A007304 -id:A007304 - cp's OEIS frontend

A096918 Intermediate prime factor of n-th product of 3 distinct primes.

3, 3, 3, 5, 3, 3, 5, 5, 3, 5, 3, 7, 5, 5, 3, 7, 3, 5, 5, 3, 5, 7, 7, 3, 5, 3, 7, 7, 3, 5, 11, 5, 5, 3, 7, 5, 3, 7, 3, 5, 11, 7, 7, 3, 7, 5, 11, 3, 11, 5, 7, 5, 3, 13, 7, 5, 5, 3, 7, 13, 3, 11, 7, 5, 3, 5, 11, 7, 3, 5, 7, 13, 7, 3, 7, 5, 5, 3, 11, 11, 3, 5, 17, 7, 3, 7, 13, 7, 5, 3, 11, 5, 5, 11, 5

Offset: 1

Reinhard Zumkeller, Jul 15 2004

nonn

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

Cf. A007304, A096917, A096919.

f[n_]:=Last/@FactorInteger[n]=={1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];f3[n_]:=First/@FactorInteger[n][[2,1]];lst={};Do[If[f[n],AppendTo[lst,f3[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 A096918(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 sorted(primefactors(bisection(f)))[1] # Chai Wah Wu, Aug 30 2024

A096917(n)*a(n)*A096919(n) = A007304(n).

A096917(n) < a(n) < A096919(n).

A096919 Greatest prime factor of n-th product of 3 distinct primes.

Original entry on oeis.org

5, 7, 11, 7, 13, 17, 7, 11, 19, 13, 23, 11, 11, 17, 29, 13, 31, 19, 13, 37, 23, 11, 17, 41, 17, 43, 19, 13, 47, 19, 13, 29, 31, 53, 23, 23, 59, 17, 61, 37, 17, 11, 19, 67, 29, 41, 19, 71, 13, 43, 31, 29, 73, 17, 13, 31, 47, 79, 23, 19, 83, 23, 37, 53, 89, 37, 17, 41, 97, 59

Offset: 1

Reinhard Zumkeller, Jul 15 2004

nonn

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

Cf. A006530, A007304, A096917, A096918.

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,f2[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 A096919(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 max(primefactors(bisection(f))) # Chai Wah Wu, Aug 30 2024

A096917(n)*A096918(n)*a(n) = A007304(n).
A096917(n) < A096918(n) < a(n).
a(n) = A006530(A007304(n)).

A128896 Triangular numbers that are products of three distinct primes.

Original entry on oeis.org

66, 78, 105, 190, 231, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051, 15753

Offset: 1

Zak Seidov, Apr 20 2007

nonn

			a(1)=T(11)=66=2*3*11, a(2)=T(12)=78=2*3*13, a(3)=T(14)=105=3*5*7, a(4)=T(19)=190=2*5*19, a(5)=T(21)=231=3*7*11, a(6)=T(28)=406=2*7*29.
T(15) = 120 = 2^3*3*5. The triangular 120 has three prime factors but is not a product of these factors. Thus, 120 is not in this sequence.

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

Cf. A000217, A068443, A069903, A076551, A127637.

Select[Table[n(n+1)/2,{n,1,210}],Transpose[FactorInteger[ # ]][[2]]=={1,1,1}&]
Select[Accumulate[Range[200]],PrimeNu[#]==PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = T(k) = k*(k+1)/2 = p*q*r for some k,p,q,r, where T(k) is triangular number and p, q, r are distinct primes.

Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - R. J. Mathar, Apr 22 2007

Name clarified by Tanya Khovanova, Sep 06 2022

A157346 Products of 3 distinct Sophie Germain primes.

