A014612 -id:A014612 - cp's OEIS frontend

A145784 Numbers with property that their number of prime factors counted with multiplicity is a multiple of 3.

1, 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 64, 66, 68, 70, 75, 76, 78, 92, 96, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 144, 147, 148, 153, 154, 160, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 216, 222

Offset: 1

Reinhard Zumkeller, Oct 19 2008

nonn

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 02 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A010872, A373975 (characteristic function).
Subsequences: A000578, A014612, A046316, A121307, A373589, A373590, A373597, A373837.
Cf. also A028260, A214195, A297845.

a145784 n = a145784_list !! (n-1)
a145784_list = filter ((== 0) . a010872 . a001222) [1..]
-- Reinhard Zumkeller, May 26 2012

Join[{1}, Select[Range[2,230], Mod[Total[Transpose[FactorInteger[#]][[2]]], 3] == 0 &]] (* T. D. Noe, May 21 2012 *)

isok(k) = !(bigomega(k) % 3); \\ Amiram Eldar, May 16 2025

A010872(A001222(a(n))) = 0.

A110188 3-almost primes p * q * r not relatively prime to p+q+r.

Original entry on oeis.org

8, 18, 27, 30, 42, 50, 66, 70, 78, 98, 102, 105, 110, 114, 125, 130, 138, 154, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 242, 246, 255, 258, 266, 282, 285, 286, 290, 310, 318, 322, 338, 343, 354, 357, 366, 370, 374, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Jonathan Vos Post, Jul 15 2005

A110187 is the converse, 3-almost primes p * q * r which are relatively prime to p+q+r.

			a(1) = 8 because 8 = 2^3, which has a prime factor 2 in common with prime 2 + 2 + 2 = 6.
30 is in the sequence, since 30 = 2 * 3 * 5, which is in fact divisible by 2 + 3 + 5 = 10.

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

Cf. A014612, A110187, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\4, forprime(q=2, min(p, lim\2\p), my(pq=p*q, t); forprime(r=2, min(lim\pq, q), t=r*pq; if(gcd(t, p+q+r)>1, listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

Original entry on oeis.org

13, 15, 117, 126, 270, 2576, 8820, 16560, 21168, 46848, 295245, 441600, 846720, 4078080, 80663040, 40590720, 2173236480, 4011724800, 21122906112, 40915058688, 274148425728, 63769149440, 2707602702336, 6167442456576, 21586195906560, 29798871072768, 420127895977984, 631722992467968

Offset: 1

Jonathan Vos Post, Jun 16 2005

An emirp ("prime" spelled backwards) is a prime whose (base 10) reversal is also prime, but which is not a palindromic prime. The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (A006567).
An emirpimes ("semiprime" spelled backwards) is a semiprime whose (base 10) reversal is a different semiprime. A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (A097393).
An "emirp tsomla-3" ("3-almost prime" spelled backwards) is the k=3 sequence of the series for which k=1 are emirps and k=2 are emirpimes, a list of these being A109023. The union of these for k=1 through k = 13 is A109019.
The primes correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (A000040). The 2-almost prime numbers correspond to semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... (A001358).
The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... (A014613).
The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).
The Mathematica code for this was written by Ray Chandler, who has coauthorship credit for this sequence.

			a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.
a(2) = 15 because 15 is the smallest emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, since 15 reversed is 51 and both 15 and 51 are semiprimes (i.e. 15 = 3 * 5 and 51 = 3 * 17).
a(3) = 117 because 117 is the smallest "emirp tsomla-3" ("3-almost prime" spelled backwards) since 117 reversed is 711 and 117 = 3^2 * 13 and 711 = 3^2 * 79.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

kAlmost[n_] := Plus @@ Last /@ FactorInteger@n; fQ[n_] := Block[{id = IntegerDigits@n, k = kAlmost@n}, If[id != Reverse@id && k == kAlmost@FromDigits@Reverse@id, k, -1]]; t = Table[0, {20}]; Do[ a = fQ@n; If[a < 20 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 10, 150000000}] (* Robert G. Wilson v, Jan 06 2008 *)
Table[Select[Range[41*10^5],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==n&][[1]],{n,14}] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Oct 15 2023 *)

a(14)-a(16) from Robert G. Wilson v, Jan 06 2008
a(17)-a(24) from Donovan Johnson, Nov 17 2008
a(25)-a(28) from Michael S. Branicky, Jun 04 2024

A124268 Primes indexed by 3-almost primes.

