A080257 -id:A080257 - cp's OEIS frontend

A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Clark Kimberling

The sequence of numbers divisible by a prime number of primes coincides with this up to 210, which has 4 prime factors. - Lior Manor, Aug 23 2001
A085970(n) = Max{k: a(k)<=n}.
Numbers n such that LCM of proper divisors of n equals neither 1 nor n. - Labos Elemer, Dec 01 2004
a(n) provides bases b in which automorphic numbers m^2 ending with m in base b exist. In the complement there aren't any automorphic numbers. - Martin Renner, Dec 07 2011
Numbers with at least 2 distinct prime factors. - Jonathan Sondow, Oct 17 2013
There exists an equiangular n-gon whose edge lengths form a permutation of 1, 2, ..., n if and only if n is in the sequence (see Woeginger's survey and Munteanu & Munteanu). - Jonathan Sondow, Oct 17 2013
Numbers that are the product of two relatively prime factors. These numbers are used in testing a sequence for multiplicativity. - Michael Somos, Jun 02 2015
A theorem from Donald McCarthy: Let d be any positive integer which is not a prime power; then there exists a finite group whose order is divisible by d but which contains no subgroup of order d (see link and A340511). - Bernard Schott, Dec 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 8719 terms from Daniel Forgues)
Donald McCarthy, Sylow's theorem is a sharp partial converse to Lagrange's theorem, Mathematische Zeitschrift, 113, 383-384 (1970).
Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Wikipedia, Prime power
G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.
Günter Ziegler and Brady Haran, Cannons and Sparrows, Numberphile video (2018).

Cf. A000040, A000961 (complement), A001221, A014963, A020500, A085970.
Cf. A340511.
Subsequence of A080257.

a024619 n = a024619_list !! (n-1)
a024619_list = filter ((== 0) . a010055) [1..]
-- Reinhard Zumkeller, Nov 17 2011

IsA024619:=func< n | not IsPrime(n) and not (t and IsPrime(b) where t, b, A024619(n)%20%5D;%20//%20_Klaus%20Brockhaus">:=IsPower(n)) >; [ n: n in [2..200] | IsA024619(n) ]; // _Klaus Brockhaus, Feb 25 2011

a := proc(n) numtheory[factorset](n); if 1 < nops(%) then n else NULL fi end:
seq(a(i), i=1..110); # Peter Luschny, Aug 11 2009

Select[Range@111, Length@FactorInteger@# > 1 &] (* Robert G. Wilson v, Dec 07 2005 *)

is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A024619(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

def A024619_list(n) :
    return [k for k in (2..n) if not k.is_prime() and not k.is_prime_power()]
A024619_list(112)  # Peter Luschny, Feb 03 2012 [corrected by Terry D. Grant, Sep 16 2020]

A001221(a(n)) > 1.
A014963(a(n)) = 1.
A020500(a(n)) = 1. - Benoit Cloitre, Aug 26 2003
A010055(a(n)) = 0. - Reinhard Zumkeller, Nov 17 2011
a(n) ~ n. - Charles R Greathouse IV, Mar 21 2013
a(n) ~ n - pi(n) [See Panaitopol]. - N. J. A. Sloane, Sep 27 2020
A118887(a(n)) > 0. - Jonathan Sondow, Oct 17 2013

A033942 Positive integers with at least 3 prime factors (counted with multiplicity).

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144

Offset: 1

Jeff Burch

nonn

A001055(a(n)) > 2; e.g., for a(3)=18 there are 4 factorizations: 1*18 = 2*9 = 2*3*3 = 3*6. - Reinhard Zumkeller, Dec 29 2001
A001222(a(n)) > 2; A054576(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
Also numbers such that no permutation of all divisors exists with coprime adjacent elements: A109810(a(n))=0. - Reinhard Zumkeller, May 24 2010
A211110(a(n)) > 3. - Reinhard Zumkeller, Apr 02 2012
A060278(a(n)) > 0. - Reinhard Zumkeller, Apr 05 2013
Volumes of rectangular cuboids with each side > 1. - Peter Woodward, Jun 16 2015
If k is a term then so is k*m for m > 0. - David A. Corneth, Sep 30 2020
Numbers k with a pair of proper divisors of k, (d1,d2), such that d1 < d2 and gcd(d1,d2) > 1. - Wesley Ivan Hurt, Jan 01 2021