Original entry on oeis.org

30, 66, 110, 138, 165, 174, 230, 246, 290, 318, 345, 410, 435, 498, 506, 530, 534, 615, 638, 678, 759, 786, 795, 830, 890, 902, 957, 1038, 1074, 1130, 1146, 1166, 1245, 1265, 1310, 1334, 1335, 1353, 1398, 1434, 1506, 1595, 1686, 1695, 1730, 1749, 1758, 1790

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

			30 = 2*3*5; 2,3 and 5 are distinct Sophie Germain primes.
66 = 2*3*11; 2,3 and 11 are distinct Sophie Germain primes.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001358, A005384, A111206, A157342, A006881, A157344, A157345, A007304.

lst={};Do[If[Plus@@Last/@FactorInteger[n]==3,a=Length[First/@FactorInteger[n]];If[a==3,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];e=b[[3]];If[PrimeQ[2*c+1]&&PrimeQ[2*d+1]&&PrimeQ[2*e+1],AppendTo[lst,n]]]],{n,7!}];lst
With[{sgps=Select[Prime[Range[100]],PrimeQ[2#+1]&]},Take[Union[ Times@@@ Subsets[sgps,{3}]],60]] (* Harvey P. Dale, Aug 10 2011 *)

A157347 Products of 3 distinct non-Sophie Germain primes.

Original entry on oeis.org

1547, 1729, 2261, 2821, 3367, 3689, 3913, 4123, 4199, 4277, 4403, 4921, 5117, 5369, 5551, 5593, 5719, 6097, 6251, 6461, 6643, 6851, 7021, 7189, 7259, 7657, 7847, 7973, 8029, 8113, 8177, 8449, 8687, 8827, 8911, 9139, 9191, 9331, 9373, 9401, 9443, 9503

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

			1547 = 7*13*17 is a term: its prime factors 7, 13, and 17 are not Sophie Germain primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..5600

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

S:=[ p: p in PrimesUpTo(120) | not IsPrime(2*p+1) ]; T:=[ q: a, b, c in S | a lt b and b lt c and q lt 10000 where q is a*b*c ]; Sort(~T); T; // Klaus Brockhaus, Apr 11 2009

lst={};Do[If[Plus@@Last/@FactorInteger[n]==3,a=Length[First/@FactorInteger[n]];If[a==3,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];e=b[[3]];If[ !PrimeQ[2*c+1]&&!PrimeQ[2*d+1]&&!PrimeQ[2*e+1],AppendTo[lst,n]]]],{n,8!}];lst

Entries verified by Klaus Brockhaus, Apr 11 2009

A178254 Number of permutations of the proper divisors of n such that no adjacent elements have a common divisor greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 2, 2, 6, 1, 4, 1, 6, 6, 0, 1, 4, 1, 4, 6, 6, 1, 0, 2, 6, 2, 4, 1, 36, 1, 0, 6, 6, 6, 0, 1, 6, 6, 0, 1, 36, 1, 4, 4, 6, 1, 0, 2, 4, 6, 4, 1, 0, 6, 0, 6, 6, 1, 0, 1, 6, 4, 0, 6, 36, 1, 4, 6, 36, 1, 0, 1, 6, 4, 4, 6, 36, 1, 0, 0, 6, 1, 0, 6, 6, 6, 0, 1, 0, 6, 4, 6, 6, 6, 0, 1, 4, 4, 0, 1, 36, 1

Offset: 1

Reinhard Zumkeller, May 24 2010

nonn

Depends only on prime signature;
range = {0, 1, 2, 4, 6, 36};
a(A033987(n)) = 0; a(A037144(n)) > 0;
a(A008578(n))=1; a(A168363(n))=2; a(A054753(n))=4; a(A006881(n))=6; a(A007304(n))=36.

			Proper divisors for n=21 are: 1, 3, and 7:
a(39) = #{[1,3,7], [1,7,3], [3,1,7], [3,7,1], [7,1,3], [7,3,1]} = 6;
proper divisors for n=12 are: 1, 2, 3, 4, and 6:
a(12) = #{[2,3,4,1,6], [4,3,2,1,6], [6,1,2,3,4], [6,1,4,3,2]} = 4;
proper divisors for n=42: 1, 2, 3, 6, 7, 14, and 21:
a(42) = #{[2,21,1,6,7,3,14], [2,21,1,14,3,7,6], [3,14,1,6,7,2,21], [3,14,1,21,2,7,6], [6,1,14,3,7,2,21], [6,1,21,2,7,3,14], ...} = 36, see the appended file for the list of all permutations.

