A014613 -id:A014613 - cp's OEIS frontend

A091538 Triangle built from m-primes as columns.

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

Offset: 0

Wolfdieter Lang, Feb 13 2004

m-primes (also called m-almost primes) are the numbers which have precisely m prime factors counting multiple factors. 1 is included as 0-prime.

The number N>=1 appears in column no. m = A001222(N).

			From _Michael De Vlieger_, May 24 2017: (Start)
Chart a(n,m) read by antidiagonals:
  n | m ->
  ------------------------------------------------
  0 |    1     0     0     0     0     0     0 ... (A000007)
  1 |    2     3     5     7    11    13    17     (A000040)
  2 |    4     6     9    10    14    15    21     (A001358)
  3 |    8    12    18    20    27    28    30     (A014612)
  4 |   16    24    36    40    54    56    60     (A014613)
  5 |   32    48    72    80   108   112   120     (A014614)
  6 |   64    96   144   160   216   224   240     (A046306)
  7 |  128   192   288   320   432   448   480     (A046308)
  8 |  256   384   576   640   864   896   960     (A046310)
       ...
Triangle begins:
  0 |    1
  1 |    0    2
  2 |    0    3    4
  3 |    0    5    6    8
  4 |    0    7    9   12   16
  5 |    0   11   10   18   24   32
  6 |    0   13   14   20   36   48    64
  7 |    0   17   15   27   40   72    96   128
  8 |    0   19   21   28   54   80   144   192   256
       ...
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Wolfdieter Lang, First 11 rows.

The column sequences (without leading zeros) are: A000007, A000040 (primes), A001358, A014612-4, A046306, A046308, A046310, A046312, A046314, A069272-A069281 for m=0..20, respectively.

A078840 is this table with the zeros omitted.

With[{nn = 11}, Function[s, Function[t, Table[Function[m, If[m == 1, Boole[k == 1], t[[m, k]]]][n - k + 1], {n, nn}, {k, n, 1, -1}]]@ Map[Position[s, #][[All, 1]] &, Range[0, nn]]]@ PrimeOmega@ Range[2^nn]] (* or *)
a = {1}; Do[Block[{r = {Prime@ n}}, Do[AppendTo[r, SelectFirst[ Range[a[[-(n - i)]] + 1, 2^n], PrimeOmega@ # == i &]], {i, 2, n - 1}]; a = Join[a, {0}, If[n == 1, {}, r], {2^n}]], {n, 11}]; a (* Michael De Vlieger, May 24 2017 *)

from math import isqrt, comb, prod
from sympy import prime, primerange, integer_nthroot, primepi
def A091538(n):
    a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
    r = n-comb(a,2)
    w = a-r
    if r==0: return int(w==1)
    if r==1: return prime(w)
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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(w+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,r)))
    return bisection(f,w,w) # Chai Wah Wu, Jun 11 2025

For n>=m>=1: a(n, m)= (n-m+1)-th member in the strictly monotonically increasing sequence of numbers N satisfying: N=product(p(k)^(e_k), k=1..) with p(k) := A000040(k) (k-th prime) such that sum(e_k, k=1..) = m, where the e_k are nonnegative. if m=0 : a(n, 0)=1 if n=0 else 0. If 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

A111344 Pierpont 4-almost primes: numbers with exactly 4 prime divisors, not necessarily distinct, of the form 2^K*3^L + 1.

Original entry on oeis.org

513, 13825, 32769, 59050, 110593, 157465, 177148, 186625, 262145, 279937, 497665, 1259713, 1327105, 2097153, 2125765, 2519425, 4718593, 4782970, 5668705, 6718465, 17915905, 18874369, 22674817, 33554433, 38263753, 56623105

