A033273 -id:A033273 - cp's OEIS frontend

A055212 Number of composite divisors of n.

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

Offset: 1

Leroy Quet, Jun 23 2000

Trivially, there is only one run of three consecutive 0's. However, there are infinitely many runs of three consecutive 1's and they are at positions A056809(n), A086005(n), and A115393(n) for n >= 1. - Timothy L. Tiffin, Jun 21 2021

			a[20] = 3 because the composite divisors of 20 are 4, 10, 20.

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

Complement of A083399.
Cf. A000005, A001221, A008578 (indices of 0's), A033273, A056809, A086005, A115393, A137944, A137945, A344713.
Cf. A001620, A077761.

a055212 = subtract 1 . a033273  -- Reinhard Zumkeller, Sep 15 2015

Table[ Count[ PrimeQ[ Divisors[n] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n],?CompositeQ],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 09 2018 *)
a[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

a(n) = numdiv(n) - omega(n) - 1; \\ Michel Marcus, Oct 17 2015

a(n) = A033273(n) - 1.
a(n) = tau(n)-omega(n)-1, where tau=A000005 and omega=A001221. - Reinhard Zumkeller, Jun 13 2003
G.f.: -x/(1 - x) + Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 2 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Dec 07 2023

A055079 Smallest number with exactly n nonprime divisors.

Original entry on oeis.org

1, 4, 8, 12, 30, 24, 36, 48, 60, 72, 2048, 192, 120, 216, 180, 288, 240, 432, 576, 420, 360, 864, 1296, 900, 960, 1728, 720, 840, 1080, 3456, 9216, 1260, 1440, 6912, 34359738368, 1680, 2160, 10368, 2880, 15552, 15360, 3600, 4620, 2520, 4320, 31104

Offset: 1

Labos Elemer, Jun 13 2000

a(n)<=2^n; see A057838 for the indices n where a(n)=2^n.

			a(5) = 30 because it is the first integer which has five nonprime divisors (1, 6, 10, 15 and 30; the divisors 2, 3 and 5 are prime).
a(35) = 2^35 = 34359738368.
a(71) = 2^71 = 2361183241434822606848.
a(191) = 2^191 = 3138550867693340381917894711603833208051177722232017256448.

Ray Chandler, Table of n, a(n) for n=1..4438

Cf. A001221, A001222, A058060, A058061, A000005, A033273, A057838.

a055079 n = head [x | x <- [1..], a033273 x == n]
-- Reinhard Zumkeller, Dec 16 2013

a = Table[0, {100} ]; Do[ c = Count[ PrimeQ[ Divisors[ n ] ], False]; If[ c < 101 && a[[ c ]] == 0, a[[ c ]] = n], {n, 2, 10077696} ];
Table[SelectFirst[Table[{n,Count[Divisors[n],?(!PrimeQ[#]&)]},{n,10000}],#[[2]]==k&],{k,34}][[;;,1]] (* The program generates the first 34 terms of the sequence. *) (* _Harvey P. Dale, Mar 04 2024 *)

sme(n) = {k = 1; while (sumdiv(k, d, ! isprime(d)) != n, k++); k;} \\ Michel Marcus, Dec 13 2013

a(n)=Min{k; A000005(k)-A001221(k)=A033273(k)=n}

More terms from Robert G. Wilson v, Nov 20 2000

Edited by Ray Chandler, Aug 12 2010

A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

Original entry on oeis.org

0, 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, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6, 5, 2, 1, 10, 1, 5, 2, 7, 1, 5, 2, 4, 4, 2, 2, 13

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Because for all d|n, dA003415(d) < A003415(n), it follows that the terms here are one less than in A319686.

Differs from A033273(n) = A000005(n) - A001221(n) at n = 1, 112, 156, 224, 280, 312, 336, 342, 380, 448, 468, 525, 560, 608, 624, 660, 672, 684, 756, 760, 780, 784, 840, 870, 896, 936, 984, 1008, ...

