A007304 -id:A007304 - cp's OEIS frontend

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

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

A179642 Product of exactly 5 primes, 3 of which are distinct.

Original entry on oeis.org

120, 168, 180, 252, 264, 270, 280, 300, 312, 378, 396, 408, 440, 450, 456, 468, 520, 552, 588, 594, 612, 616, 680, 684, 696, 700, 702, 728, 744, 750, 760, 828, 882, 888, 918, 920, 945, 952, 980, 984, 1026, 1032, 1044, 1064, 1100, 1116, 1128, 1144, 1160

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

			120=2^3*3*5, 168=2^3*3*7, 180=2^2*3^2*5, 252=2^2*3^2*7, 264=2^3*3*11, 270=2*3^3*5

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Union of A189975 and A179643.

Cf. A006881, A007304, A085986, A085987, A178739.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3} || Sort[Last/@FactorInteger[n]]=={1,2,2}; Select[Range[2000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));forprime(p=2,sqrt(lim\12),forprime(q=p+1,sqrt(lim\p^2\2),t=(p*q)^2;forprime(r=2,lim\t,if(p==r||q==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

A215217 Smaller member of a pair of sphenic twins, consecutive integers, each the product of three distinct primes.

Original entry on oeis.org

230, 285, 429, 434, 609, 645, 741, 805, 902, 969, 986, 1001, 1022, 1065, 1085, 1105, 1130, 1221, 1245, 1265, 1309, 1310, 1334, 1406, 1434, 1442, 1462, 1490, 1505, 1533, 1581, 1598, 1605, 1614, 1634, 1729, 1742, 1833, 1885, 1886, 1946, 2013, 2014, 2054, 2085

Offset: 1

Martin Renner, Aug 06 2012

nonn

455 is not a term of the sequence, since 455 = 5*7*13 is sphenic, i.e., the number of distinct prime factors is 3, though 456 = 2^3*3*19 has 3 distinct prime factors but is not sphenic, because the number of prime factors with repetition is 5 > 3.

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

Cf. A007304, A066509, A140077.

twinLow [] = []
twinLow [_] = []
twinLow (n : (m : ns))
    | m == n + 1 = n : twinLow (m : ns)
    | otherwise = twinLow (m : ns)
a215217 n = (twinLow a007304_list) !! (n - 1)
-- Peter Dolland, May 31 2019

Sphenics:= select(t -> (map(s->s[2],ifactors(t)[2])=[1,1,1]), {$1..10000}):
Sphenics intersect map(`-`,Sphenics,1); # Robert Israel, Aug 13 2014

Select[Range[2500], (PrimeNu[#] == PrimeOmega[#] == PrimeNu[#+1] == PrimeOmega[#+1] == 3)&] (* Jean-François Alcover, Apr 11 2014 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,2500}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)

is_a033992(n) = omega(n)==3 && bigomega(n)==3
is(n) = is_a033992(n) && is_a033992(n+1) \\ Felix Fröhlich, Jun 10 2019

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

A123322 Products of 8 distinct primes (squarefree 8-almost primes).

Original entry on oeis.org

9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 18888870, 20030010, 20281170, 20930910, 21111090, 21411390, 21637770, 21951930, 23130030, 23393370, 23993970, 24534510, 25555530, 25571910

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046310.

			a(1) = 9699690 = 2*3*5*7*11*13*17*19 = A002110(8).

Robert G. Wilson v, Table of n, a(n) for n = 1..29422 (first 10000 terms from Rick Shepherd)
Index to sequences related to prime signature

Cf. A001221, A001222, A005117, A046310, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123321 (k=7), A115343 (k=9).

N:= 3*10^7: # to get all terms  <= N
pmax:= floor(N/mul(ithprime(i),i=1..7)):
Primes:= select(isprime,[2,seq(i,i=3..pmax,2)]):
sort(select(`<`,map(convert,combinat:-choose(Primes,8),`*`),N)); # Robert Israel, Dec 18 2018

f8Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f8Q[n], AppendTo[lst, n]], {n, 10!, 11!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Take[ Sort[ Times @@@ Subsets[ Prime@ Range@ 15, {8}]], 22] (* Robert G. Wilson v, Dec 18 2018 *)

is(n)=issquarefree(n)&&omega(n)==8 \\ Charles R Greathouse IV, Feb 01 2017, corrected (following an observation from Zak Seidov) by M. F. Hasler, Dec 19 2018

is(n) = my(f = factor(n)); omega(f) == 8 && bigomega(f) == 8 \\ David A. Corneth, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123322(n):
    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,0,1,1,8)))
    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

Edited by Robert Israel, Dec 18 2018

A157352 Products (semiprimes) of two distinct safe primes.