R. Zumkeller, Table of n, a(n) for n = 1..10000
R. Zumkeller, Example: n=42

Cf. A109810.

A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

Original entry on oeis.org

1152, 3200, 6272, 8748, 15488, 21632, 36992, 46208, 54675, 67712, 107163, 107648, 123008, 175232, 215168, 236672, 264627, 282752, 312500, 359552, 369603, 445568, 476288, 574592, 632043, 645248, 682112, 703125, 789507, 798848, 881792, 1013888

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Numbers with same prime signature.
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688.

Cf. A085548, A085967, A085969.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
      do od; k
    end:
seq (a(n), n=1..32);  # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,7}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/7), t=p^7;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A179689(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 f(x): return n+x-sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Title edited by Daniel Forgues, Jan 22 2011

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7qr with p, q and r distinct primes.

Original entry on oeis.org

1920, 2688, 4224, 4480, 4992, 6528, 7040, 7296, 8320, 8832, 9856, 10880, 11136, 11648, 11904, 12160, 14208, 14720, 15232, 15744, 16512, 17024, 18048, 18304, 18560, 19840, 20352, 20608, 21870, 22656, 23424, 23680, 23936, 25728, 25984, 26240, 26752, 27264

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes List of Prime Signatures
OEIS Wiki, Numbers with same prime signature.

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 1, 1]
      do od; k
    end:
seq (a(n), n=1..40); # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,7}; Select[Range[30000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\6)^(1/7), t1=p^7;forprime(q=2, lim\t1, if(p==q, next);t2=t1*q;forprime(r=q+1, lim\t2, if(p==r,next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primerange, primepi, integer_nthroot
def A179696(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 f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**7)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,7)[0]+1))+sum(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Title edited by Daniel Forgues, Jan 22 2011

A267891 Numbers with 8 odd divisors.

Original entry on oeis.org

105, 135, 165, 189, 195, 210, 231, 255, 270, 273, 285, 297, 330, 345, 351, 357, 375, 378, 385, 390, 399, 420, 429, 435, 455, 459, 462, 465, 483, 510, 513, 540, 546, 555, 561, 570, 594, 595, 609, 615, 621, 627, 645, 651, 660, 663, 665, 690, 702, 705, 714, 715, 741, 750, 756, 759, 770, 777, 780, 783, 795, 798, 805, 837

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly eight odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 8 subparts. - Omar E. Pol, Dec 29 2016
Numbers n such that A000265(n) has prime signature {7} or {3,1} or {1,1,1}, i.e., is in A092759 or A065036 or A007304. - Robert Israel, Mar 15 2018
Numbers that can be formed in exactly 7 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Column 8 of A266531.
Cf. A000265, A001227, A007304, A038547, A065036, A092759, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, this sequence, A267892, A267893.

[n: n in [1..1000] | #[d: d in Divisors(n) | IsOdd(d)] eq 8]; // Bruno Berselli, Apr 04 2016

filter:= proc(n) local r;
  r:= n/2^padic:-ordp(n,2);
  numtheory:-tau(r)=8
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 15 2018

Select[Range@ 840, Length@ Select[Divisors@ #, OddQ] == 8 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 8; \\ after Michel Marcus

A001227(a(n)) = 8.

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

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

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
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.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

cp's OEIS Frontend

A096918 Intermediate prime factor of n-th product of 3 distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A096919 Greatest prime factor of n-th product of 3 distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A128896 Triangular numbers that are products of three distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A157346 Products of 3 distinct Sophie Germain primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A157347 Products of 3 distinct non-Sophie Germain primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A178254 Number of permutations of the proper divisors of n such that no adjacent elements have a common divisor greater than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7*q*r with p, q and r distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7qr with p, q and r distinct primes.