			The proper divisors of 112 are [1, 2, 4, 7, 8, 14, 16, 28, 56]. Applying arithmetic derivative A003415 to these, we obtain values [0, 1, 4, 1, 12, 9, 32, 32, 92], of which only 7 are distinct: 0, 1, 4, 9, 12, 32, and 92, thus a(112) = 7.

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

One less than A319686.
Cf. A003415.
Cf. also A304793, A305611, A316555, A316556, A319695 for similarly constructed sequences.

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Most[Divisors[n]]]; Array[a, 100] (* Amiram Eldar, Apr 17 2024 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319685(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((dA003415(d)), mapput(m,s,s); k++)); (k); };

a(n) = A319686(n)-1.

A294894 Number of divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

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

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=25, with divisors [1, 5, 25], both B(5,x) and B(25,x) are irreducible, so only 1 is counted and a(25)=1.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294891, A294892, A294893.
Cf. also A294884, A294904.
Differs from A033273 for the first time at n=25.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294894(n) = sumdiv(n,d,(0==A283991(d)));

a(n) = Sum_{d|n} (1-A283991(d)).
a(n) + A294893(n) = A000005(n).
a(n) = 1 + A294892(n) - A283991(n).

A059992 Numbers with an increasing number of nonprime divisors.

Original entry on oeis.org

1, 4, 8, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 75600, 83160, 110880, 151200, 166320, 221760, 277200, 332640

Offset: 1

Robert G. Wilson v, Mar 08 2001

nonn

Positions of records in A033273.
From Michael De Vlieger, Jan 04 2025: (Start)
Conjecture: This sequence includes all highly composite numbers (from A002182) except 2 and 6, but there are other terms in this sequence (e.g., a(3) = 8, a(9) = 72) that are not highly composite.
Conjecture: a(n)/A007947(a(n)) is in A301414. (End)

			a(4)=12 because twelve has 4 nonprime divisors {1, 4, 6 and 12} whereas a(3)=8 has only 3; and twelve is the first number greater than eight which exhibits this property.

Amiram Eldar, Table of n, a(n) for n = 1..571 (terms 1..146 from Ray Chandler)
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.

Cf. A055079, A002182, A033273, A180040.

l = 0; Do[ c = Count[PrimeQ[ Divisors[n] ], False]; If[c > l, l = c; Print[n] ], {n, 1, 10^6} ]

lista(nn) = {my(m=0, nb); for (n=1, nn, nb = sumdiv(n, d, !isprime(d)); if (nb > m, m = nb; print1(n, ", ")););} \\ Michel Marcus, Jul 16 2019

Alternate description and b-file from Ray Chandler, Aug 07 2010

A087652 Product of the nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 32, 9, 10, 1, 288, 1, 14, 15, 512, 1, 972, 1, 800, 21, 22, 1, 55296, 25, 26, 243, 1568, 1, 27000, 1, 16384, 33, 34, 35, 1679616, 1, 38, 39, 256000, 1, 74088, 1, 3872, 6075, 46, 1, 42467328, 49, 12500, 51, 5408, 1, 1417176, 55, 702464, 57, 58

Offset: 1

Reinhard Zumkeller, Sep 25 2003

nonn

			For n = 12: nonprime divisors = {4,6,12}: a(12) = 4*6*12 = 288.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A000005, A001358, A007947, A007955, A023890, A033273.

a[n_] := n^(DivisorSigma[0, n]/2) / Times @@ FactorInteger[n][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

a(n) = my(p=1); fordiv(n, d, if (!isprime(d), p*=d)); p; \\ Michel Marcus, Aug 05 2017

a(n) = 1 if n = 1 or n is prime.
a(n) = n if n = 1 or n is semiprime (A001358).
From Wesley Ivan Hurt, Jun 08 2020: (Start)
a(n) = Product_{d|n, d nonprime} d.
If n is squarefree, then a(n) = n^(d(n)/2-1), where d(n) is the number of divisors of n (A000005). (End)
a(p^e) = p^((e^2+e-2)/2) for p prime, e > 0. - Bernard Schott, Jun 08 2020
a(n) = A007955(n)/A007947(n). - Amiram Eldar, Feb 01 2025

A081707 a(n) = tau(n) - bigomega(n) = A000005(n) - A001222(n).

