cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A007304 Sphenic numbers: products of 3 distinct primes.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438
Offset: 1

Views

Author

Simon Plouffe

Keywords

Comments

Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005
Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007
Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012
Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012
n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020

Examples

			From _Gus Wiseman_, Nov 05 2020: (Start)
Also Heinz numbers of strict integer partitions into three parts, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are counted by A001399(n-6) = A069905(n-3), with ordered version A001399(n-6)*6. The sequence of terms together with their prime indices begins:
     30: {1,2,3}     182: {1,4,6}     286: {1,5,6}
     42: {1,2,4}     186: {1,2,11}    290: {1,3,10}
     66: {1,2,5}     190: {1,3,8}     310: {1,3,11}
     70: {1,3,4}     195: {2,3,6}     318: {1,2,16}
     78: {1,2,6}     222: {1,2,12}    322: {1,4,9}
    102: {1,2,7}     230: {1,3,9}     345: {2,3,9}
    105: {2,3,4}     231: {2,4,5}     354: {1,2,17}
    110: {1,3,5}     238: {1,4,7}     357: {2,4,7}
    114: {1,2,8}     246: {1,2,13}    366: {1,2,18}
    130: {1,3,6}     255: {2,3,7}     370: {1,3,12}
    138: {1,2,9}     258: {1,2,14}    374: {1,5,7}
    154: {1,4,5}     266: {1,4,8}     385: {3,4,5}
    165: {2,3,5}     273: {2,4,6}     399: {2,4,8}
    170: {1,3,7}     282: {1,2,15}    402: {1,2,19}
    174: {1,2,10}    285: {2,3,8}     406: {1,4,10}
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • "Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.

Links

Crossrefs

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.
Cf. A162143 (a(n)^2).
For the following, NNS means "not necessarily strict".
A014612 is the NNS version.
A046389 is the restriction to odds (NNS: A046316).
A075819 is the restriction to evens (NNS: A075818).
A239656 gives first differences.
A285508 lists terms of A014612 that are not squarefree.
A307534 is the case where all prime indices are odd (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338557 is the case where all prime indices are even (NNS: A338556).
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers.
A008289 counts strict partitions by sum and length.
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
Cf. A000212, A000217, A001840, A101271, A284825, A321773, A337599, A337605.

Programs

  • Haskell
    a007304 n = a007304_list !! (n-1)
a007304_list = filter f [1..] where
f u = p < q && q < w && a010051 w == 1 where
p = a020639 u; v = div u p; q = a020639 v; w = div v q
-- Reinhard Zumkeller, Mar 23 2014
  • Maple
    with(numtheory): a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n),n=1..450); # Emeric Deutsch
A007304 := proc(n)
    option remember;
    local a;
    if n =1 then
        30;
    else
        for a from procname(n-1)+1 do
            if bigomega(a)=3 and nops(factorset(a))=3 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Dec 06 2016
is_a := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A007304List := upto -> select(is_a, [seq(1..upto)]):  # Peter Luschny, Apr 14 2025
  • Mathematica
    Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *)
With[{upto=500},Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]],{3}],#<=upto&]]] (* Harvey P. Dale, Jan 08 2015 *)
Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==3&] (* Gus Wiseman, Nov 05 2020 *)
  • PARI
    for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2,(lim)^(1/3),forprime(q=p+1,sqrt(lim\p),t=p*q;forprime(r=q+1,lim\t,listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); Set(v) \\ Charles R Greathouse IV, Jan 21 2025
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A007304(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)))
    kmin, kmax = 0,1
    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 # Chai Wah Wu, Aug 29 2024
  • SageMath
    def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

Formula

A008683(a(n)) = -1.
A000005(a(n)) = 8. - R. J. Mathar, Aug 14 2009
A002033(a(n)-1) = 13. - Juri-Stepan Gerasimov, Oct 07 2009, R. J. Mathar, Oct 14 2009
A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010
A050326(a(n)) = 5, subsequence of A225228. - Reinhard Zumkeller, May 03 2013
a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

Extensions

More terms from Robert G. Wilson v, Jan 04 2006
Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009

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

Views

Author

Reinhard Zumkeller, Jul 15 2004

Keywords

Links

Crossrefs

Cf. A020639, A007304, A096918, A096919.

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jul 15 2004

Keywords

Links

Crossrefs

Cf. A007304, A096917, A096919.

