A014614 -id:A014614 - 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

A120036 Number of 5-almost primes 5ap such that 2^n < 5ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 8, 21, 41, 91, 199, 403, 873, 1767, 3740, 7709, 15910, 32759, 67185, 138063, 281566, 576165, 1173435, 2390366, 4860357, 9873071, 20033969, 40612221, 82266433, 166483857, 336713632, 680482316, 1374413154, 2774347425

Offset: 0

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

nonn

The partial sum equals the number of Pi_5(2^n) = 0, 0, 0, 0, 1, 2, 7, 15, 36, 77, 168, 367, 770, 1643,..

			(2^5, 2^6] there is one semiprime, namely 48. 32 was counted in the previous entry.

Cf. A014614, A114453, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j))^(1/3)]}, {l, k, PrimePi[(n/(Prime@i*Prime@j*Prime@k))^(1/2)]}]; t = Table[ FiveAlmostPrimePi[2^n], {n, 0, 37}]; Rest@t - Most@t

A206349 Even numbers k such that 6k+1, 12k+1, 18k+1, 36k+1 and 72k+1 are all primes.

Original entry on oeis.org

380, 506, 3796, 6006, 8976, 9186, 10920, 12896, 14476, 14800, 15386, 32326, 38460, 39536, 40420, 41456, 43430, 60076, 74676, 76986, 82530, 87390, 99486, 107926, 112840, 126996, 127920, 144326, 179566, 181986, 188526, 193006, 194616, 205200, 217520, 230370

Offset: 1

José María Grau Ribas, Feb 06 2012

(6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) is a Carmichael number for all n in this sequence.

More precisely, these products are in A112428 = A002997 intersect A014614. - M. F. Hasler, Apr 14 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Volume 45, Number 4 (1939), pp. 269-274.

Select[Range[250000], PrimeQ[6 #+1] && PrimeQ[12 #+1] && PrimeQ[18 #+1] && PrimeQ[36 #+1] && PrimeQ[72 #+1] && Mod[#,2] == 0&]

is_A206349(n,c=72)=!bittest(n,0)&&!until(bittest(c\=2, 0)&&9>c+=3, isprime(n*c+1)||return) \\ M. F. Hasler, Apr 14 2015

A046304 Divisible by at least 5 primes (counted with multiplicity).

Original entry on oeis.org

32, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 272, 280, 288, 300, 304, 312, 320, 324, 336, 352, 360, 368, 378, 384, 392, 396, 400, 405, 408, 416, 420, 432, 440, 448, 450, 456

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

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

Subsequence of A033987.

Cf. A014614.

Select[Range[500],PrimeOmega[#]>4&] (* Harvey P. Dale, Apr 16 2013 *)

is(n)=bigomega(n)>4 \\ Charles R Greathouse IV, Sep 17 2015

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A046304(n):
    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 almostprimepi(n, k):
        if k==0: return int(n>=1)
        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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    def f(x): return n+1+sum(almostprimepi(x,k) for k in range(1,5))
    return bisection(f,n,n) # Chai Wah Wu, Mar 29 2025

Product p_i^e_i with Sum e_i >= 5.

a(n) = n + O(n (log log n)^3/log n). - Charles R Greathouse IV, Apr 07 2017

A046318 Odd numbers divisible by exactly 5 primes (counted with multiplicity).

Original entry on oeis.org

243, 405, 567, 675, 891, 945, 1053, 1125, 1323, 1377, 1485, 1539, 1575, 1755, 1863, 1875, 2079, 2205, 2295, 2349, 2457, 2475, 2511, 2565, 2625, 2925, 2997, 3087, 3105, 3125, 3213, 3267, 3321, 3465, 3483, 3591, 3675, 3807, 3825, 3861, 3915, 4095, 4125

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

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

Cf. A014614.

A067820 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 5.

Original entry on oeis.org

32, 944, 15470, 57967, 632148, 14845324, 69921004, 888781058, 2674685524, 10077383364, 21117216104, 393370860205, 3157222675953, 5509463413255, 24819420480104, 361385490681003, 441826936079342

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

a(16) > 3*10^13. - Brian Trial, May 13 2017
All multiples of 32 greater than 32 are of form 2^5*m and have at least 6 factors. Thus this sequence will be limited to a run of at most 31 integers. - Brian Trial, May 13 2017
a(18) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			a(3)=15470 because 15470 is the start of a record breaking run of 3 consecutive integers (15470 to 15472) each having 5 prime factors; i.e. bigomega(n)=A001222(n)=5 for n = 15470, ..., 15472.

Cf. A067813, A067814, A067821, A067822.

