A060278 -id:A060278 - cp's OEIS frontend

A037143 Numbers with at most 2 prime factors (counted with multiplicity).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118

Offset: 1

N. J. A. Sloane

nonn

A001222(a(n)) <= 2; A054576(a(n)) = 1. - Reinhard Zumkeller, Mar 10 2006
Products of two noncomposite numbers. - Juri-Stepan Gerasimov, Apr 15 2010
Also, numbers with permutations of all divisors only with coprime adjacent elements: A109810(a(n)) > 0. - Reinhard Zumkeller, May 24 2010
A060278(a(n)) = 0. - Reinhard Zumkeller, Apr 05 2013
1 together with numbers k such that sigma(k) + phi(k) - d(k) = 2k - 2. - Wesley Ivan Hurt, May 03 2015
Products of two not necessarily distinct terms of A008578 (the same relation between A000040 and A001358). - Flávio V. Fernandes, May 28 2021

Felix Fröhlich, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Union of A008578 and A001358. Complement of A033942.
Cf. A063928, A001222, A054576, A109810, A060278.
A101040(a(n))=1 for n>1.
Subsequence of A037144. - Reinhard Zumkeller, May 24 2010
A098962 and A139690 are subsequences.

a037143 n = a037143_list !! (n-1)
a037143_list = 1 : merge a000040_list a001358_list where
   merge xs'@(x:xs) ys'@(y:ys) =
         if x < y then x : merge xs ys' else y : merge xs' ys
-- Reinhard Zumkeller, Dec 18 2012

with(numtheory): A037143:=n->`if`(bigomega(n)<3,n,NULL): seq(A037143(n), n=1..200); # Wesley Ivan Hurt, May 03 2015

Select[Range[120], PrimeOmega[#] <= 2 &] (* Ivan Neretin, Aug 16 2015 *)

is(n)=bigomega(n)<3 \\ Charles R Greathouse IV, Apr 29 2015

from math import isqrt
from sympy import primepi, primerange
def A037143(n):
    def f(x): return int(n-2+x-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

More terms from Henry Bottomley, Aug 15 2001

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

A023891 Sum of composite divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 6, 0, 12, 9, 10, 0, 22, 0, 14, 15, 28, 0, 33, 0, 34, 21, 22, 0, 54, 25, 26, 36, 46, 0, 61, 0, 60, 33, 34, 35, 85, 0, 38, 39, 82, 0, 83, 0, 70, 69, 46, 0, 118, 49, 85, 51, 82, 0, 114, 55, 110, 57, 58, 0, 157, 0, 62, 93, 124, 65, 127, 0, 106, 69, 129, 0

Offset: 1

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000203, A035322, A035321, A060278, A023890.

Array[ Plus @@ (Select[ Divisors[ # ], (!PrimeQ[ # ] && #>1)& ])&, 75 ]
a[n_] := DivisorSigma[1, n] - Plus @@ FactorInteger[n][[;; , 1]] - 1; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = sumdiv(n, d, d*!isprime(d)) - 1; \\ Michel Marcus, Jun 12 2019

a(n) = A023890(n) - 1. - Sean A. Irvine, Jun 11 2019

A035321 Sum of composite divisors of n that are not primes nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 0, 10, 0, 18, 0, 14, 15, 0, 0, 24, 0, 30, 21, 22, 0, 42, 0, 26, 0, 42, 0, 61, 0, 0, 33, 34, 35, 72, 0, 38, 39, 70, 0, 83, 0, 66, 60, 46, 0, 90, 0, 60, 51, 78, 0, 78, 55, 98, 57, 58, 0, 153, 0, 62, 84, 0, 65, 127, 0, 102, 69, 129, 0, 168, 0, 74, 90, 114, 77

Offset: 1

N. J. A. Sloane

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A035322, A023891, A060278.