Programs

  • Mathematica
    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 *)
  • Python
    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

Formula

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

A176170 Smallest prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Keywords

Comments

FactorInteger[210]=2*3*5*7,...

Links

Crossrefs

Cf. A020639, A046386, A006881, A007304, A070647, A096916, A096917, A096918, A096919

Programs

  • Mathematica
    f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f1[n]]],{n,0,2*7!}];lst
  • PARI
    n=0; i=0; while(i<10000,n++;if((4 == bigomega(n))&&(4 == omega(n)),i++;write("b176170.txt", i, " ", factor(n)[1,1]))); \\ Antti Karttunen, Dec 06 2017

Formula

a(n) = A020639(A046386(n)). - Antti Karttunen, Dec 06 2017

A176171 Second prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 5, 5, 3, 5, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 3, 3, 3, 5, 7, 3, 5, 3, 5, 3, 5, 5, 3, 5, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 5, 7, 3, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 7, 5, 3, 5, 3, 3, 7, 3, 5, 3, 5, 3, 3, 5, 5, 3, 5, 3, 5, 3, 3, 3
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Keywords

Comments

FactorInteger[210]=2*3*5*7,...

Crossrefs

Cf. A006881, A007304, A070647, A096916, A096917, A096918, A096919, A176170

Programs

  • Mathematica
    f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f2[n]]],{n,0,2*7!}];lst

A176172 3rd prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

5, 5, 5, 7, 5, 7, 5, 5, 7, 7, 7, 11, 5, 7, 5, 7, 5, 11, 7, 7, 7, 5, 11, 5, 7, 13, 7, 7, 5, 11, 13, 11, 7, 5, 7, 7, 5, 7, 13, 7, 5, 11, 11, 17, 7, 7, 11, 5, 7, 11, 11, 5, 11, 7, 5, 13, 7, 13, 17, 5, 7, 13, 11, 13, 7, 5, 11, 7, 7, 11, 19, 5, 11, 11, 7, 11, 7, 13, 5, 11, 17, 7, 13, 11, 7, 5, 7, 7, 5
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Keywords

Comments

FactorInteger[210]=2*3*5*7,...

Links

Crossrefs

Cf. A006881, A007304, A070647, A096916, A096917, A096918, A096919, A176170, A176171

Programs

  • Maple
    N:= 10000: # to use products <= N
Primes:= select(isprime, [2,seq(i,i=3..N/30)]):
P4:= NULL:
for ia from 1 to nops(Primes) do
  a:= Primes[ia];
  for ib from 1 to ia-1 do
    b:= Primes[ib];
    if 6*a*b > N then break fi;
    for ic from 1 to ib-1 do
      c:= Primes[ic];
      if 2*a*b*c > N then break fi;
      for id from 1 to ic-1 do
        d:= Primes[id];
        if a*b*c*d > N then break fi;
        R[a*b*c*d]:= b;
        P4:= P4, a*b*c*d;
od od od od:
P4:= sort([P4]):
map(t -> R[t], P4); # Robert Israel, May 14 2019
  • Mathematica
    f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f3[n]]],{n,0,2*7!}];lst

A176173 4th|largest prime-factor of n-th product of 4 distinct primes.

Original entry on oeis.org

7, 11, 13, 11, 17, 13, 19, 23, 17, 11, 19, 13, 29, 13, 31, 23, 37, 17, 11, 17, 29, 41, 19, 43, 31, 17, 19, 13, 47, 13, 19, 23, 37, 53, 23, 41, 59, 17, 23, 43, 61, 17, 29, 19, 47, 19, 13, 67, 29, 31, 19, 71, 13, 31, 73, 17, 53, 29, 23, 79, 23, 31, 37, 19, 59, 83, 23, 61, 37, 17
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Keywords

Comments

FactorInteger[210]=2*3*5*7,...

Crossrefs

Cf. A006881, A007304, A070647, A096916, A096917, A096918, A096919, A176170, A176171, A176172

Programs

  • Mathematica
    f0[n_]:=Last/@FactorInteger[n]=={1,1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=First/@FactorInteger[n][[2,1]];f3[n_]:=First/@FactorInteger[n][[3,1]];f4[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f0[n],AppendTo[lst,f4[n]]],{n,0,2*7!}];lst
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.