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 1-5 of 5 results.

A079856 Duplicate of A073329.

Original entry on oeis.org

2, 10, 60, 420, 4290, 53130, 903210, 17687670, 406816410, 11125544430
Offset: 1

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Author

Keywords

A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 12, 16, 11, 10, 18, 24, 32, 13, 14, 20, 36, 48, 64, 17, 15, 27, 40, 72, 96, 128, 19, 21, 28, 54, 80, 144, 192, 256, 23, 22, 30, 56, 108, 160, 288, 384, 512, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024, 31, 26, 44, 81, 120, 224, 432, 640, 1152
Offset: 0

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Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

Comments

An n-almost-prime is a positive integer that has exactly n prime factors.
This sequence is a rearrangement of the natural numbers. - Robert G. Wilson v, Feb 11 2006
Each antidiagonal begins with the n-th prime and ends with 2^n.
From Eric Desbiaux, Jun 27 2009: (Start)
(A001222 gives this sequence)
A001221 gives another table:
1
- 2 3 4 5 7 8 9 11 ... A000961
- 6 10 12 14 15 18 20 21 ... A007774
- 30 42 60 66 70 78 84 90 ... A033992
- 210 330 390 420 462 510 546 570 ... A033993
- 2310 2730 3570 3990 4290 4620 4830 5460 ... A051270
Antidiagonals begin with A000961 and end with A002110.
Diagonal is A073329 which is last term in n-th row of A048692. (End)

Examples

			Table begins:
  1
  -  2  3   5   7  11  13  17  19  23  29 ...
  -  4  6   9  10  14  15  21  22  25  26 ...
  -  8 12  18  20  27  28  30  42  44  45 ...
  - 16 24  36  40  54  56  60  81  84  88 ...
  - 32 48  72  80 108 112 120 162 168 176 ...
  - 64 96 144 160 216 224 240 324 336 352 ...

Links

Crossrefs

Cf. A078840, A078841, A078842, A078843, A078844, A078445, A078846, A109636.
T(1, k)=A000040(k), T(2, k)=A001358(k), T(3, k)=A014612(k), T(4, k)=A014613(k), T(5, k)=A014614(k), T(6, k)=A046306(k), T(7, k)=A046308(k), T(8, k)=A046310(k), T(9, k)=A046312(k), T(10, k)=A046314(k).
T(11, k)=A069272(k), T(12, k)=A069273(k), T(13, k)=A069274(k), T(14, k)=A069275(k), T(15, k)=A069276(k), T(16, k)=A069277(k), T(17, k)=A069278(k), T(18, k)=A069279(k), T(19, k)=A069280(k), T(20, k)=A069281(k).
T(k, 1)=A000079(k), T(k, 2)=A007283(k), T(k, 3)=A116453(k), T(k, k)=A101695(k), T(k, k+1)=A078841(k).
A091538 is this sequence with zeros inserted, making a square array.

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 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ AlmostPrime[k, n - k + 1], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v *)
mx = 11; arr = NestList[Take[Union@Flatten@Outer[Times, #, primes], mx] &, primes = Prime@Range@mx, mx]; Prepend[Flatten@Table[arr[[k, n - k + 1]], {n, mx}, {k, n}], 1] (* Ivan Neretin, Apr 30 2016 *)
(* The next code skips the initial 1. *)
width = 15; (seq = Table[
  Rest[NestList[1 + NestWhile[# + 1 &, #, ! PrimeOmega[#] == z &] &,
  2^z, width - z + 1]] - 1, {z, width}]) // TableForm
Flatten[Map[Reverse[Diagonal[Reverse[seq], -width + #]] &, Range[width]]]
(* Peter J. C. Moses, Jun 05 2019 *)
Grid[Table[Select[Range[200], PrimeOmega[#] == n &], {n, 0, 7}]]
(* Clark Kimberling, Nov 17 2024 *)
  • PARI
    T(n,k)=if(k<0,0,s=1; while(sum(i=1,s,if(bigomega(i)-n,0,1))
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, prime
def A078840_T(n,k):
    if n == 1: return prime(k)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(k-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Extensions

Edited by Robert G. Wilson v, Feb 11 2006

A125666 Table read by ascending antidiagonals: n-th row of table consists of the positive integers divisible by exactly n distinct primes.

