 4almostprime(prime(1)) = 53 - 24 = 29.
a(2) = prime(4almostprime(2)) - 4almostprime(prime(2)) = 89 - 36 = 53.
a(3) = prime(4almostprime(3)) - 4almostprime(prime(3)) = 151 - 54 = 97.
It is mere coincidence that the first 4 values are all primes.

Robert G. Wilson v, Table of n, a(n) for n=1..1000

Cf. Primes indexed by 4-almost primes = A124282. 4-almost primes indexed by primes = A124283. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)).

Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124283.

FourAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@ Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[ FourAlmostPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2];
Table[ Prime@ FourAlmostPrime@ n - FourAlmostPrime@ Prime@ n, {n, 52}]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, prime
def A124284(n):
    def f(x): return int(x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=prime(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return prime(m)-r # Chai Wah Wu, Aug 17 2024

a(n) = prime(4almostprime(n)) - 4almostprime(prime(n)) = A000040(A014613(n)) -A014613(A000040(n)).

More terms from Robert G. Wilson v, Aug 31 2007

A124309 5-almost primes indexed by primes.

Original entry on oeis.org

48, 72, 108, 120, 180, 208, 270, 280, 368, 420, 450, 520, 592, 612, 660, 700, 760, 828, 920, 952, 976, 1032, 1064, 1128, 1242, 1288, 1323, 1372, 1380, 1428, 1575, 1624, 1674, 1700, 1752, 1768, 1880, 1976, 2028, 2096, 2178, 2196, 2312, 2328, 2384, 2394, 2475

Offset: 1

Jonathan Vos Post, Oct 25 2006

			a(1) = 5almostprime(prime(1)) = 5almostprime(2) = 48 = 2^4 * 3.
a(2) = 5almostprime(prime(2)) = 5almostprime(3) = 72 = 2^3 * 3^2.
a(3) = 5almostprime(prime(3)) = 5almostprime(5) = 108 = 2^2 * 3^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A124308 Primes indexed by 5-almost primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040, A001358] at n.

Cf. A000040, A014614, A114405, A114415, A124282-A124284, A124309.

list(lim)=my(v=List(),u=v); forprime(p=2,lim\16, forprime(q=2,min(lim\(8*p),p), forprime(r=2,min(lim\(4*p*q),q), forprime(s=2,min(lim\(2*p*q*r),r), forprime(t=2,min(lim\(p*q*r*s),s), listput(v,p*q*r*s*t)))))); v=Set(v); forprime(p=2,#v, listput(u,v[p])); v=0; Vec(u) \\ Charles R Greathouse IV, Feb 10 2017

a(n) = 5almostprime(prime(n)) = A014614(A000040(n)).

a(16)-a(47) from Giovanni Resta, Jun 13 2016

A124308 Primes indexed by 5-almost primes.

Original entry on oeis.org

131, 223, 359, 409, 593, 613, 659, 953, 997, 1049, 1069, 1223, 1283, 1543, 1601, 1693, 1733, 1747, 1811, 1987, 2003, 2069, 2503, 2593, 2693, 2713, 2789, 2801, 2903, 3079, 3181, 3221, 3301, 3323, 3541, 3571, 3727, 4003, 4127, 4283

Offset: 1

Jonathan Vos Post, Oct 25 2006

			a(1) = prime(5almostprime(1)) = prime(32 = 2^5) = 131.
a(2) = prime(5almostprime(2)) = prime(48 = 2^4 * 3) = 223.
a(3) = prime(5almostprime(3)) = prime(72 = 2^3 * 3^2) = 359.
a(4) = prime(5almostprime(4)) = prime(80 = 2^4 * 5) = 409.

Cf. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040, A001358] at n.

Cf. A000040, A014614, A114405, A114415, A122824, A124269, A124270, A124282-A124284, A124309, A124310.

Prime[#]&/@Select[Range[600],PrimeOmega[#]==5&] (* Harvey P. Dale, Nov 20 2015 *)

a(n) = prime(5almostprime(n)) = A000040(A014614(n)). {p such that p is prime and omega(primepi(p)) = 5} = {p such that p is in A000040 and A001222(A000720(p)) = 5}.

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A106349 Primes indexed by semiprimes.

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A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

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A124268 Primes indexed by 3-almost primes.

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A124269 3-almost primes indexed by primes.

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A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

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A124282 Primes indexed by 4-almost primes.

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A124283 4-almost primes indexed by primes.

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