Original entry on oeis.org

35, 55, 77, 115, 161, 235, 253, 295, 329, 413, 415, 517, 535, 581, 649, 749, 835, 895, 913, 1081, 1135, 1169, 1177, 1253, 1315, 1357, 1589, 1735, 1795, 1837, 1841, 1909, 1915, 1969, 2335, 2395, 2429, 2461, 2497, 2513, 2515, 2681, 2773, 2815, 2893, 2935

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

35=5*7; 5 and 7 are safe primes, 55=5*11; 5 and 11 are safe primes,...

			a(1) = 35 since 35 = 5 * 7, and (5 - 1)/2 = 2 and (7 - 1)/2 = 3 are both prime, thus 5 and 7 are distinct safe primes.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358, A005384, A005385, A006881, A007304, A111206, A157342, A157344, A157345, A157346, A157347.

lst={};Do[If[Plus@@Last/@FactorInteger[n]==2,a=Length[First/@FactorInteger[n]];If[a==2,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];If[PrimeQ[(c-1)/2]&&PrimeQ[(d-1)/2],AppendTo[lst,n]]]],{n,7!}];lst
Select[Select[Range@ 3000, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)
Module[{upto=3000,sp},sp=Select[Prime[Range[PrimePi[upto/5]]],PrimeQ[(#-1)/2]&];Select[Union[Times@@@Subsets[sp,{2}]],#<+upto&]] (* Harvey P. Dale, Aug 25 2017 *)

Example corrected by Harvey P. Dale, Aug 25 2017

A259349 Numbers n such that n-1, n, and n+1 are all products of 6 distinct primes (i.e. belong to A067885).

Original entry on oeis.org

1990586014, 1994837494, 2129658986, 2341714794, 2428906514, 2963553594, 3297066410, 3353808094, 3373085990, 3623442746, 3659230730, 3809238770, 3967387346, 4058711734, 4144727994, 4196154390, 4502893746, 4555267690, 4653623534

Offset: 1

James G. Merickel, Jun 24 2015

nonn

A subsequence of A169834 and A067885.
The rudimentary method employed by the PARI program below reaches the limit of its usefulness here. Contrast it with the method required for A259350, which is over 4.5 orders of magnitude faster than the analog of this (and may still be some distance best).
a(1)=A093550(6) (that sequence's 5th term, with offset 2). The program arbitrarily makes use of this knowledge, but will run (slower) without it.

			1990586013 = 3*13*29*67*109*241,
1990586014 = 2*23*37*43*59*461, and
1990586015 = 5*11*17*19*89*1259; and no smaller trio of this kind exists, making the middle value a(1).

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

Cf. A093550, A169834, A364265, A248201, A248202, A248203, A248204, A259350, A259801.
For products of 1, 2, 3, 4, 5, and 6 distinct primes see A000040, A006881, A007304, A046386, A046387, and A067885, resp.
See A364265 for a closely related sequence. - N. J. A. Sloane, Jul 18 2023

{
\\Program initialized with known a(1).\\
\\The purpose of vector s and value u\\
\\is to skip bad values modulo 36.\\
k=1990586014;s=[4,4,8,8,8,4];u=1;
while(1,
  if(issquarefree(k),
    if(issquarefree(k-1),
      if(issquarefree(k+1),
        if(omega(k)==6,
          if(omega(k-1)==6,
            if(omega(k+1)==6,
              print1(k" ")))))));
  k+=s[u];if(u==6,u=1,u++))
}

{n: A001221(n-1) = A001221(n) = A001221(n+1) = A001222(n-1) = A001222(n) = A001222(n+1) = 6}. - R. J. Mathar, Jul 18 2023

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Inverse Moebius transform of A056671.

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) =  9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A096917 Smallest prime factor of the n-th product of 3 distinct primes.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 5, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 5, 2, 2, 2, 3, 2, 5, 2, 3, 2, 2, 3, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Jul 15 2004

nonn

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

Cf. A020639, A007304, A096918, A096919.

f[n_]:=Last/@FactorInteger[n]=={1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f[n],AppendTo[lst,f1[n]]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 10 2010 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A096917(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    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 min(primefactors(bisection(f))) # Chai Wah Wu, Aug 30 2024

a(n)*A096918(n)*A096919(n) = A007304(n).
a(n) < A096918(n) < A096919(n).
a(n) = A020639(A007304(n)).

cp's OEIS Frontend

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

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A179642 Product of exactly 5 primes, 3 of which are distinct.

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A215217 Smaller member of a pair of sphenic twins, consecutive integers, each the product of three distinct primes.

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

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

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A123322 Products of 8 distinct primes (squarefree 8-almost primes).

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PARI

PARI

Python

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A157352 Products (semiprimes) of two distinct safe primes.

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