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A014612.
A101040(a(n))=0.
A033987 is a subsequence; complement of A037143. - Reinhard Zumkeller, May 24 2010
Subsequence of A080257.
See also A002808 for 'Composite numbers' or 'Divisible by at least 2 primes'.

a033942 n = a033942_list !! (n-1)
a033942_list = filter ((> 2) . a001222) [1..]
-- Reinhard Zumkeller, Oct 27 2011

with(numtheory): A033942:=n->`if`(bigomega(n)>2, n, NULL): seq(A033942(n), n=1..200); # Wesley Ivan Hurt, Jun 23 2015

Select[ Range[150], Plus @@ Last /@ FactorInteger[ # ] > 2 &] (* Robert G. Wilson v, Oct 12 2005 *)
Select[Range[150],PrimeOmega[#]>2&] (* Harvey P. Dale, Jun 22 2011 *)

is(n)=bigomega(n)>2 \\ Charles R Greathouse IV, May 04 2013

from sympy import factorint
def ok(n): return sum(factorint(n).values()) > 2
print([k for k in range(145) if ok(k)]) # Michael S. Branicky, Sep 10 2022

from math import isqrt
from sympy import primepi, primerange
def A033942(n):
    def f(x): return int(n+primepi(x)+sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Numbers of the form Product p_i^e_i with Sum e_i >= 3.

a(n) ~ n. - Charles R Greathouse IV, May 04 2013

Corrected by Patrick De Geest, Jun 15 1998

A347440 Number of factorizations of n with alternating product < 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 3, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 0, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 1, 1, 0, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.
Also the number of factorizations of n with alternating sum < 0.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) factorizations for n = 6, 12, 24, 30, 48, 72, 96, 120:
  2*3  2*6  3*8      5*6   6*8      8*9      2*48         2*60
       3*4  4*6      2*15  2*24     2*36     3*32         3*40
            2*12     3*10  3*16     3*24     4*24         4*30
            2*2*2*3        4*12     4*18     6*16         5*24
                           2*2*2*6  6*12     8*12         6*20
                           2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                    2*2*3*6  2*2*4*6      10*12
                                    2*3*3*4  2*3*4*4      2*2*5*6
                                             2*2*2*12     2*3*4*5
                                             2*2*2*2*2*3  2*2*2*15
                                                          2*2*3*10

PlanetMath, alternating sum

Positions of 0's are A000430.
Positions of 2's are A054753.
Positions of non-0's are A080257.
Positions of 1's are A332269.
The weak version (<= 1 instead of < 1) is A339846, ranked by A028982.
The reciprocal version is A339890.
The additive version is A344608, ranked by A119899.
The even-sum additive version is A344743, ranked by A119899 /\ A300061.
Allowing any integer alternating product gives A347437, additive A347446.
The equal version (= 1 instead of < 1) is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The complement (>= 1 instead of < 1) is counted by A347456.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A008549, A058622, A119620, A294175, A330972, A347442, A347447, A347454.

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[Select[facs[n],altprod[#]<1&]],{n,100}]

a(2^n) = A344608(n).

a(n) = A339846(n) - A347438(n).