One less than A178637.

pp := array(1..100); for i from 1 to 100 do pp[i] := 0; od: for i from 1 to 25 do for j from 1 to 6 do t1 := ithprime(i)^j; if t1<100 then pp[t1] := 1; fi; od: od: pp[1] := 1; A035321 := proc(n) local i,d,t1,t2; t1 := 0; for d from 1 to n do if n mod d = 0 and pp[d] = 0 then t1 := t1+d; fi; od; t1; end;

Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]>1)& ])&, 80 ]

A035321(n) = sumdiv(n,d,(omega(d)>1)*(d)); \\ Antti Karttunen, Aug 06 2018

a(n) = A178637(n) - 1. - Antti Karttunen, Aug 06 2018

Description corrected by Jack Brennen, Mar 28 2001

A035322 Sum of composite divisors of n that are less than n and are not primes nor prime powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 10, 0, 0, 0, 18, 0, 0, 0, 14, 0, 31, 0, 0, 0, 0, 0, 36, 0, 0, 0, 30, 0, 41, 0, 22, 15, 0, 0, 42, 0, 10, 0, 26, 0, 24, 0, 42, 0, 0, 0, 93, 0, 0, 21, 0, 0, 61, 0, 34, 0, 59, 0, 96, 0, 0, 15, 38, 0, 71, 0, 70, 0, 0, 0, 123, 0, 0, 0, 66, 0

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A035321, A023891, A060278.

Table[ Plus @@ Select[ Divisors[ n ], (Length[ FactorInteger[ # ] ]>1 && #Harvey P. Dale, Jan 09 2019 *)

Description corrected by Jack Brennen, Mar 28 2001

A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.

Original entry on oeis.org

532, 945, 2624, 5704, 6536, 229648, 497696, 652970, 685088, 997408, 1481504, 11177984, 32869504, 52813084, 132612224, 224841856, 2140668416, 2404135424, 2550700288, 6469054976, 9367192064, 19266023936, 23414463358, 31381324288, 45812547584, 55620289024

Offset: 1

Naohiro Nomoto

From Peter Luschny, Dec 14 2009: (Start)
A term of this sequence is a Zumkeller number (A083207) since the set of its divisors can be partitioned into two disjoint parts so that the sums of the two parts are equal.
1 + sigma*(k) = sigma'(k) + k
sigma*(k) := Sum_{1 < d < k, d|k, d not prime}, (A060278),
sigma'(k) := Sum_{1 < d < k, d|k, d prime}, (A105221). (End)

			532 = 1 - 2 + 4 - 7 + 14 - 19 + 28 + 38 + 76 + 133 + 266.

Donovan Johnson, Table of n, a(n) for n = 1..34 (terms <= 10^12)
Donovan Johnson, 82 terms > 10^12.
Peter Luschny, Zumkeller Numbers.

Cf. A083207, A105221, A060278, A000203, A027751, A010051, A002808.

import Data.List (partition)
a048055 n = a048055_list !! (n-1)
a048055_list = [x | x <- a002808_list,
               let (us,vs) = partition ((== 1) . a010051) $ a027751_row x,
               sum us + x == sum vs]
-- Reinhard Zumkeller, Apr 05 2013

with(numtheory): A048055 := proc(n) local k;
if sigma(n)=2*(n+add(k,k=select(isprime,divisors(n))))
then n else NULL fi end: seq(A048055(i),i=1..7000);
# Peter Luschny, Dec 14 2009

zummableQ[n_] := DivisorSigma[1, n] == 2*(n + Total[Select[Divisors[n], PrimeQ]]); n = 2; A048055 = {}; While[n < 10^6, If[zummableQ[n], Print[n]; AppendTo[A048055, n]]; n++]; A048055 (* Jean-François Alcover, Dec 07 2011, after Peter Luschny *)

from sympy import divisors, primefactors
A048055 = []
for n in range(1,10**4):
    s = sum(divisors(n))
    if not s % 2 and 2*n <= s and (s-2*n)/2 == sum(primefactors(n)):
        A048055.append(n) # Chai Wah Wu, Aug 20 2014

a(15)-a(19) from Donovan Johnson, Dec 07 2008
a(20)-a(24) from Donovan Johnson, Jul 06 2010
a(25)-a(26) from Donovan Johnson, Feb 09 2012

cp's OEIS Frontend

A037143 Numbers with at most 2 prime factors (counted with multiplicity).

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

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A023891 Sum of composite divisors of n.

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A035321 Sum of composite divisors of n that are not primes nor prime powers.

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A035322 Sum of composite divisors of n that are less than n and are not primes nor prime powers.

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A048055 Numbers k such that (sum of the nonprime proper divisors of k) - (sum of prime divisors of k) = k.

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