Offset: 1

Jonathan Vos Post, Nov 08 2005

nonn

			a(1) = 513 = (2^9)*(3^0)+1 = 3 * 3 * 3 * 19.
a(2) = 13825 = (2^9)*(3^3)+1 = 5 * 5 * 7 * 79.
a(3) = 32769 = (2^15)*(3^0)+1 = 3 * 3 * 11 * 331.
a(4) = 59050 = (2^0)*(3^10)+1 = 2 * 5 * 5 * 1181.
a(10) = 279937 = (2^7)*(3^7)+1 = 7 * 7 * 29 * 197 (lots of sevens).
a(24) = 33554433 = (2^25)*(3^0) = 3 * 11 * 251 * 4051.
a(60) = 31381059610 = (2^0)*(3^22)+1 = 2 * 5 * 5501 * 570461.
a(168) = 16677181699666570 = (2^0)*(3^34)+1 = 2 * 5 * 956353 * 1743831169.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2500
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A014613 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

is(n)=bigomega(n)==4 && n-1 == 2^valuation(n-1,2)*3^valuation(n-1,3) \\ Charles R Greathouse IV, Feb 01 2017

list(lim)=my(v=List(),L=lim\1-1); for(e=0,logint(L,3), my(t=3^e); while(t<=L, if(bigomega(t+1)==4, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Extended by Ray Chandler, Nov 08 2005

Name edited by Charles R Greathouse IV, Feb 01 2017

A376704 4-brilliant numbers: numbers which are the product of four primes having the same number of decimal digits.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 14641, 17303, 20449, 22627, 24167, 25289, 26741, 28561, 29887, 30613, 31603, 34969, 35321, 36179

Offset: 1

Paolo Xausa, Oct 02 2024

			35321 is a term because 35321 = 11 * 13 * 13 * 19, and these four prime factors have the same number of digits.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Dario Alpern, 4-Brilliant Numbers.

Subsequence of A014613.

Cf. A078972, A376703, A376705, A376706.

A376704Q[k_] := With[{f = FactorInteger[k]}, Total[f[[All, 2]]] == 4 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1];
Select[Range[40000], A376704Q] (* or *)
dlist4[d_] := Sort[Times @@@ DeleteDuplicates[Map[Sort, Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 4]]]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Flatten[Array[dlist4, 2]]

A046317 Odd numbers divisible by exactly 4 primes (counted with multiplicity).

Original entry on oeis.org

81, 135, 189, 225, 297, 315, 351, 375, 441, 459, 495, 513, 525, 585, 621, 625, 693, 735, 765, 783, 819, 825, 837, 855, 875, 975, 999, 1029, 1035, 1071, 1089, 1107, 1155, 1161, 1197, 1225, 1269, 1275, 1287, 1305, 1365, 1375, 1395, 1425, 1431, 1449, 1521

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

Zak Seidov, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A014613.

Select[Range[1,1601,2],PrimeOmega[#]==4&] (* Harvey P. Dale, Aug 15 2014 *)

Offset corrected by Zak Seidov, May 03 2020

A086046 Sum of first n 4-almost primes.

Original entry on oeis.org

16, 40, 76, 116, 170, 226, 286, 367, 451, 539, 629, 729, 833, 959, 1091, 1226, 1362, 1502, 1652, 1804, 1960, 2144, 2333, 2529, 2727, 2931, 3141, 3361, 3586, 3814, 4046, 4280, 4528, 4778, 5038, 5314, 5608, 5904, 6201, 6507, 6815, 7130, 7458, 7788, 8128

Offset: 1

Shyam Sunder Gupta, Aug 24 2003

Elements in this sequence can themselves be 4-almost primes. a(1) = 16 = 2^4. a(2) = 40 = 2^3 * 5. a(19) = 1652 = 2^2 * 7 * 59. a(20) = 1804 = 2^2 * 11 * 41. a(31) = 4046 = 2 * 7 * 17^2. a(37) = 5608 = 2^3 * 701. a(39) = 6201 = 3^2 * 13 * 53. a(40) = 6507 = 3^3 * 241. a(42) = 7130 = 2 * 5 * 23 * 31. a(43) = 7458 = 2 * 3 * 11 * 113. Does this happen infinitely often? - Jonathan Vos Post, Dec 11 2004

			a(2)=40 because sum of first two 4-almost primes i.e. 16+24 is 40.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Accumulate[Select[Range[1000],PrimeOmega[#]==4&]] (* Harvey P. Dale, Feb 07 2014 *)

a(n) = sum_{i=1..n} A014613(i). - R. J. Mathar, Sep 14 2012

A046330 Palindromes with exactly 4 prime factors (counted with multiplicity).

Original entry on oeis.org

88, 232, 414, 424, 444, 484, 525, 585, 636, 666, 676, 686, 808, 858, 868, 999, 1881, 2002, 2332, 2442, 2662, 3003, 3663, 3773, 3993, 4114, 4444, 4774, 5005, 5115, 5225, 6116, 6556, 6666, 7007, 7227, 8668, 9999, 10101, 10701, 11011, 12321, 13431

Offset: 1

Patrick De Geest, Jun 15 1998

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A014613, A046317, A046410.