Original entry on oeis.org

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

Offset: 1

Vladeta Jovovic, Apr 03 2003

nonn

Number of divisors of n that are not positive powers of primes (cf. A000961). - Benoit Cloitre, May 03 2003; corrected Dec 16 2008 at the suggestion of Ray Chandler.
a(n) = 1 iff n is in A000961. - Robert Israel, Nov 23 2015
a(n) = 2 iff n is in A006881. - Altug Alkan, Nov 23 2015
a(n) = 3 iff n is in A054753. - Michel Marcus, Nov 24 2015

			After first statement in comment section, a(60) = 8 because we have: 1,6,10,12,15,20,30,60. The divisors 2,3,4,5 are excluded from the count. - _Geoffrey Critzer_, Nov 22 2015

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

Cf. A033273(n) = tau(n) - omega(n) = A000005(n) - A001221(n).

Cf. A000961.

seq(numtheory:-tau(n)-numtheory:-bigomega(n), n=1..300); # Robert Israel, Nov 23 2015

Table[DivisorSigma[0, n] - PrimeOmega[n], {n, 1, 105}] (* Geoffrey Critzer, Nov 22 2015 *)

first(m)=vector(m,n,numdiv(n) - bigomega(n)) \\ Anders Hellström, Nov 22 2015

A087802 a(n) = Sum_{d|n, d nonprime} mu(d), where mu = A008683.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3

Offset: 1

Reinhard Zumkeller, Oct 11 2003

nonn

A064372 and this sequence first differ at term 64: A064372(64)=2 and a(64)=1. - Rick L. Shepherd, Mar 07 2004

			Divisors of n=42: {1,2,3,6,7,14,21,42}, a(42) = mu(1) + mu(6) + mu(14) + mu(21) + mu(42) = 1+1+1+1-1 = 3.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001221, A008683 (mu), A023890, A033273. Different from A079553.

Table[Total[MoebiusMu[#]&/@Select[Divisors[n],!PrimeQ[#]&]],{n,120}] (* Harvey P. Dale, Oct 14 2014 *)

A087802(n) = sumdiv(n,d,if(!isprime(d),moebius(d)))

a(n) = if n=1 then 1, else A001221(n). - Vladeta Jovovic, Oct 17 2003

A327394 Number of stable divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. Maximum stable divisor is A327393.

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences related to prime indices in the factorization of n

See link for additional cross-references.
Cf. A000005, A033273, A303362, A316476, A327393.
Inverse Möbius transform of A378442.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]

A378442(n)={my(v=apply(primepi, factor(n)[, 1])); for(j=2, #v, for(i=1, j-1, if(v[j]%v[i]==0, return(0)))); 1}; \\ From the function "ok" in A316476 by Andrew Howroyd, Aug 26 2018
A327394(n) = sumdiv(n,d,A378442(d)); \\ Antti Karttunen, Nov 27 2024

a(n) = Sum_{d|n} A378442(d). - Antti Karttunen, Nov 27 2024

More terms from Antti Karttunen, Nov 27 2024

A333748 Number of nonprime divisors of n that are <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Cf. A018252, A033273, A038548, A063962, A069288, A069291, A333749, A333750, A333751.

Table[DivisorSum[n, 1 &, # <= Sqrt[n] && !PrimeQ[#] &], {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, (d^2<=n) && !isprime(d)); \\ Michel Marcus, Apr 03 2020

G.f.: Sum_{k>=1} x^(A018252(k)^2) / (1 - x^A018252(k)).

cp's OEIS Frontend

A055212 Number of composite divisors of n.

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A055079 Smallest number with exactly n nonprime divisors.

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A319685 Number of distinct values obtained when arithmetic derivative (A003415) is applied to proper divisors of n.

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A294894 Number of divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

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A059992 Numbers with an increasing number of nonprime divisors.

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A087652 Product of the nonprime divisors of n.

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A081707 a(n) = tau(n) - bigomega(n) = A000005(n) - A001222(n).

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