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.

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

Views

Author

Jonathan Vos Post, Oct 25 2006

Keywords

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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}.

A124317 Semiprimes indexed by 3-almost primes.

Original entry on oeis.org

22, 34, 51, 57, 82, 85, 87, 123, 133, 134, 146, 158, 201, 205, 209, 214, 221, 226, 237, 295, 305, 309, 321, 327, 341, 361, 365, 371, 394, 395, 413, 447, 478, 481, 497, 501, 529, 533, 543, 545, 551, 554, 559, 583, 597, 614, 623, 635, 689, 699, 734, 763, 766
Offset: 1

Views

Author

Jonathan Vos Post, Oct 26 2006

Keywords

Comments

Note that a(10)-a(9) = a(30)-a(29) = 1, achieving the minimum possible, due to a combination of the appropriate semiprime gap (A065516) and 3-almost prime gap (A114403).

Examples

			a(1) = semiprime(3almostprime(1)) = semiprime(8 = 2^3) = 22 = 2 * 11.
a(2) = semiprime(3almostprime(2)) = semiprime(12 = 2^2 * 3) = 34 = 2 * 17.
a(3) = semiprime(3almostprime(3)) = semiprime(18 = 2 * 3^2) = 51 = 3 * 17.

Links

Crossrefs

Cf. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. 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, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124318, A124319.

Programs

  • Mathematica
    p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[2][[Take[p[3], 60]]] (* Giovanni Resta, Jun 13 2016 *)
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124317(n):
    def f(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def g(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = m, g(m)+m
    while r != k:
        r, k = k, g(k)+m
    return r # Chai Wah Wu, Aug 17 2024

Formula

a(n) = semiprime(3almostprime(n)) = A001358(A014612(n)).

Extensions

Data corrected by Giovanni Resta, Jun 13 2016

A124318 3-almost primes indexed by semiprimes.

Original entry on oeis.org

20, 28, 44, 45, 66, 68, 98, 99, 110, 114, 147, 148, 153, 165, 170, 188, 207, 222, 238, 244, 245, 261, 273, 284, 310, 322, 343, 356, 357, 363, 374, 387, 388, 399, 429, 438, 465, 475, 477, 494, 498, 506, 531, 549, 555, 590, 595, 596, 602, 603, 628, 639, 642
Offset: 1

Views

Author

Jonathan Vos Post, Oct 26 2006

Keywords

Examples

			a(1) = 3almostprime(semiprime(1)) = 3almostprime(4 = 2^2) = 20 = 2^2 * 5.
a(2) = 3almostprime(semiprime(2)) = 3almostprime(6 = 2 * 3) = 28 = 2^2 * 7.
a(3) = 3almostprime(semiprime(3)) = 3almostprime(9 = 3^2) = 44 = 2^2 * 11.
a(4) = 3almostprime(semiprime(4)) = 3almostprime(10 = 2 * 5) = 45 = 3^2 * 5.

Crossrefs

Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. 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, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124319.

Programs

  • Mathematica
    p[k_] := Select[Range[1000], PrimeOmega[#] == k &]; p[3][[Take[p[2], 60]]] (* Giovanni Resta, Jun 13 2016 *)
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124318(n):
    def g(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    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)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = m, g(m)+m
    while r != k:
        r, k = k, g(k)+m
    return r # Chai Wah Wu, Aug 17 2024

Formula

a(n) = 3almostprime(semiprime(n)) = A014612(A001358(n)).

Extensions

a(22)-a(53) from Giovanni Resta, Jun 13 2016

A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

Original entry on oeis.org

2, 6, 7, 12, 16, 17, -11, 24, 23, 20, -1, 10, 48, 40, 39, 26, 14, 4, -1, 51, 60, 48, 48, 43, 31, 39, 22, 15, 37, 32, 39, 60, 90, 82, 68, 63, 64, 58, 66, 51, 53, 48, 28, 34, 42, 24, 28, 39, 87, 96, 106, 124, 124, 135, 131, 131, 88, 91, 72, 96, 103, 83, 83, 81, 91
Offset: 1

Views

Author

Jonathan Vos Post, Oct 26 2006

Keywords

Examples

			a(1) = semiprime(3almostprime(1)) - 3almostprime(semiprime(1)) = 22 - 20 = 2.
a(2) = semiprime(3almostprime(2)) - 3almostprime(semiprime(2)) = 34 - 28 = 6.
a(3) = semiprime(3almostprime(3)) - 3almostprime(semiprime(3)) = 51 - 44 = 7.
a(4) = semiprime(3almostprime(4)) - 3almostprime(semiprime(4)) = 57 - 45 = 12.
a(7) = semiprime(3almostprime(7)) - 3almostprime(semiprime(7)) = 87 - 98 = -11, which is the first negative value in the commutators we have seen in these related set of sequences, exposing an incorrect assumption.

Crossrefs

Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. 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, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124318.

Programs

  • Mathematica
    p[k_] := p[k] = Select[Range[1000], PrimeOmega[#] == k &]; p[2][[ Take[p[3], 70]]] - p[3][[Take[p[2], 70]]] (* Giovanni Resta, Jun 13 2016 *)
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124319(n):
    def f(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def g(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, g(n)+n
        while m != k:
            m, k = k, g(k)+n
        return m
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=A001358(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return A001358(m)-r # Chai Wah Wu, Aug 17 2024

Extensions

a(18) corrected and a(22)-a(65) from Giovanni Resta, Jun 13 2016
Showing 1-4 of 4 results.

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