A030078 -id:A030078 - cp's OEIS frontend

A054397 Numbers m such that there are precisely 5 groups of order m.

8, 12, 18, 20, 27, 50, 52, 68, 98, 116, 125, 135, 148, 164, 171, 212, 242, 244, 273, 292, 297, 333, 338, 343, 356, 388, 399, 404, 436, 452, 459, 548, 578, 596, 621, 628, 651, 657, 692, 722, 724, 741, 772, 777, 783, 788, 825, 855, 875, 916, 932, 964, 981

Offset: 1

N. J. A. Sloane, May 21 2000

nonn

For m = 2*p^2 (p prime), there are precisely 5 groups of order m, so A079704 and A143928 (p odd prime) are two subsequences. - Bernard Schott, Dec 10 2021
For m = p^3, p prime, there are also 5 groups of order m, so A030078, where these groups are described, is another subsequence. - Bernard Schott, Dec 11 2021
For m squarefree, there are 5 groups of order m if and only if all of the following hold: 3|m, there are exactly two prime factors p,q of m such that p,q = 1 mod 3, no other relations of the form p' = 1 mod q' hold for p',q' prime factors of m. - Robin Jones, May 27 2025

			For m = 8, the 5 groups of order 8 are C8, C4 x C2, D8, Q8, C2 x C2 x C2 and for m = 12 the 5 groups of order 12 are C3 : C4, C12, A4, D12, C6 x C2 where C, D, Q  mean cyclic, dihedral, quaternion groups of the stated order and A is the alternating group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 03 2017

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..2035, (terms 1..120 from Muniru A Asiru and Georg Fischer).
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), this sequence (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

Cf. A384370 (squarefree numbers in this sequence).

A054397 := Filtered([1..2015], n -> NumberSmallGroups(n) = 5); # Muniru A Asiru, Nov 03 2017

Select[Range[10^4], FiniteGroupCount[#] == 5 &] (* Robert Price, May 23 2019 *)

Sequence is { k | A000001(k) = 5 }. - Muniru A Asiru, Nov 03 2017

More terms from Christian G. Bower, May 25 2000

A325249 Sum of the omega-sequence of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 4, 3, 5, 1, 8, 1, 5, 5, 5, 1, 8, 1, 8, 5, 5, 1, 9, 3, 5, 4, 8, 1, 7, 1, 6, 5, 5, 5, 7, 1, 5, 5, 9, 1, 7, 1, 8, 8, 5, 1, 10, 3, 8, 5, 8, 1, 9, 5, 9, 5, 5, 1, 12, 1, 5, 8, 7, 5, 7, 1, 8, 5, 7, 1, 10, 1, 5, 8, 8, 5, 7, 1, 10, 5, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The omega-sequence of 180 is (5,3,2,2,1) with sum 13, so a(180) = 13.

Positions of m's are A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A118914, A181819, A182850, A323023, A325274.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Table[Total[omseq[n]],{n,100}]

a(n) = A056239(A325248(n)).

a(n!) = A325274(n).

A179665 a(n) = prime(n)^9.

Original entry on oeis.org

512, 19683, 1953125, 40353607, 2357947691, 10604499373, 118587876497, 322687697779, 1801152661463, 14507145975869, 26439622160671, 129961739795077, 327381934393961, 502592611936843, 1119130473102767

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Product_{n >= 2, m_n = (a(n) mod 4) - 2} ((a(n) + 1) / (a(n) - 1))^m_n = 209865342976 / 209844223875. - Dimitris Valianatos, May 13 2020

			a(1) = 512 since the ninth power of the first prime is 2^9 = 512. - _Wesley Ivan Hurt_, Mar 27 2014

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Will Nicholes, Prime Signatures.

Cf. A001248, A030078, A030514, A050997, A030516, A092759, A179645.

Cf. A013667, A013676, A085969.

[p^9: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A179665:=n->ithprime(n)^9; seq(A179665(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^9&, 30]
Prime[Range[30]]^9 (* Harvey P. Dale, Jul 20 2024 *)

a(n)=prime(n)^9 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A000040(n)^9 = A001017(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(9) = 0.0020044675... (A085969). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(9)/zeta(18) = A013667/A013676.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(9) = 1/A013667. (End)

A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 17, 1, 4, 7, 7, 4, 12, 1, 13, 4, 4

Offset: 1

Labos Elemer, Jul 19 2001

nonn

Let us say that two divisors d_1 and d_2 of n are adjacent divisors if d_1/d_2 or d_2/d_1 is a prime. Then a(n) is the number of all pairs of adjacent divisors of n. - Vladimir Shevelev, Aug 16 2010
Equivalent to the preceding comment: a(n) is the number of edges in the directed multigraph on tau(n) vertices, vertices labeled by the divisors d_i of n, where edges connect vertex(d_i) and vertex(d_j) if the ratio of the labels is a prime. - R. J. Mathar, Sep 23 2011
a(A001248(n)) = 2. - Reinhard Zumkeller, Dec 02 2014
Depends on the prime signature of n as follows: a(A025487(n)) = 0, 1, 2, 4, 3, 7, 4, 10, 12, 5, 12, 13, 20, 6, 17, 16, 28, 7, 22, 33, 19 ,32, 24, 36, 8, 27, 46, ... (n>=1). - R. J. Mathar, May 28 2017

			n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2+2+3 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
E. Pérez Herrero, Psychedelic Geometry Blogspot, CURIOUS SERIES-002