Original entry on oeis.org

2, 6, 3, 30, 10, 4, 210, 42, 12, 5, 2310, 330, 60, 14, 7, 30030, 2730, 390, 66, 15, 8, 510510, 39270, 3570, 420, 70, 18, 9, 9699690, 570570, 43890, 3990, 462, 78, 20, 11, 223092870, 11741730, 690690, 46410, 4290, 510, 84, 21, 13, 6469693230, 281291010
Offset: 1

Views

Author

Leroy Quet, Jan 29 2007

Keywords

Comments

Concatenated sequence is a permutation of the integers >= 2.
The chosen encoding of the table by *rising* antidiagonals is contrary to the OEIS standard which rather expects falling antidiagonals: as a consequence, displaying this sequence as a table (2nd link after the list of terms above) will list the integers with given number of prime divisors in columns rather than rows. - M. F. Hasler, Jun 06 2024

Examples

			The table begins:
  n\k|     1     2    3    4    5    6  ...
  ---+-------------------------------------
   1 |     2,    3,   4,   5,   7,   8, ...
   2 |     6,   10,  12,  14,  15, ...
   3 |    30,   42,  60,  66, ...
   4 |   210,  330, 390, ...
   5 |  2310, 2730, ...
   6 | 30030,  ...
  ...|   ...

Crossrefs

Cf. A001221, A002110 (col 1), A246655 (row 1), A007774 (row 2), A033992 (row 3), A033993 (row 4), A051270 (row 5), A074969 (row 6), A176655 (row 7), A348072 (row 8), A348073 (row 9), A073329 (diag), compare to A048692.

Programs

  • Mathematica
    f[n_, m_] := f[n, m] = Block[{c = m, k = If[m == 1, Product[Prime[i], {i, n}], f[n, m - 1] + 1]},While[Length@FactorInteger[k] != n, k++ ];k];Table[f[d - m + 1, m], {d, 10}, {m, d}] // Flatten (* Ray Chandler, Feb 08 2007 *)
  • PARI
    A125666(n, k=0)={if(k, for(m=vecprod(primes(n)), oo, omega(m)!=n || k-- || return(m)), A125666(A004736(n), A002260(n)))} \\ M. F. Hasler, Jun 06 2024

Extensions

Extended by Ray Chandler, Feb 08 2007

A048692 Triangle read by rows in which row n contains first n numbers with exactly n distinct prime factors.

Original entry on oeis.org

2, 6, 10, 30, 42, 60, 210, 330, 390, 420, 2310, 2730, 3570, 3990, 4290, 30030, 39270, 43890, 46410, 51870, 53130, 510510, 570570, 690690, 746130, 870870, 881790, 903210, 9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410
Offset: 1

Views

Author

Amarnath Murthy, Aug 20 2002

Keywords

Examples

			    2;
    6,  10;
   30,  42,  60;
  210, 330, 390, 420;
  ...

Links

Crossrefs

Cf. A002110 (first column).
Main diagonal gives A073329.
Extending the rows to give a square array, we get A125666.

Programs

  • Mathematica
    f[n_] := Flatten[Table[ # [[1]]] & /@ FactorInteger[n]]; (* for n=7 *) Take[ Select[ Range[10^7], Length[f[ # ]] == 7 & ], 7]
Module[{nn=8,dpf=Table[{n,PrimeNu[n]},{n,2 10^7}]},Flatten[Table[Select[dpf,#[[2]]==n&,n],{n,nn}],1][[All,1]]] (* The program generates the first 36 terms of the sequence. *) (* Harvey P. Dale, Sep 09 2022 *)

Extensions

Edited, corrected and extended by Robert G. Wilson v, Aug 22 2002
More terms from David A. Corneth, Jan 09 2021

A340467 a(n) is the n-th squarefree number having n prime factors.

Original entry on oeis.org

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330
Offset: 1

Views

Author

Alois P. Heinz, Jan 08 2021

Keywords

Comments

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

Examples

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Links

Crossrefs

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

Programs

  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    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) # Chai Wah Wu, Aug 31 2024

Formula

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.
Showing 1-5 of 5 results.

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