Original entry on oeis.org

19, 37, 61, 71, 103, 107, 113, 181, 193, 197, 229, 239, 307, 317, 337, 349, 379, 383, 397, 479, 521, 523, 557, 571, 601, 619, 641, 643, 683, 691, 733, 787, 853, 857, 883, 887, 971, 977, 1013, 1019, 1021, 1033, 1039, 1091, 1109, 1123, 1151, 1187, 1279

Offset: 1

Jonathan Vos Post, Oct 23 2006

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(3almostprime(1)) = prime(8) = 19.
a(2) = prime(3almostprime(2)) = prime(12) = 37.
a(3) = prime(3almostprime(3)) = prime(18) = 61.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A122824, A124269, A124270.

Prime[#]&/@Select[Range[400],PrimeOmega[#]==3&] (* Harvey P. Dale, Mar 19 2020 *)

a(n) = prime(3almostprime(n)) = A000040(A014612(n)). {p such that p is prime and omega(primepi(p)) = 3} = {p such that p is in A000040 and A001222(A000720(p)) = 3}.

A124269 3-almost primes indexed by primes.

Original entry on oeis.org

12, 18, 27, 30, 50, 63, 75, 78, 102, 124, 130, 164, 172, 175, 190, 231, 246, 258, 279, 286, 292, 332, 345, 369, 404, 418, 425, 430, 435, 452, 524, 539, 574, 578, 606, 610, 638, 652, 663, 692, 722, 725, 775, 782, 795, 801, 854, 906, 916, 927, 938, 963, 969

Offset: 1

Jonathan Vos Post, Oct 23 2006

Primes indexed by 3-almostprimes = A124268. 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) = 3almostprime(prime(1)) = 3almostprime(2) = 12 = 2^2 * 3.
a(2) = 3almostprime(prime(2)) = 3almostprime(3) = 18 = 2 * 3^2.
a(3) = 3almostprime(prime(3)) = 3almostprime(5) = 27 = 3^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A122824, A124268, A124270.

From R. J. Mathar, Oct 15 2010: (Start)
A014612 := proc(n) option remember; if n = 1 then 8; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then return a; end if; end do; end if; end proc:
A124269 := proc(n) A014612(ithprime(n)) ; end proc: seq(A124269(n),n=1..80) ; (End)

p3 = Select[Range[1000], PrimeOmega[#] == 3 &]; p3[[Prime@ Range@ PrimePi@ Length@ p3]] (* Giovanni Resta, Jun 13 2016 *)

a(n) = 3almostprime(prime(n)) = A014612(A000040(n)).

More terms from R. J. Mathar, Oct 15 2010

A132195 Number of three-prime Carmichael numbers less than 10^n.

Original entry on oeis.org

1, 7, 12, 23, 47, 84, 172, 335, 590, 1000, 1858, 3284, 6083, 10816, 19539, 35586, 65309, 120625, 224763, 420658, 790885, 1494738

Offset: 3

Jonathan Vos Post, Nov 19 2007

a(n) = C_3(n) in Table 1, p. 34 of Chick (2007-2008) = card{c such that c is in A002997 INTERSECTION A014612 and c <= 10^n}.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 220.

J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv:0711.2915 [math.NT], 2007-2008, Table 1, p. 34.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

For k-prime Carmichael numbers up to 10^n for k = 3,4,...,11, see A132195, A174612, A174613, A174614, A174615, A174616, A174617, A299710, A299711.

Cf. A002997, A006931, A055553.

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A120806 Positive integers k such that k+d+1 is prime for all divisors d of k.

Original entry on oeis.org

1, 3, 5, 9, 11, 29, 35, 39, 41, 65, 125, 179, 191, 239, 281, 419, 431, 641, 659, 749, 755, 809, 905, 935, 989, 1019, 1031, 1049, 1229, 1289, 1451, 1469, 1481, 1829, 1859, 1931, 2129, 2141, 2339, 2519, 2549, 2969, 3161, 3299, 3329, 3359, 3389, 3539, 3821, 3851

Offset: 1

Walter Kehowski, Jul 06 2006

nonn

No a(n) can be even, since a(n)+2 must be prime. If a(n) is a prime, then it is a Sophie Germain twin prime (A045536). The only square is 9. Let the degree of n be the sum of the exponents in its prime factorization. By convention, degree(1)=0. Then every a(n) has degree less than or equal to 3. Let the weight of n be the number of its distinct prime factors. By convention, weight(1)=0. Clearly, w<=d is always true, with d=w only when the number is squarefree. Let [w,d] be the set of all integers with weight w and degree d. Then only the following possibilities occur: 1. [0,0] => a(1)=1. 2. [1,1] => Sophie Germain twin prime: 3, 5, 11, 29, A005384, A045536. 3. [1,2] => a(4)=9 is the only occurrence. 4. [1,3] => 5^3, 71^3 and 303839^3 are the first few cubes, A000578, A120808. 5. [2,2] => 5*7, 3*13 and 5*13 are the first few semiprimes, A001358, A120807. 6. [2,3] => 11*13^2, 61^2*89 and 13^2*12671 are the first few examples, A014612, A054753, A120809. 7. [3,3] => 5*11*17, 5*53*1151, 5*11*42533 are the first few 3-almost primes, A007304, A120810.