Cf. A001221, A001248, A027750.

a062799 = sum . map a001221 . a027750_row
-- Reinhard Zumkeller, Dec 02 2014

read("transforms") ;
A001221 := proc(n)
        nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n),n=1..80)] ;
ones := [seq(1,n=1..80)] ;
DIRICHLET(ones,omega) ; # R. J. Mathar, Sep 23 2011
N:= 1000: # to get a(1) to a(N)
B:= Vector(N,t-> nops(numtheory:-factorset(t))):
A:= Vector(N):
for d from 1 to N do
  md:= d*[$1..floor(N/d)];
  A[md]:= map(`+`,A[md],B[d])
od:
convert(A,list); # Robert Israel, Oct 21 2015

f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
omega[n_] := Length[FactorInteger[n]]; SetAttributes[omega, Listable]; omega[1] := 0; A062799[n_] := Plus @@ omega[Divisors[n]] (* Enrique Pérez Herrero, Sep 08 2009 *)

a(n)=my(f=factor(n)[,2],s);forvec(v=vector(#f,i,[0,f[i]]),s+=sum(i=1,#f,v[i]>0));s \\ Charles R Greathouse IV, Oct 15 2015

vector(100, n, sumdiv(n, k, omega(k))) \\ Altug Alkan, Oct 15 2015

a(n) = Sum_{d|n} A001221(d), that is, where d runs over divisors of n.
For squarefree s (i.e., s in A005117), a(s) = omega(s)*2^(omega(s)-1), where omega(n) = A001221(n). Also, for n>1, a(n) <= omega(n)*A000005(n) - 1. - Enrique Pérez Herrero, Sep 08 2009
Let n=Product_{i=1..omega(n)} p(i)^e(i). a(n) = d[Product_{i=1..omega(n)} (1 + e(i)*x)]/dx|x=1. In other words, a(n) = Sum_{m>=1} A146289(n,m)*m. - Geoffrey Critzer, Feb 10 2015
a(A000040(n)) = 1; a(A001248(n)) = 2; a(A030078(n)) = 3; a(A030514(n)) = 4; a(A050997(n)) = 5. - Altug Alkan, Oct 17 2015
a(n) = Sum_{prime p|n} A000005(n/p). - Max Alekseyev, Aug 11 2016
G.f.: Sum_{k>=1} omega(k)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s)^2*primezeta(s) where primezeta(s) = Sum_{prime p} p^(-s). - Benedict W. J. Irwin, Jul 16 2018

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

Original entry on oeis.org

1, 8, 27, 32, 125, 128, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 6859, 7776, 8192, 9261, 10648, 10976, 12167, 13824, 16000, 16807, 17496, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 31104, 32768, 35937

Offset: 1

Amiram Eldar, Jul 03 2020

nonn

This sequence is a permutation of A355038.
This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).
a(n)/rad(a(n)) is a permutation of the squares.
a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

			8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.

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

Intersection of A001694 and A268335.
Intersection of A036966 and A268335.
A355038 in ascending order.
A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.
Cf. A007947, A065487.

Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

from math import isqrt, prod
from sympy import factorint
def afind(N): # all terms up to limit N
    cands = (n**2*prod(factorint(n**2)) for n in range(1, isqrt(N//2)+2))
    return sorted(c for c in cands if c <= N)
print(afind(4*10**4)) # Michael S. Branicky, Jun 16 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).

A056595 Number of nonsquare divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 6, 1, 3, 2, 4, 1, 7, 1, 3, 3, 3, 3, 5, 1, 3, 3, 6, 1, 7, 1, 4, 4, 3, 1, 7, 1, 4, 3, 4, 1, 6, 3, 6, 3, 3, 1, 10, 1, 3, 4, 3, 3, 7, 1, 4, 3, 7, 1, 8, 1, 3, 4, 4, 3, 7, 1, 7, 2, 3, 1, 10, 3, 3, 3, 6, 1, 10, 3, 4, 3, 3, 3, 9, 1, 4, 4, 5, 1, 7, 1

Offset: 1

Labos Elemer, Jul 21 2000

nonn

a(A000430(n))=1; a(A030078(n))=2; a(A030514(n))=2; a(A006881(n))=3; a(A050997(n))=3; a(A030516(n))=3; a(A054753(n))=4; a(A000290(n))=A055205(n). - Reinhard Zumkeller, Aug 15 2011

			a(36)=5 because the set of divisors of 36 has tau(36)=nine elements, {1, 2, 3, 4, 6, 9, 12, 18, 36}, five of which, that is {2, 3, 6, 12, 18}, are not perfect squares.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000188, A046951.
See A194095 and A194096 for record values and where they occur.
Cf. A001620, A013661.

