A033993 -id:A033993 - cp's OEIS frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A259349 (requires squarefree). Subsequence of A273879.
Cf. A364266 (5 distinct factors).
See also A001221, A001222, A005117.
Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 6 then
        if omega(k+1) = 6 then
            if omega(k+2) = 6 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

More terms from David A. Corneth, Jul 18 2023

A136153 Composites one larger than a prime, with exactly four distinct prime factors.

Original entry on oeis.org

390, 420, 462, 510, 570, 660, 770, 798, 840, 858, 930, 1020, 1050, 1092, 1110, 1218, 1230, 1260, 1290, 1302, 1320, 1410, 1428, 1430, 1482, 1554, 1560, 1610, 1638, 1710, 1722, 1848, 1890, 1914, 1932, 1950, 1974, 1980, 2030, 2040, 2070, 2090, 2100, 2130

Offset: 1

Enoch Haga, Dec 16 2007

			a(0)=390 because 30 follows the prime 29 and has four prime factors 2, 3, 5 and 13.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A136151, A136152, A136154, A136155.

Select[Prime[Range[400]]+1,PrimeNu[#]==4&] (* Harvey P. Dale, Aug 15 2021 *)

isok(n) = (omega(n)==4) && isprime(n-1); \\ Michel Marcus, Jun 08 2014

Equals A008864 INTERSECT A033993. - R. J. Mathar, Feb 20 2008

Edited by R. J. Mathar, Feb 20 2008

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A258402 a(n) = (n^2 + 4n + 6) n^2.

Original entry on oeis.org

0, 11, 72, 243, 608, 1275, 2376, 4067, 6528, 9963, 14600, 20691, 28512, 38363, 50568, 65475, 83456, 104907, 130248, 159923, 194400, 234171, 279752, 331683, 390528, 456875, 531336, 614547, 707168, 809883, 923400, 1048451, 1185792, 1336203, 1500488, 1679475

Offset: 0

Garrett Frandson, Jun 05 2015

Consider a natural number r such that r has 15 proper divisors and 4 prime factors. (Note that these prime factors do not have to be distinct). The difference between these two values, say d(r), is in this case 11. Where n is a positive integer, d(r^n) = (n^2 + 4*n + 6) * n^2.

The integers which satisfy the proper-divisor-prime-factor requirement are those of A033993.

			The smallest integer which satisfies this is 210: It has 15 proper divisors (1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105) and 4 prime factors (2, 3, 5, 7), so d(210) = 11. The square of 210, 44100, we would expect to have a difference of 72 between the number of its proper divisors and prime factors, and with respectively 80 and 8, d(44100) = 72 indeed. Checking this with further integer powers of 210 will continue to generate terms in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A033993.

[(n^2+4*n+6)*n^2: n in [0..40]]; // Vincenzo Librandi, Jun 06 2015

I:=[0,11, 72,243,608]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jun 06 2015

Table[(n^2 + 4n + 6) * n^2, {n, 0, 39}] (* Alonso del Arte, Jun 06 2015 *)
CoefficientList[Series[x (11 + 17 x - 7 x^2 + 3 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2015 *)
LinearRecurrence[{5,-10,10,-5,1},{0,11,72,243,608},40] (* Harvey P. Dale, May 05 2018 *)

a(n)=(n^2+4*n+6)*n^2 \\ Charles R Greathouse IV, Jun 15 2015

From Vincenzo Librandi, Jun 06 2015: (Start)
G.f.: x*(11 + 17*x - 7*x^2 + 3*x^3)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)
E.g.f.: exp(x)*x*(11 + 25*x + 10*x^2 + x^3). - Stefano Spezia, Oct 28 2023

More terms from Alonso del Arte, Jun 06 2015

A316351 Numbers k such that k^2 + 1 has exactly four distinct prime factors.

Original entry on oeis.org

47, 73, 83, 123, 133, 157, 173, 177, 183, 187, 191, 203, 213, 217, 233, 237, 242, 253, 255, 265, 273, 278, 293, 302, 307, 313, 317, 319, 327, 333, 337, 343, 353, 377, 387, 395, 401, 403, 411, 413, 421, 423, 437, 438, 467, 473, 477, 483, 487, 489, 497, 499, 507

Offset: 1

Gordon Elliot Michaels, Jun 29 2018

nonn

			For k = 133, k^2 + 1 = 17690 = 2*5*29*61 which has 4 distinct prime factors, so 133 is a term.
For k = 157, k^2 + 1 = 24650 = 2*5*5*17*29 which has 4 distinct prime factors, so 157 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A002522, A033993.

Select[Range@510, PrimeNu[#^2 + 1] == 4 &] (* Robert G. Wilson v, Jul 15 2018 *)

isok(n) = omega(n^2+1) == 4; \\ Michel Marcus, Jun 30 2018

A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

Original entry on oeis.org

22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423

Offset: 1

Metin Sariyar, Oct 09 2021

a(12) <= 2535332433. - David A. Corneth, Oct 10 2021

a(12) >= 10^9. - Michel Marcus, Oct 11 2021

			22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655.

Cf. A323083.

Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)

isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021

a(8)-a(9) from Amiram Eldar, Oct 09 2021
a(10)-a(11) from Michel Marcus, Oct 10 2021
a(12) confirmed by Martin Ehrenstein, Oct 28 2021
a(13)-a(14) from Martin Ehrenstein, Oct 30 2021

A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

Original entry on oeis.org

660, 1092, 1140, 1155, 1260, 1320, 1365, 1380, 1428, 1540, 1560, 1740, 1785, 1820, 1860, 1980, 1995, 2184, 2220, 2340, 2380, 2415, 2436, 2460, 2508, 2580, 2604, 2660, 2805, 2820, 2856, 2860, 2940, 3003, 3036, 3060, 3108, 3120, 3135, 3180, 3192, 3220, 3300

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 25 2008

nonn

			Examples of triples: 660.779.1021, 660.989.1189, 660.2989.3061, 660.4331.4381, 660.12091.12109, 660.27221.27229, 660.108899.108901
1092.1325.1717, 1092.1595.1933, 1092.6035.6133, 1092.8245.8317, 1092.33115.33133, 1092.74525.74533, 1092.298115.298117

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A056866 Orders of non-solvable groups. A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors. A138605 Short legs of more than 3 primitive Pythagorean triangles. A033993 Numbers that are divisible by exactly four different primes.

PythagoreanAs[a_]:=(q={};k=0;Do[y=(a^2+b^2)^0.5;c=IntegerPart[y];If[c==y,p=0;If[GCD[a,b,c]==1,AppendTo[q,a.b.c];k++ ]],{b,a+1,a^2}];PrependTo[q,k];q);lst={};Do[If[PythagoreanAs[n][[1]]==7,Print[n];AppendTo[lst,n]],{n,6*10^2,2*10^3}];lst

a^2+b^2=c^2

More terms from Ray Chandler, Jan 22 2020

A295644 Rectangular array, by antidiagonals; row 1 is the ordered list of all k having at most 2 unitary divisors; for n > 1, row n is the ordered list of all k having 2^n unitary divisors.

Original entry on oeis.org

1, 2, 6, 3, 10, 30, 4, 12, 42, 210, 5, 14, 60, 330, 2310, 7, 15, 66, 390, 2730, 30030, 8, 18, 70, 420, 3570, 39270, 510510, 9, 20, 78, 462, 3990, 43890, 570570

Offset: 1

Clark Kimberling, Jun 26 2018

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
row 1: A000961
row 2: A007774
row 3: A033992
row 4: A033993
col 1: A231209

			Northwest corner:
     1    2    3    4    5    7    8    9   11
     6   10   12   14   15   18   20   21   22
    30   42   60   66   70   78   84   90  102
   210  330  390  420  462  510  546  570  630
  2310 2730 3570 3990 4290 4620 4830 5460 5610

Cf. A034444.

As an array, essentially the same as A125666.

z = 10000;
t = Table[2^PrimeNu[n], {n, 1, z}] ;(*  A035555 *)
r[n_] := Flatten[Position[t, 2^n]]; r[1] = Join[{1}, r[1]];
v[n_, k_] := r[n][[k]];
TableForm[Table[v[n, k], {n, 1, 5}, {k, 1, 15}]]  (* A295644 array *)
Table[v[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten  (* A295644 sequence *)

A350416 Numbers with exactly 9 semiprime divisors.

Original entry on oeis.org

6300, 8820, 9900, 11700, 12600, 14700, 15300, 17100, 17640, 18900, 19404, 19800, 20700, 21780, 22050, 22932, 23400, 25200, 26100, 26460, 27900, 29400, 29700, 29988, 30420, 30492, 30600, 31500, 33300, 33516, 34200, 35100, 35280, 36300, 36900, 37800, 38700, 38808

Offset: 1

Wesley Ivan Hurt, Dec 29 2021

Numbers with exactly four distinct prime divisors (cf. A033993), one of which has multiplicity 1 and the others at least 2. - David A. Corneth, Jun 10 2022

			6300 is in the sequence as 4, 6, 9, 10, 14, 15, 21, 25, 35 are the exactly 9 of its semiprime divisors. - _David A. Corneth_, Jun 10 2022

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), A350373 (k=6), A350374 (k=7), A350375 (k=8), this sequence (k=9).

Cf. A033993, A220264.

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 9; Select[Range[40000], q] (* Amiram Eldar, Dec 30 2021 *)
spd9Q[n_]:=Count[Divisors[n],?(PrimeOmega[#]==2&)]==9; Select[Range[ 40000],spd9Q] (* _Harvey P. Dale, Jun 09 2022 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 9; \\ Michel Marcus, Dec 30 2021

is(n)= if(n==1, return(0)); my(f = vecsort(factor(n)[,2])); #f == 4 && f[1] == 1 && f[2]>=2 \\ David A. Corneth, Jun 10 2022

A046453 Composite numbers with four prime factors (not necessarily distinct) whose concatenation yields a palindrome.

Original entry on oeis.org

16, 81, 625, 2401, 3975, 4719, 5565, 6591, 8745, 10731, 14399, 14641, 16863, 20111, 34391, 36501, 39963, 41019, 46179, 47229, 48231, 60813, 61761, 62991, 69027, 109179, 113631, 116475, 143991, 158985, 159777, 162409, 164145, 167571, 183159, 184065

Offset: 1

Patrick De Geest, Jul 15 1998

			164145 = 3 * 5 * 31 * 353 -> 3531353 is palindromic.

Intersection of A033993 and A046447.

pal4Q[n_]:=Module[{ds=Flatten[IntegerDigits/@(Table[First[#],{Last[ #]}]&/@ FactorInteger[n])]},PrimeOmega[n]==4&&ds==Reverse[ds]]; Select[ Range[190000],pal4Q] (* Harvey P. Dale, Nov 12 2011 *)

One more term from Harvey P. Dale, Nov 12 2011

cp's OEIS Frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

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A136153 Composites one larger than a prime, with exactly four distinct prime factors.

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A064040 Integers whose number of distinct prime divisors is prime.

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A258402 a(n) = (n^2 + 4*n + 6) * n^2.

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A316351 Numbers k such that k^2 + 1 has exactly four distinct prime factors.

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A348266 k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.

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Mathematica

PARI

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A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

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A258402 a(n) = (n^2 + 4n + 6) n^2.