			a(11) = 125 since divisors(125) = {1, 5, 25, 125} and the set of all n+d+1 is {127, 131, 151, 251} and these are all primes.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000578, A001358, A005384, A007304, A014612, A054753, A120776, A120807, A120808, A120809, A120810.

with(numtheory); L:=[1]: for w to 1 do for k from 1 to 12^6 while nops(L)<=1000 do x:=2*k+1; if andmap(isprime,[x+2,2*x+1]) then S:=divisors(x) minus {1,x}; Q:=map(z-> x+z+1, S); if andmap(isprime,Q) then L:=[op(L),x]; print(nops(L),ifactor(x)); fi; fi; od od; L;

q[k_] := AllTrue[Divisors[k], PrimeQ[k + # + 1] &]; Select[Range[5000], q] (* Amiram Eldar, Aug 05 2024 *)

is(n)=fordiv(n,d,if(!isprime(n+d+1),return(0)));1; \\ Joerg Arndt, Nov 07 2015

A285508 Numbers with exactly three prime factors, not all distinct.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 343, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452

Offset: 1

Kalle Siukola, Apr 20 2017

Cubes of primes together with products of a prime and the square of a different prime.

Numbers k for which A001222(k) = 3, but A001221(k) < 3. - Antti Karttunen, Apr 20 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Kalle Siukola, Python program

Cf. A001221, A001222.
Setwise difference of A014612 and A007304.
Union of A030078 and A054753.

N:= 1000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/4,2)]): nP:= nops(P):
sort(select(`<=`,[seq(seq(P[i]*P[j]^2,i=1..nP),j=1..nP)],N)); # Robert Israel, Oct 20 2024

Select[Range[452], PrimeOmega[#] == 3 && PrimeNu[#] < 3 &] (* Giovanni Resta, Apr 20 2017 *)

isA285508(n) = ((omega(n) < 3) && (bigomega(n) == 3));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285508(n),write("b285508.txt", k, " ", n);k=k+1));
\\ Antti Karttunen, Apr 20 2017

from sympy import primefactors, primeomega
def omega(n): return len(primefactors(n))
def bigomega(n): return primeomega(n)
print([n for n in range(1, 501) if omega(n)<3 and bigomega(n) == 3]) # Indranil Ghosh, Apr 20 2017 and Kalle Siukola, Oct 25 2023

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A285508(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
    def f(x): return int(n+x-sum(primepi(x//(k**2))-(a<<1)+primepi(isqrt(x//k))-1 for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024

;; With my IntSeq-library.
(define A285508 (MATCHING-POS 1 1 (lambda (n) (and (= 3 (A001222 n)) (< (A001221 n) 3))))) ;; Antti Karttunen, Apr 20 2017

A307534 Heinz numbers of strict integer partitions with 3 parts, all of which are odd.

Original entry on oeis.org

110, 170, 230, 310, 374, 410, 470, 506, 590, 670, 682, 730, 782, 830, 902, 935, 970, 1030, 1034, 1054, 1090, 1265, 1270, 1298, 1370, 1394, 1426, 1474, 1490, 1570, 1598, 1606, 1670, 1705, 1790, 1826, 1886, 1910, 1955, 1970, 2006, 2110, 2134, 2162, 2255, 2266

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

The enumeration of these partitions by sum is given by A001399.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   110: {1,3,5}
   170: {1,3,7}
   230: {1,3,9}
   310: {1,3,11}
   374: {1,5,7}
   410: {1,3,13}
   470: {1,3,15}
   506: {1,5,9}
   590: {1,3,17}
   670: {1,3,19}
   682: {1,5,11}
   730: {1,3,21}
   782: {1,7,9}
   830: {1,3,23}
   902: {1,5,13}
   935: {3,5,7}
   970: {1,3,25}
  1030: {1,3,27}
  1034: {1,5,15}
  1054: {1,7,11}

Cf. A001221, A001222, A001399, A005117, A007304, A014612, A037144, A051037, A056239, A080193, A080257, A112798, A143207, A304636.

Select[Range[1000],SquareFreeQ[#]&&PrimeNu[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, nextprime
def A307534(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
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1) for a,k in filter(lambda x:x[0]&1,enumerate(primerange(2,integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024

cp's OEIS Frontend

A145784 Numbers with property that their number of prime factors counted with multiplicity is a multiple of 3.

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A110188 3-almost primes p * q * r not relatively prime to p+q+r.

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A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

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A124268 Primes indexed by 3-almost primes.

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A124269 3-almost primes indexed by primes.

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A132195 Number of three-prime Carmichael numbers less than 10^n.

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A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

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A120806 Positive integers k such that k+d+1 is prime for all divisors d of k.