A120944 -id:A120944 - cp's OEIS frontend

A000469 1 together with products of 2 or more distinct primes.

1, 6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158

Offset: 1

Dan Bentley (dtb(AT)research.att.com)

Nonprime squarefree numbers.

Except for 1, composite n such that the squarefree part of n is greater than phi(n). - Benoit Cloitre, Apr 06 2002

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

Cf. A005117, A007913, A000010, A010051, A239508, A239509, A120944 (composite squarefree numbers, same sequence apart from the first term).

a000469 n = a000469_list !! (n-1)
a000469_list = filter ((== 0) . a010051) a005117_list
-- Reinhard Zumkeller, Mar 21 2014

select(numtheory:-issqrfree and not isprime, [$1..1000]); # Robert Israel, Aug 06 2015

lst={}; Do[If[SquareFreeQ[n], If[ !PrimeQ[n], AppendTo[lst,n]]], {n,200}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009 *)
With[{upto=200},Complement[Select[Range[upto],SquareFreeQ],Prime[ Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Oct 01 2011 *)
Select[Range[200], !PrimeQ[#] && PrimeOmega[#] == PrimeNu[#] &] (* Carlos Eduardo Olivieri, Aug 06 2015 *)

for(n=0,64, if(isprime(n), n+1, if(issquarefree(n),print(n))))

for(n=1,160,if(core(n)*(1-isprime(n))>eulerphi(n),print1(n,",")))

from math import isqrt
from sympy import primepi, mobius
def A000469(n):
    def f(x): return n+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 02 2024

n such that A007913(n)>A000010(n). - Benoit Cloitre, Apr 06 2002
N-floor(N/p1) - floor(N/(p2) - ... - floor(N/p(i) + floor(N/(c2) + floor(N/(c3)+ ... + floor(N/c(j)-1 where N is any number; p1,p2 are the primes with p(i) being the first prime > square root of N and c2, c3 are the numbers other than 1 in this sequence with c(j) <= N will yield the number of primes less than or equal to N other than p1, p2, ..., p(i). - Ben Paul Thurston, Aug 15 2007
A005171(a(n))*A008966(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009
Sum(n=1, Infinity, 1/a(n)^s) = Zeta(s)/Zeta(2s) - PrimeZeta(s). - Enrique Pérez Herrero, Mar 31 2012
n such that A001221(n) = A001222(n), n nonprime. - Carlos Eduardo Olivieri, Aug 06 2015
a(n) = kn + O(n/log n) where k = Pi^2/6. - Charles R Greathouse IV, Aug 02 2024

A130092 Numbers with at least two factors having in their canonical prime factorization equal exponents.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

Complement of A130091; A120944 and A085986 are subsequences;

a(n)>A130091(n) for n<=150, a(n) < A130091(n) for n>150.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization

t[n_] := FactorInteger[n][[All, 2]];s = Select[Range[400], Union[t[#]] == Sort[t[#]] &] (* A130091 *)
Complement[Range[Max[s]], s]  (* A130092 *)
(* Clark Kimberling, Mar 12 2015 *)

A360519 Let C consist of 1 together with all numbers with at least two distinct prime factors; this is the lexicographically earliest infinite sequence {a(n)} of distinct elements of C such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2).

Original entry on oeis.org

1, 6, 10, 35, 21, 12, 20, 55, 33, 18, 14, 77, 99, 15, 40, 22, 143, 39, 24, 28, 91, 65, 30, 34, 119, 63, 36, 26, 221, 51, 42, 38, 95, 45, 48, 44, 187, 85, 50, 46, 69, 57, 76, 52, 117, 75, 70, 58, 87, 93, 62, 56, 105, 111, 74, 68, 153, 123, 82, 80, 115, 161, 84, 60, 145, 203, 98, 54, 129, 215, 100, 66, 141

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 01 2023

nonn

In other words, C contains all positive numbers except powers of primes p^k, k>=1.
This is a modified version of the Enots Wolley sequence A336957. The modification ensures that the sequence does not contain the prime 2.
Let Ker(k), the kernel of k, denote the set of primes dividing k. Thus Ker(36) = {2,3}, Ker(1) = {}.
Theorem: a(1)=1, a(2)=6; thereafter, a(n) is the smallest number m not yet in the sequence such that
(i) Ker(m) intersect Ker(a(n-1)) is nonempty,
(ii) Ker(m) intersect Ker(a(n-2)) is empty, and
(iii) The set Ker(m) \ Ker(a(n-1)) is nonempty.
Conjecture: The sequence is a permutation of C.

Scott R. Shannon, Table of n, a(n) for n = 1..100000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing squarefree composites (in A120944) in green, numbers neither prime power nor squarefree (in A126706) in blues, highlighting powerful numbers (in A001694 and A286708) in large dark blue.
Scott R. Shannon, White on black graph of first 10^5 terms [Some aspects of the graph are more visible when black and white are swapped. The green line is a(n) = n]
Scott R. Shannon, Image of the first 1 million terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
N. J. A. Sloane, Table showing a(1)-a(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111).