Select[Range[14000],PalindromeQ[#]&&PrimeOmega[#]==4&] (* Harvey P. Dale, Jul 07 2024 *)

A114435 Indices of 4-almost prime triangular numbers.

Original entry on oeis.org

8, 16, 20, 23, 26, 36, 40, 45, 49, 50, 51, 53, 59, 60, 62, 65, 68, 69, 74, 76, 77, 83, 88, 89, 91, 92, 100, 103, 105, 110, 114, 115, 117, 123, 126, 129, 131, 136, 139, 146, 149, 150, 151, 154, 156, 165, 169, 182, 185, 186, 187, 194, 196, 197, 198, 206, 210

Offset: 1

Jonathan Vos Post, Feb 13 2006

			a(1) = 8 because T(8) = TriangularNumber(8) = 8*(8+1)/2 = 36 = 2^2 * 3^2 is a 4-almost prime.
a(2) = 16 because T(16) = 16*(16+1)/2 = 136 = 2^3 * 17 is a 4-almost prime.
a(3) = 20 because T(20) = 20*(20+1)/2 = 210 = 2 * 3 * 5 * 7 (210 = primorial 4#).
a(4) = 23 because T(23) = 23*(23+1)/2 = 276 = 2^2 * 3 * 23.
a(5) = 26 because T(26) = 26*(26+1)/2 = 351 = 3^3 * 13.
a(6) = 36 because T(36) = 36*(36+1)/2 = 666 = 2 * 3^2 * 37.
a(27) = 100 because T(100) = 100*(100+1)/2 = 5050 = 2 * 5^2 * 101.
a(57) = 210 because T(210) = 210*(210+1)/2 = 22155 = 3 * 5 * 7 * 211 (again, 210 = primorial 4#).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000217, A001222, A014613, A069904.

Flatten[Position[Accumulate[Range[800]], ?(PrimeOmega[#]== 4 &)]] (* _Vincenzo Librandi, Apr 09 2014 *)

is(n)=my(t=bigomega(n/gcd(n,2))); if(t<3, bigomega((n+1)/gcd(n+1,2))+t==4, t==3 && isprime((n+1)/gcd(n+1,2))) \\ Charles R Greathouse IV, Jun 14 2017

{a(n)} = {k such that A001222(A000217(k)) = 4}. {a(n)} = {k such that k*(k+1)/2 has exactly 4 prime factors, with multiplicity}. {a(n)} = {k such that A000217(k) is an element of A014613}.

{ m : A069904(m) = 4 }. - Alois P. Heinz, Aug 05 2019

A124941 Numbers k such that k and k+4 are 4-almost primes.

Original entry on oeis.org

36, 56, 84, 100, 132, 136, 152, 228, 340, 344, 372, 376, 472, 484, 488, 532, 546, 564, 568, 580, 621, 632, 686, 708, 770, 804, 808, 820, 846, 852, 856, 868, 872, 950, 1012, 1192, 1204, 1206, 1208, 1274, 1304, 1326, 1336, 1444, 1524, 1550, 1572, 1576, 1690

Offset: 1

Zak Seidov, Nov 13 2006

nonn

			36=2^2*3^2, 40=2^3*5; 56=2^3*7, 60=2^2*3*5.

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

Cf. A014613, A124940, A124942, A157483, A180117.

Select[Range[1690], PrimeOmega[#]==PrimeOmega[#+4]==4 &] (* James C. McMahon, Dec 07 2024 *)

isok(n) = (bigomega(n) == 4) && (bigomega(n+4) == 4); \\ Michel Marcus, Oct 11 2013

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A091538 Triangle built from m-primes as columns.

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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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A111344 Pierpont 4-almost primes: numbers with exactly 4 prime divisors, not necessarily distinct, of the form 2^K*3^L + 1.

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A376704 4-brilliant numbers: numbers which are the product of four primes having the same number of decimal digits.

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A046317 Odd numbers divisible by exactly 4 primes (counted with multiplicity).

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A086046 Sum of first n 4-almost primes.

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A046330 Palindromes with exactly 4 prime factors (counted with multiplicity).

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