A107464 -id:A107464 - cp's OEIS frontend

A007304 Sphenic numbers: products of 3 distinct primes.

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

Simon Plouffe

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

			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)

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.

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.

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

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

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 *)

for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011

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

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

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

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

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

More terms from Robert G. Wilson v, Jan 04 2006

Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009

A107953 Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).

Original entry on oeis.org

31, 175, 703, 2415, 7551, 22143, 61951, 167167, 438271, 1122303, 2818047, 6959103, 16941055, 40730623, 96862207, 228130815, 532676607, 1234173951, 2839543807, 6491734015, 14755561471, 33361494015, 75061264383, 168124481535, 375004332031, 833223655423

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 28 2005

This sequence is one of a triple sequence A(n,m,l) of the number of fuzzy subsets of a set with n+m+l objects of 3 kinds. There are n,m and l objects for each kind respectively. Here a(n)= A(n,2,1). The sequence A107464 is one other example of A(n,m,l) for m=l=1.

			a(3) = 2^4*((9/2)*16 + 21*3 + 16) - 1 = 2415 which is the number of distinct chains in the power set lattice (or fuzzy subsets) of a set X_(n+3) with 3 kinds of objects, n of one kind, 2 of another and one of yet another.

V. Murali, On the number of fuzzy subsets of an (n+3)-element set of specification n^1 2^1 1, Rhodes University Preprint, 2005.

Colin Barker, Table of n, a(n) for n = 0..1000
Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A107464, A007047, A107392.

Table[2^(n+1) (n^2/2 (n+13)+21n+16)-1,{n,0,30}] (* or *) LinearRecurrence[ {9,-32,56,-48,16},{31,175,703,2415,7551},30] (* Harvey P. Dale, Feb 10 2015 *)

Vec((48*x^3-120*x^2+104*x-31)/((x-1)*(2*x-1)^4) + O(x^100)) \\ Colin Barker, Jan 15 2015

a(n) = 2^(n+1) * ((n^2/2)*(n+13) + 21*n + 16) - 1.
G.f.: (48*x^3-120*x^2+104*x-31) / ((x-1)*(2*x-1)^4). - Colin Barker, Jan 15 2015
a(0)=31, a(1)=175, a(2)=703, a(3)=2415, a(4)=7551, a(n)=9*a(n-1)- 32*a(n-2)+ 56*a(n-3)-48*a(n-4)+16*a(n-5). - Harvey P. Dale, Feb 10 2015

a(5) corrected Jun 01 2005

Incorrect term deleted by Colin Barker, Jan 15 2015

A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

Original entry on oeis.org

79, 527, 2415, 9263, 31871, 101759, 307455, 890111, 2490367, 6774783, 18001919, 46886911, 120029183, 302678015, 753205247, 1852375039, 4507828223, 10866393087, 25970081791, 61583917055, 144997089279, 339159810047

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), May 30 2005

This sequence is an example of another line in a triple sequence A(n,m,l) with n a nonnegative integer, m = 2 and l = 1. It is related to sequences A107464, A107953 which are part of the same triple sequence with different parameter values for m and l.

			a(2) = 8 * ( (16 + 184)/6 + (316 + 370)/3 + 40 ) - 1 = 2415. This is the number of fuzzy subsets of a set of (2+4) elements of which 2 are of one kind, 3 are of another kind and 1 of a kind distinct from the other two.

V. Murali, On the enumeration of fuzzy subsets of X_(n+4) of specification n^1 3^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 12 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (11,-50,120,-160,112,-32).

Cf. A007047, A107392, A107464, A107953.

a[n_] := 2^n(n^4 + 23n^3 + 158n^2 + 370n + 240)/3 - 1; Table[ a[n], {n, 0, 21}] (* Robert G. Wilson v, May 31 2005 *)
LinearRecurrence[{11,-50,120,-160,112,-32},{79,527,2415,9263,31871,101759},40] (* Harvey P. Dale, Aug 15 2025 *)

a(n) = 2^(n+1)*( (n^4 + 23*n^3)/6 + (79*n^2 + 185*n)/3 + 40 ) - 1.

G.f.: (128*x^4-432*x^3+568*x^2-342*x+79) / ((x-1)*(2*x-1)^5). [Colin Barker, Dec 10 2012]

a(6)-a(21) from Robert G. Wilson v, May 31 2005

A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

Original entry on oeis.org

191, 1471, 7551, 31871, 119231, 410303, 1327103, 4090623, 12130303, 34842623, 97435647, 266313727, 713637887, 1879523327, 4875091967, 12474187775, 31531728895, 78832992255, 195135799295, 478649778175, 1164351373311

Offset: 0

Venkat Murali (v.murali(AT)ru.ac.za), Jun 01 2005

This sequence is another example, together with A107953 and A107954, of a triple sequence A(n,m,l) with n a nonnegative integer, m = 4 and l = 1.

			a(3) = (2^(3+1))*(1/24)*(3^5 + 36 * 3^4 + 431 * 3^3 + 2088 * 3^2 + 3972 * 3 + 2304) - 1 = 31871. This is the number of chains in the power set lattice (which is also the number of fuzzy subsets) of X_(n+5).

Venkat Murali, On the enumeration of fuzzy subsets of an (n+5)-element set X_(n+5) of specification n^1 4^1 1, Rhodes University JRC-Abstract-Report, In Preparation, 15 pages 2005.

Venkat Murali, Home page.
Index entries for linear recurrences with constant coefficients, signature (13,-72,220,-400,432,-256,64).

Cf. A007047, A107392, A107464, A107953, A107954.

a(n) = (2^(n+1))*(1/24)*(n^5 + 36*n^4 + 431*n^3 + 2088*n^2 + 3972*n + 2304) - 1,

G.f.: (320*x^5-1360*x^4+2400*x^3-2180*x^2+1012*x-191) / ((x-1)*(2*x-1)^6). [Colin Barker, Dec 10 2012]

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A007304 Sphenic numbers: products of 3 distinct primes.

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A107953 Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).

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A107954 Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.

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A107955 Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.

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