Cf. A098550, A336957.

For a number of sequences related to this, see A361102 (the sequence C) and the following entries.

with(numtheory);
N:= 10^4: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N, datatype=integer[4]):
A[1]:=1; A[2]:=6;
for n from 3 do
  for k from 10 to N do
    if B[k] = 0 and igcd(k, A[n-1]) > 1 and igcd(k, A[n-2]) = 1 then
          if nops(factorset(k) minus factorset(A[n-1])) > 0 then
       A[n]:= k;
       B[k]:= 1;
       break;
          fi;
    fi
  od:
  if k > N then break; fi;
od:
s1:=[seq(A[i], i=1..n-1)];

nn = 2^12; c[_] = False;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[
 Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}];
 u = 10; i = a[1]; j = a[2];
Do[k = u;
  While[Nand[! PrimePowerQ[k], ! c[k],
    CoprimeQ[i, k], ! CoprimeQ[j, k], ! Divisible[j, f[k]]], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, f[k]}];
  If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]]
  , {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 03 2023 *)

A010553 a(n) = tau(tau(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 4, 2

Offset: 1

N. J. A. Sloane

Ramanujan (1915) posed the problem of finding the extreme large values of a(n). Buttkewitz et al. determined the maximal order of log a(n).

Every number eventually appears. Sequence A193987 gives the least term where each number appears. - T. D. Noe, Aug 10 2011

S. Ramanujan, Highly composite numbers. Proc. London Math. Soc., series 2, 14 (1915), 347-409. Republished in Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78-128.

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 2000 terms from Enrique Pérez Herrero)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta, A problem of Ramanujan, Erdos and Katai on the iterated divisor function, arXiv:1108.1815 [math.NT], Aug 08 2011.

Cf. A000005, A036450, A193987 (least number k such that tau(tau(k)) = n), A335831.

Maple
```
with(numtheory): f := n->tau(tau(n));
```

Table[Nest[DivisorSigma[0, #] &, n, 2], {n, 81}] (* Michael De Vlieger, Dec 24 2015 *)

A010553(n)=numdiv(numdiv(n)); \\ Enrique Pérez Herrero, Jul 13 2010

a(n) = A000005(A000005(n)). a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 3 for pq = product of two distinct primes (A006881), a(pq...z) = k + 1 for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = A000005(k+1) for p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). - Jaroslav Krizek, Jul 17 2009
a(A007947(n)) = 1 + A001221(n); (n>1). - Enrique Pérez Herrero, May 30 2010
Asymptotically, Max_{i<=n} log(tau(tau(i))) = sqrt(log(n))/log_2(n) * (c + O(log_3(n)/log_2(n)) where c = 8*Sum_{j>=1} log^2 (1 + 1/j)) ~ 2.7959802335... [Buttkewitz et al.].

A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 3, 4, 3, 7, 3, 7, 4, 7, 3, 12, 3, 7, 7, 6, 3, 12, 3, 12, 7, 7, 3, 15, 4, 7, 7, 12, 3, 15, 3, 12, 7, 7, 7, 13, 3, 7, 7, 15, 3, 15, 3, 12, 12, 7, 3, 18, 4, 12, 7, 12, 3, 15, 7, 15, 7, 7, 3, 28, 3, 7, 12, 8, 7, 15, 3, 12, 7, 15, 3, 28, 3, 7, 12, 12, 7, 15, 3, 18, 6, 7, 3, 28, 7, 7, 7, 15, 3

Offset: 1

Amarnath Murthy, Jun 13 2001

nonn

a(1) = 1, a(p) = 3 for p = primes (A000040), a(pq) = 7 for pq = product of two distinct primes (A006881), a(pq...z) = 2^(k+1)-1 = A000225(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = sigma(k+1) = A000203(k+1) for p^k = prime powers (A000961(n) for n > 1). Sequence {1,3,4,12} is finite sequence of numbers n such that sigma(tau(n)) = n. [Jaroslav Krizek, Jul 16 2009]
For semiprime n, a(n) is either 4 or 7. Also a(n) = d(n) + omega(n) + mu(n), the sum of three core sequences A000005, A001221 and A008683. When n is semiprime, a(n) is completely defined by the Mobius function as: a(n) = 4 + 3*mu(n). a(n) also has the fractal-like identities a(d(n)) = d(n) and a(n) = sigma(a(d(n))). - Wesley Ivan Hurt, Sep 02 2013
If n is a triprime (A014612), d(n) is 4, 6, or 8 and a(n) = sigma(d(n)) is 7, 12, or 15 respectively. Then a(n) = -d(n)^2/4 + 5*d(n) - 9. - Wesley Ivan Hurt, Sep 08 2013

			sigma(d(12)) = sigma(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A001221, A008683, A062068.