A119313 Numbers with a prime as third-smallest divisor.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 42, 45, 46, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 74, 75, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).

			a(1) = A087134(3) = 6.
From _Gus Wiseman_, Oct 19 2019: (Start)
The sequence of terms together with their divisors begins:
    6: {1,2,3,6}
   10: {1,2,5,10}
   12: {1,2,3,4,6,12}
   14: {1,2,7,14}
   15: {1,3,5,15}
   18: {1,2,3,6,9,18}
   21: {1,3,7,21}
   22: {1,2,11,22}
   24: {1,2,3,4,6,8,12,24}
   26: {1,2,13,26}
   30: {1,2,3,5,6,10,15,30}
   33: {1,3,11,33}
   34: {1,2,17,34}
   35: {1,5,7,35}
   36: {1,2,3,4,6,9,12,18,36}
   38: {1,2,19,38}
   39: {1,3,13,39}
   42: {1,2,3,6,7,14,21,42}
   45: {1,3,5,9,15,45}
   46: {1,2,23,46}
(End)

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

Complement of A119314.
Subsequences: A006881, A000469, A008588.
A subset of A002808 and A080257.
Numbers whose third-largest divisor is prime are A328338.
Second-smallest divisor is A020639.
Third-smallest divisor is A292269.
Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.

q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* Gus Wiseman, Oct 15 2019 *)
Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

Name edited by Gus Wiseman, Oct 19 2019

A211110 Number of partitions of n into divisors > 1 of n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 12, 1, 3, 3, 10, 1, 15, 1, 16, 3, 3, 1, 80, 2, 3, 5, 20, 1, 94, 1, 36, 3, 3, 3, 280, 1, 3, 3, 158, 1, 154, 1, 28, 25, 3, 1, 1076, 2, 29, 3, 32, 1, 255, 3, 262, 3, 3, 1, 7026, 1, 3, 32, 202, 3, 321, 1, 40, 3, 302, 1, 12072, 1

Offset: 0

Reinhard Zumkeller, Apr 01 2012

nonn

a(A000040(n)) = 1; a(A002808(n)) > 1;
a(A001248(n)) = 2; a(A080257(n)) > 2;
a(A006881(n)) = 3; a(A033942(n)) > 3.

			a(10) = #{10, 5+5, 2+2+2+2+2} = 3;
a(11) = #{11} = 1;
a(12) = #{12, 6+6, 6+4+2, 6+3+3, 6+2+2+2, 4+4+4, 4+4+2+2, 4+3+3+2, 4+2+2+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 12;
a(13) = #{13} = 1;
a(14) = #{14, 7+7, 2+2+2+2+2+2+2} = 3;
a(15) = #{15, 5+5+5, 3+3+3+3+3} = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A211111, A018818, A027750.

Cf. A210442.

a211110 n = p (tail $ a027750_row n) n where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

a[n_] := Module[{b, l}, l = Rest[Divisors[n]]; b[m_, i_] := b[m, i] = If[m==0, 1, If[i<1, 0, b[m, i-1] + If[l[[i]]>m, 0, b[m-l[[i]], i]]]]; b[n, Length[l]]]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

isokp(p, n) = {for (k=1, #p, if ((p[k]==1) || (n % p[k]), return (0));); return (1);}
a(n) = {my(nb = 0); forpart(p=n, if (isokp(p,n), nb++)); nb;} \\ Michel Marcus, Jun 30 2015

A220571 Composite numbers that are Brazilian.

Original entry on oeis.org

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

Bernard Schott, Dec 16 2012

nonn

There are just two differences of members with A080257:
1) the term 6 is missing here because 6 is not a Brazilian number.
2) the new term 121 is present although 121 has only 3 divisors, because 121 = 11^2 = 11111_3 is a composite number which is Brazilian. 121 is the lone square of a prime which is Brazilian: Theorem 5, page 37 of Quadrature article in links.
There is an infinity of Brazilian composite numbers (Theorem 1, page 32 of Quadrature article in links: every even number >= 8 is a Brazilian number).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A125134, A190300.

Select[Range[4, 10^2], And[CompositeQ@ #, Module[{b = 2, n = #}, While[And[b < n - 1, Length@ Union@ IntegerDigits[n, b] > 1], b++]; b < n - 1]] &] (* Michael De Vlieger, Jul 30 2017, after T. D. Noe at A125134 *)

A307534 Heinz numbers of strict integer partitions with 3 parts, all of which are odd.