a056595 n = length [d | d <- [1..n], mod n d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011

A056595 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not issqr(d) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A056595(n),n=1..40) ; # R. J. Mathar, Aug 18 2024

Table[Count[Divisors[n],?(#!=Floor[Sqrt[#]]^2&)],{n,110}] (* _Harvey P. Dale, Jul 10 2013 *)
a[1] = 0; a[n_] := Times @@ (1 + (e = Last /@ FactorInteger[n])) - Times @@ (1 + Floor[e/2]); Array[a, 100] (* Amiram Eldar, Jul 22 2019 *)

a(n)=sumdiv(n,d,!issquare(d)) \\ Charles R Greathouse IV, Aug 28 2016

a(n) = A000005(n) - A046951(n) = tau(n) - tau(A000188(n)).

Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A060800 a(n) = p^2 + p + 1 where p runs through the primes.

Original entry on oeis.org

7, 13, 31, 57, 133, 183, 307, 381, 553, 871, 993, 1407, 1723, 1893, 2257, 2863, 3541, 3783, 4557, 5113, 5403, 6321, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 16257, 17293, 18907, 19461, 22351, 22953, 24807, 26733, 28057, 30103, 32221

Offset: 1

Jason Earls, Apr 27 2001

Terms are divisible by 3 iff p is of the form 6*m+1 (A002476). - Michel Marcus, Jan 15 2017

			a(3) = 31 because 5^2 + 5 + 1 = 31.

Harry J. Smith, Table of n, a(n) for n = 1..1000
R. J. Mathar, No common terms in the sequences sigma(p^i) and sigma(p^(i+1)) as p runs through the primes.

Cf. A001248, A131991, A131992, A131993, A253905.

Cf. A008864, A000203. - Zak Seidov, Feb 13 2016

[p^2+p+1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 20 2014

A060800:= n -> map (p -> p^(2)+p+1, ithprime(n)):
seq (A060800(n), n=1..41); # Jani Melik, Jan 25 2011

#^2 + # + 1&/@Prime[Range[200]] (* Vincenzo Librandi, Mar 20 2014 *)

{ n=0; forprime (p=2, prime(1000), write("b060800.txt", n++, " ", p^2 + p + 1); ) } \\ Harry J. Smith, Jul 13 2009

a(n) = A036690(n) + 1.
a(n) = 1 + A008864(n)*A000040(n) = (A030078(n) - 1)/A006093(n). - Reinhard Zumkeller, Aug 06 2007
a(n) = sigma(prime(n)^2) = A000203(A000040(n)^2). - Zak Seidov, Feb 13 2016
a(n) = A000203(A001248(n)). - Michel Marcus, Feb 15 2016
Product_{n>=1} (1 - 1/a(n)) = zeta(3)/zeta(2) (A253905). - Amiram Eldar, Nov 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001

A347460 Number of distinct possible alternating products of factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 5, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
  1  4  8    12   24   30    36   48    60    120
     1  2    3    6    10/3  9    12    15    30
        1/2  3/4  8/3  5/6   4    16/3  20/3  40/3
             1/3  2/3  3/10  1    3     15/4  15/2
                  3/8  2/15  4/9  3/4   12/5  24/5
                  1/6        1/4  1/3   3/5   10/3
                             1/9  3/16  5/12  5/6
                                  1/12  4/15  8/15
                                        3/20  3/10
                                        1/15  5/24
                                              2/15
                                              3/40
                                              1/30

Positions of 1's are 1 and A000040.
Positions of 2's appear to be A001358.
Positions of 3's appear to be A030078.
Dominates A038548, the version for reverse-alternating product.
Counting only integers gives A046951.
The even-length case is A072670.
The version for partitions (not factorizations) is A347461, reverse A347462.
The odd-length case is A347708.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A292886 counts knapsack factorizations, by sum A293627.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
Cf. A002033, A119620, A143823, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@facs[n]]],{n,100}]

A133530 Sum of third powers of three consecutive primes.

Original entry on oeis.org

160, 495, 1799, 3871, 8441, 13969, 23939, 43415, 66347, 104833, 149365, 199081, 252251, 332207, 458079, 581237, 733123, 885655, 1047691, 1239967, 1453843, 1769795, 2189429, 2647943, 3035701, 3348071, 3612799, 3962969, 4786309

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=160 because 2^3+3^3+5^3=160.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133531, A133532, A133533.

a = 3; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a, {n, 1, 100}]

a(n) = A133534(n) + A030078(n+2). - Michel Marcus, Nov 08 2013

cp's OEIS Frontend

A054397 Numbers m such that there are precisely 5 groups of order m.

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A325249 Sum of the omega-sequence of n.

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A179665 a(n) = prime(n)^9.

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A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

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A080257 Numbers having at least two distinct or a total of at least three prime factors.

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A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

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