A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
seq (A062069(n), n=1..40); # Jani Melik, Jan 25 2011

Table[DivisorSigma[1, DivisorSigma[0, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

v=[]; for(n=1,150,v=concat(v, sigma(numdiv(n)))); v

{ for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013

More terms from Jason Earls, Jun 19 2001

A182853 Squarefree composite integers and powers of squarefree composite integers.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Matthew Vandermast, Jan 04 2011

Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Numbers such that A001221(n) > 1 and A071625(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 3. See also A182854, A182855.
Subsequence of A072774 and A182851.
Cf. A120944.

isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
isok(n) = isoka(n) || (ispower(n,,&k) && isoka(k)); \\ Michel Marcus, Jun 24 2017

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A182853(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    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 19 2024

(define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 05 2016

A228299 Composite squarefree numbers n such that p+d(n) divides n+d(n), where p are the prime factors of n and d(n) the number of divisors of n.

Original entry on oeis.org

21098, 134930, 343027, 361730, 387127, 751394, 793595, 1344517, 1430449, 1579394, 1794854, 3542797, 5022254, 7930117, 9241627, 12122947, 21089129, 21928717, 49825117, 70233329, 78795074, 90079589, 95208734, 110995807, 124648303, 124964219, 144871634

Offset: 1

Paolo P. Lava, Aug 20 2013

nonn

Subsequence of A120944.

			Prime factors of 21098 are 2, 7, 11 and 137 while d(21098) = 16. We have that 21098 + 16 = 21114 and 21114 / (2 + 16) =  1173, 21114 / (7 + 16) = 918, 21114 / (11 + 16) = 782 and 21114 / (137 + 16) = 138.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000005, A228300-A228302.

with (numtheory); P:=proc(q) local a,i,ok,n;
for n from 1 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n+tau(n))/(a[i][1]+tau(n)),integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^6);

More terms from Michel Marcus, Sep 21 2013

Deleted first term from Paolo P. Lava, Sep 23 2013

A228302 Composite squarefree numbers n such that p+d(n) divides n-d(n) for all prime factors p of n, where d(n) is the number of divisors of n.

Original entry on oeis.org

4958, 51653, 55583, 1251574, 4909102, 5430797, 5785073, 6096931, 13892243, 14058781, 14809517, 16699426, 27391073, 32426566, 32673383, 38669686, 43459682, 44762461, 53638783, 69836866, 74975761, 75226313, 85607461, 96973703, 105139141, 122864065

Offset: 1

Paolo P. Lava, Aug 20 2013

nonn

Subsequence of A120944.

			Prime factors of 51653 are 7, 47 and 157 while d(51653) = 8. We have that 51653 - 8 = 51645 and 51645 / (7 + 8) =  3443, 51645 / (47 + 8) = 939 and 51645 / (157 + 8) = 313.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000005, A228299-A228301.

with(numtheory);  P:=proc(q) local a,i,ok,p,n;
for n from 1 to q do if not isprime(n) and issqrfree(n) then
a:=ifactors(n)[2]; ok:=1; for i from 1 to nops(a) do
if not type((n-tau(n))/(a[i][1]+tau(n)),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^9);

More terms from Michel Marcus, Sep 21 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A337030 a(n) is the number of squarefree composite numbers < prime(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 4, 6, 7, 8, 11, 13, 14, 15, 16, 19, 19, 22, 24, 24, 27, 28, 31, 35, 35, 36, 38, 38, 40, 46, 48, 50, 51, 56, 56, 58, 61, 63, 64, 67, 67, 73, 73, 75, 75, 82, 90, 91, 91, 93, 96, 96, 99, 102, 105, 108, 108, 110, 111, 112, 117, 124, 126, 126, 127

Offset: 1

Hugo Pfoertner, Aug 11 2020

nonn

			a(1) = a(2) = a(3) = 0 because the only composite number < 5 is the square 4.
a(4) = 1: 6 is the first squarefree composite number < prime(4) = 7.

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

Cf. A061398, A120944.

m=0;pp=0;forprime(p=2,320,forcomposite(c=pp,p,if(issquarefree(c),m++));print1(m,", ");pp=p)

a(1) = 0; a(n+1) = a(n) + A061398(n-1) for n>1.

A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			42 is in the sequence because 42 = 2*3*7 (3 distinct prime factors).
81 is in the sequence because 81 = 3^4 (4 prime factors, 1 distinct).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Complement of A126706.
Union of A005117 and A246547.
Union of A000469 and A246655.
Union of A000961 and A120944.
Cf. A025475.

Select[Range[110], PrimePowerQ[#] || SquareFreeQ[#] &]
Select[Range[110], PrimeNu[#] == 1 || PrimeNu[#] == PrimeOmega[#] &]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A303554(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

cp's OEIS Frontend

A000469 1 together with products of 2 or more distinct primes.

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A130092 Numbers with at least two factors having in their canonical prime factorization equal exponents.

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A360519 Let C consist of 1 together with all numbers with at least two distinct prime factors; this is the lexicographically earliest infinite sequence {a(n)} of distinct elements of C such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2).

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A010553 a(n) = tau(tau(n)).

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A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

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A182853 Squarefree composite integers and powers of squarefree composite integers.

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A228299 Composite squarefree numbers n such that p+d(n) divides n+d(n), where p are the prime factors of n and d(n) the number of divisors of n.

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