Original entry on oeis.org

110, 170, 230, 310, 374, 410, 470, 506, 590, 670, 682, 730, 782, 830, 902, 935, 970, 1030, 1034, 1054, 1090, 1265, 1270, 1298, 1370, 1394, 1426, 1474, 1490, 1570, 1598, 1606, 1670, 1705, 1790, 1826, 1886, 1910, 1955, 1970, 2006, 2110, 2134, 2162, 2255, 2266

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

The enumeration of these partitions by sum is given by A001399.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   110: {1,3,5}
   170: {1,3,7}
   230: {1,3,9}
   310: {1,3,11}
   374: {1,5,7}
   410: {1,3,13}
   470: {1,3,15}
   506: {1,5,9}
   590: {1,3,17}
   670: {1,3,19}
   682: {1,5,11}
   730: {1,3,21}
   782: {1,7,9}
   830: {1,3,23}
   902: {1,5,13}
   935: {3,5,7}
   970: {1,3,25}
  1030: {1,3,27}
  1034: {1,5,15}
  1054: {1,7,11}

Cf. A001221, A001222, A001399, A005117, A007304, A014612, A037144, A051037, A056239, A080193, A080257, A112798, A143207, A304636.

Select[Range[1000],SquareFreeQ[#]&&PrimeNu[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, nextprime
def A307534(n):
    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
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1) for a,k in filter(lambda x:x[0]&1,enumerate(primerange(2,integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024

A080256 Sum of numbers of distinct and of all prime factors of n.

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 6, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 7, 2, 4, 5, 7, 4, 6, 2, 5, 4, 6, 2, 7, 2, 4, 5, 5, 4, 6, 2, 7, 5, 4, 2, 7, 4, 4, 4, 6, 2, 7, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 6, 2, 6, 6

Offset: 1

Reinhard Zumkeller, Feb 10 2003

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

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

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {my(f = factor(n)); omega(f) + bigomega(f);} \\ Amiram Eldar, Sep 28 2023

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).
Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

A088381 Numbers greater than the cube of their smallest prime factor.

Original entry on oeis.org

10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Reinhard Zumkeller, Sep 28 2003

nonn

a(n) > A020639(a(n))^3 = A088378(a(n)); complement of A088380;

a(n) > A088380(k) for n <= 28, a(n) < A088380(k) for n > 28.

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

Cf. A080257, A088378, A088380, A088383.
Cf. A020639, A138511 (subsequence).
Positions of numbers greater than 3 in A307908.

a088381 n = a088381_list !! (n-1)
a088381_list = filter f [1..] where
                      f x = p ^ 2 < div x p  where p = a020639 x
-- Reinhard Zumkeller, Jan 08 2015

filter:= n -> n > min(numtheory:-factorset(n))^3:
select(filter, [$2..200]); # Robert Israel, Aug 11 2020

isok(n) = n > factor(n)[1,1]^3; \\ Michel Marcus, Jan 08 2015

A084114 Number of divisions when calculating A084110(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 1, 1, 1, 2, 0, 3, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 1, 1, 1, 1, 4, 0, 1, 1, 2, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n) - 1 - A084113(n) = A032741(n) - A084113(n) = (A032741(n)-A084115(n))/2;

a(n) = 0 iff n is prime or a square of prime (A000430).

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

Cf. A080257.

Cf. A027750, A084110, A084113.

a084114 = g 0 1 . tail . a027750_row where
   g c _ []     = c
   g c x (d:ds) = if r > 0 then g c (x * d) ds else g (c + 1) x' ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

cp's OEIS Frontend

A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A033942 Positive integers with at least 3 prime factors (counted with multiplicity).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A347440 Number of factorizations of n with alternating product < 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A119313 Numbers with a prime as third-smallest divisor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A211110 Number of partitions of n into divisors > 1 of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

A220571 Composite numbers that are Brazilian.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A307534 Heinz numbers of strict integer partitions with 3 parts, all of which are odd.

Views