Subsequence of A014614.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==5, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]
Table[SequencePosition[PrimeOmega[Range[15*10^6]],PadRight[{},n,5],1][[All,1]],{n,6}]//Flatten (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Sep 03 2022 *)

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(13)-a(14) from Donovan Johnson, Jan 31 2009
a(15) from Brian Trial, May 13 2017
a(16)-a(17) from Toshitaka Suzuki, Aug 31 2025

A111345 Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

4375, 19684, 7077889, 7962625, 34012225, 100663297, 129140164, 452984833, 459165025, 544195585, 644972545, 918330049, 5159780353, 7346640385, 8589934593, 13947137605, 14495514625, 23219011585, 27518828545, 28991029249

Offset: 1

Jonathan Vos Post, Nov 08 2005

			a(1) = 4375 = (2^1)*(3^7)+1 = 5 * 5 * 5 * 5 * 7.
a(2) = 19684 = (2^0)*(3^9)+1 = 2 * 2 * 7 * 19 * 37.
a(3) = 7077889 = (2^18)*(3^3)+1 = 7 * 13 * 13 * 31 * 193 (prime factors each have all odd digits).
a(4) = 7962625 = (2^15)*(3^5)+1 = 5 * 5 * 5 * 11 * 5791 (again, coincidentally, prime factors each have all odd
digits).
a(7) = 129140164 = (2^0)*(3^17)+1 = 2 * 2 * 103 * 307 * 1021.
a(15) = 8589934593 = (2^33)*(3^0)+1 = 3 * 3 * 67 * 683 * 20857.
a(21) = 34359738369 = (2^35)*(3^0)+1 = 3 * 11 * 43 * 281 * 86171.
a(30) = 793437161473 = (2^11)*(3^18)+1 = 11 * 11 * 11 * 43 * 13863281.
a(32) = 847288609444 = (2^0)*(3^25)+1 = 2 * 2 * 61 * 151 * 22996651.
a(47) = 68630377364884 = (2^0)*(3^29)+1 = 2 * 2 * 523 * 6091 * 5385997.
a(48) = 70368744177665 = (2^46)*(3^0)+1 = 5 * 277 * 1013 * 1657 * 30269.
a(81) = 50031545098999708 = (2^0)*(3^35)+1 = 2 * 2 * 61 * 547 * 374857981681.
a(89) = 144115188075855873 = (2^57)*(3^0)+1 = 3 * 3 * 571 * 174763 * 160465489.
a(99) = 450283905890997364 = (2^0)*(3^37)+1 = 2 * 2 * 18427 * 107671 * 56737873.
a(113) = 4611686018427387905 = (2^62)*(3^0)+1 = 5 * 5581 * 8681 * 49477 * 384773.

Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A014614 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.
A111344 gives the Pierpont 4-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.

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)==5, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 5.

Extended by Ray Chandler, Nov 08 2005

A046331 Palindromes with exactly 5 prime factors (counted with multiplicity).

Original entry on oeis.org

252, 272, 464, 616, 656, 696, 828, 848, 888, 2552, 4004, 4554, 4664, 4884, 5445, 5775, 6006, 6996, 8118, 8228, 8778, 8888, 9009, 18081, 18981, 20202, 20502, 21012, 21812, 22022, 23432, 23532, 24642, 25552, 25652, 26862, 27272, 27672, 28182

Offset: 1

Patrick De Geest, Jun 15 1998

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A014614, A046318.

Select[Range[30000],#==IntegerReverse[#]&&PrimeOmega[#]==5&] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Jul 16 2016 *)

A109383 5-almost primes of the form semiprime + 1.

Original entry on oeis.org

112, 120, 162, 300, 304, 378, 392, 396, 408, 520, 552, 567, 592, 612, 630, 656, 675, 680, 688, 696, 700, 750, 780, 918, 924, 944, 952, 980, 990, 1044, 1100, 1116, 1136, 1140, 1160, 1168, 1170, 1242, 1264, 1272, 1300, 1323, 1352, 1372, 1380, 1386, 1416, 1470

Offset: 1

Jonathan Vos Post, Aug 25 2005

			a(1) = 112 because (3*37)+1 = (2^4) * 7 = 112.
a(2) = 120 because (7*17)+1 = (2^3) * 3 * 5 = 120.
a(3) = 162 because (7*23)+1 = 2 * (3^4) = 162.

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

Primes are in A000040. Semiprimes are in A001358. 5-almost primes are in A014614.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in this sequence.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[1500], f[ # ] == 5 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)

v=vector(10000);i=0; for(n=1,9e99, if(issemi(n)&bigomega(n+1)==5, v[i++]=n+1;if(i==#v, return))); v \\ Charles R Greathouse IV, Feb 14 2011

a(n) is in this sequence iff a(n) is in A014614 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Mar 20 2007

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A111697 5-almost primes with semiprime digits (digits 4, 6, 9 only).

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A091538 Triangle built from m-primes as columns.

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A120036 Number of 5-almost primes 5ap such that 2^n < 5ap <= 2^(n+1).

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A206349 Even numbers k such that 6k+1, 12k+1, 18k+1, 36k+1 and 72k+1 are all primes.

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A046304 Divisible by at least 5 primes (counted with multiplicity).

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A046318 Odd numbers divisible by exactly 5 primes (counted with multiplicity).

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A067820 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 5.

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A111345 Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

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