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.

A125527 Number of semiprimes <= 2^n.

Original entry on oeis.org

0, 1, 2, 6, 10, 22, 42, 82, 157, 304, 589, 1124, 2186, 4192, 8110, 15658, 30253, 58546, 113307, 219759, 426180, 827702, 1608668, 3129211, 6091437, 11868599, 23140878, 45150717, 88157689, 172235073, 336717854, 658662065, 1289149627, 2524532330
Offset: 1

Views

Author

Robert G. Wilson v, Dec 29 2006

Keywords

Links

Crossrefs

Cf. A126279, A066265, A007053, A120033, A127396.

Programs

  • Mathematica
    SemiPrimePi[n_] := Sum[ PrimePi[ n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; Table[ SemiPrimePi[2^n], {n, 47}]
  • PARI
    a(n)=my(s,i,N=2^n); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, May 12 2013
  • Perl
    use ntheory ":all"; print "$ ",semiprime_count(1 << $),"\n" for 1..48; # Dana Jacobsen, Sep 10 2018

Formula

a(n) = A072000(2^n). - R. J. Mathar, Aug 26 2011

A126279 Triangle read by rows: T(k,n) is number of numbers <= 2^n that are products of k primes.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 6, 2, 1, 11, 10, 7, 2, 1, 18, 22, 13, 7, 2, 1, 31, 42, 30, 14, 7, 2, 1, 54, 82, 60, 34, 15, 7, 2, 1, 97, 157, 125, 71, 36, 15, 7, 2, 1, 172, 304, 256, 152, 77, 37, 15, 7, 2, 1, 309, 589, 513, 325, 168, 81, 37, 15, 7, 2, 1, 564, 1124, 1049, 669, 367, 177, 83, 37
Offset: 1

Views

Author

Jonathan Vos Post & Robert G. Wilson v, Dec 22 2006

Keywords

Examples

			Triangle begins:
  1
  2 1
  4 2 1
  6 6 2 1
  11 10 7 2 1
  18 22 13 7 2 1
  31 42 30 14 7 2 1
  54 82 60 34 15 7 2 1
  97 157 125 71 36 15 7 2 1
  172 304 256 152 77 37 15 7 2 1

References

  • Adolf Hildebrand, On the number of prime factors of an integer. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 167 - 185, Academic Press, Boston, MA, 1988.
  • Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 205 - 211.

Links

Crossrefs

First column: A007053, second column: A125527, third column: A127396, 4th column: A334069. The last row reversed: A052130; the n-th row's sum: A000225 = 2^n -1.
Cf. A126280: same array but for powers of ten.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[ n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[ AlmostPrimePi[m, 2^n], {n, 16}, {m, n}] // Flatten

A120034 Number of 3-almost primes t such that 2^n < t <= 2^(n+1).

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 17, 30, 65, 131, 257, 536, 1033, 2132, 4187, 8370, 16656, 33123, 65855, 130460, 259431, 513737, 1019223, 2019783, 4003071, 7930375, 15712418, 31126184, 61654062, 122137206, 241920724, 479226157, 949313939, 1880589368, 3725662783
Offset: 0

Views

Author

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

Keywords

Comments

The partial sum equals the number of Pi_3(2^n) = A127396(n).

Examples

			(2^3, 2^4] there is one semiprime, namely 12. 8 was counted in the previous entry.

Crossrefs

Cf. A014612, A072114, A109251, A036378, A120033 - A120043.

Programs

  • Mathematica
    ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@Sqrt[n/Prime@i]}]; t = Table[ ThreePrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

A334069 Number of numbers <= 2^n that are the product of exactly four primes, not necessarily distinct.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 34, 71, 152, 325, 669, 1405, 2866, 5931, 12139, 24782, 50444, 102458, 207945, 420511, 850518, 1716168, 3460304, 6968639, 14022029, 28189833, 56631732, 113697179, 228115641, 457456902, 916899721, 1836996851, 3678943569, 7365141297, 14740076678, 29490954290
Offset: 1

Views

Author

Robert G. Wilson v, Apr 13 2020

Keywords

Examples

			a(6) = 7 because
  16 = 2 * 2 * 2 * 2,
  24 = 2 * 2 * 2 * 3,
  36 = 2 * 2 * 3 * 3,
  40 = 2 * 2 * 2 * 5,
  54 = 2 * 3 * 3 * 3,
  56 = 2 * 2 * 2 * 7, and
  60 = 2 * 2 * 3 * 5
are the seven numbers less than 2^6 = 64 that are each the product of four primes.

Links

Crossrefs

Cf. A007053, A014613, A082996, A125527, A127396, A126279.
Partial sums of A120035.

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)]}]; Array[FourAlmostPrimePi[2^#] &, 37]

Formula

a(n) = A082996(2^n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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