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 1-3 of 3 results.

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

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Author

Jonathan Vos Post, Oct 24 2006

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Formula

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

Views

Author

Jonathan Vos Post, Oct 24 2006

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • Python
    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

Formula

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

Extensions

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

Views

Author

Jonathan Vos Post, Oct 24 2006

Keywords

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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}]
  • Python
    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

Formula

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

Extensions

More terms from Robert G. Wilson v, Aug 31 2007
Showing 1-3 of 3 results.

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