1,1

Donovan Johnson, Table of n, a(n) for n = 1..1000

A008472(n)/A006530(n) is an integer, n has at least 3 distinct prime factors and n is squarefree.

n = 286 = 2*11*13 has a form of 2pq, where p and q are twin primes;

n = 5414430 = 2*3*5*7*19*23*59, sum = 2+3+5+7+19+23+59 = 118 = 2*59.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[lf[n], 1]&& !Equal[amo[n], 1], Print[{n, ba[n]}]], {n, 2, 1000000}]

(* Alternative: *)

Select[Range@ 45000, Function[n, And[Length@ # > 1, SquareFreeQ@ n, Divisible[Total@ #, Last@ #]] &[FactorInteger[n][[All, 1]] ]]] (* Michael De Vlieger, Jul 18 2017 *)

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Labos Elemer, May 13 2002

approved

1,1

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

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.

T. D. Noe, Table of n, a(n) for n = 1..10000

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 24.

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

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)

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

# Alternative:

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

# Alternative:

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 *)

(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

(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

(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

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.

nonn,easy,changed

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

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

approved

1,1

The lower prime factor p_1 is equal to 2 and the other two are twin primes: p_3 - p_2 = 2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

60 is a term since 60 = 2^2*3*5 and 2 + 3 = 5.

286 is a term since 286 = 2*11*13 and 2 + 11 = 13.

Select[Range[2500], PrimeNu[#]==3&&Part[First/@FactorInteger[#], 1]+Part[First/@FactorInteger[#], 2]==Part[First/@FactorInteger[#], 3]&]

(PARI) isok(k) = if (omega(k)==3, my(f=factor(k)[, 1]); f[1] + f[2] == f[3]); \\ Michel Marcus, Sep 19 2023

Cf. A001221, A001359, A006512, A071142.

Subsequence of A033992 and of A071140.

Stefano Spezia, Sep 19 2023

approved

1,1

A larger than usual number of terms is provided in order to distinguish this sequence from A365795, from which it first differs at n = 58 (a(58) = 3135 is missing from A365795).

First differs from A071140 at n = 140.

Paolo Xausa, Table of n, a(n) for n = 1..10000

A382469Q[k_] := Last[#] == Total[Most[#]] & [FactorInteger[k][[All, 1]]];

Select[Range[4000], A382469Q]

Positions of zeros in A382468.

Supersequence of A365795.

Cf. A006530, A027748, A071140, A221054.

Paolo Xausa, Mar 31 2025

approved

1,2

This is a superset of 2*product of twin primes, A071142.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Christian N. K. Anderson, Table of n, a(n), and equal sums of factors for n=1..10000

q[n_] := Module[{p = FactorInteger[n][[;; , 1]], sum, x}, sum = Total[p]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, p}], x][[1 + sum/2]] > 0]; Select[Range[3200], q] (* Amiram Eldar, May 31 2025 *)

(Haskell)

a221054 n = a221054_list !! (n-1)

a221054_list = filter (z 0 0 . a027748_row) $ tail a005843_list where

z u v [] = u == v

z u v (p:ps) = z (u + p) v ps || z u (v + p) ps

-- Reinhard Zumkeller, Apr 18 2013

(PARI) isok(k) = my(f=factor(k), nb=#f~); for (i=0, 2^nb-1, my(v=Vec(Vecrev(binary(i)), nb)); if (sum(k=1, nb, if (v[k], f[k, 1])) == sum(k=1, nb, if (!v[k], f[k, 1])), return(1)); ); \\ Michel Marcus, May 31 2025

Cf. A175592 (multiplicity of prime factors allowed).

Cf. A071139-A071147, especially A071140.

Cf. A008472, A037074, A071142.

Cf. A027748, A005843, A083207.

Kevin L. Schwartz and Christian N. K. Anderson, Apr 15 2013

Missing terms inserted by Michel Marcus, May 31 2025

approved

1,1

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

98420 is in the sequence because the prime divisors are 2, 5, 7, 19, 37 and the sum 2 + 5 + 7 + 19 + 37 = 70 = 2*(37 - 2).

filter:= proc(n) local P; P:= numtheory:-factorset(n);

convert(P, `+`) = 2*(max(P)-min(P))

end proc:

select(filter, [$1..10000]); # Robert Israel, Apr 09 2019

Select[Range[5500], Plus@@((pl=First/@FactorInteger[#])/2)==pl[[-1]]-pl[[1]]&]

(PARI) isok(n) = if (n>1, my(f=factor(n)[, 1]); 2*(vecmax(f) - vecmin(f)) == vecsum(f)); \\ Michel Marcus, Apr 10 2019

Cf. A071140.

Michel Lagneau, Nov 13 2011

approved

1,1

Observation : it seems that the prime divisors of a majority of numbers n are of the form {2, p, q} with q = 2^2 + p^2, but there exists more rarely numbers with more prime divisors (examples : 8715 = 3*5*7*83; 153230 = 2*5*7*11*199).

Terms which are odd: 8715, 26145, 41349, 43575, 61005, 61971, 78435, ..., . - Robert G. Wilson v, Jul 02 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..725 from Robert Israel)

8996 is in the sequence because the prime divisors are {2, 13, 173} and 173 = 13^2 + 2^2.

filter:= proc(n)

local F, f, x;

F:= numtheory:-factorset(n);

f:= max(F);

evalb(f = add(x^2, x=F minus {f}));

end proc:

select(filter, [$1..10000]); # Robert Israel, Jul 02 2014

Reap[Do[p = First /@ FactorInteger[n]; If[p[[-1]] == Plus@@(Most[p]^2), Sow[n]], {n, 9962}]][[2, 1]]

lpfQ[n_]:=With[{f=FactorInteger[n][[;; , 1]]}, Total[Most[f]^2]==Last[f]]; Select[Range[10000], lpfQ] (* Harvey P. Dale, Jul 28 2024 *)

(PARI) isok(n) = {my(f = factor(n)); f[#f~, 1] == sum(i=1, #f~ - 1, f[i, 1]^2); } \\ Michel Marcus, Jul 02 2014

Cf. A071140.

See also the related sequences A048261, A121518.

Michel Lagneau, Feb 18 2011

Corrected by T. D. Noe, Feb 18 2011

approved