A058060 -id:A058060 - cp's OEIS frontend

A058061 Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.

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

Offset: 1

Labos Elemer, Nov 23 2000

From Bernard Schott, Mar 24 2020: (Start)
a(n) = 1 iff n = p^(q-1) with p, q primes (A009087).
a(n) = 2 if n=p*q with p, q primes (A006881), or if n=p^2*q with p, q primes (A054753), or if n=p^4*q with p, q primes (A178739), or if n=p^6*q with p, q primes (A189987), or if n=p^2*q^4 with p, q primes (A189988), or if n=p^(m-1) with p prime and m is semiprime in A001358 (not exhaustive). (End)

			For n=120, d(120)=16, a(120)=4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001222, A000005, A058060, A079057 (partial sums).

Table[PrimeOmega@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = bigomega(numdiv(n)); \\ Michel Marcus, Dec 14 2013

a(n) = A001222(A000005(n)).

Additive with a(p^e) = A001222(e+1). - Amiram Eldar, Jan 15 2024

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

A057838 Numbers k such that A055079(k) = 2^k.

Original entry on oeis.org

2, 3, 11, 35, 71, 191, 419, 659, 1091, 1199, 1379, 1655, 2015, 2135, 2339, 2591, 3059, 4439, 6119, 6215, 6335, 7055, 8099, 8351, 8519, 9815, 11159, 12419, 12431, 12599, 12719, 12851, 13679, 15119, 15239, 16415, 16919, 17255, 17879, 18215, 18479

Offset: 1

Labos Elemer, Nov 24 2000

nonn

			11 is a term: 2^11 has 11 nonprime divisors; c(11)=A055079(11) could not have r = 2, 3, 4 or more distinct prime divisors because 11 + {2, 3, 4, 5, 6, 7, 8, 9, ...} values of corresponding d(c(11)) = {13, 14, 15, ...} had 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2 non-distinct prime divisors, which provides an upper bound for r ... in contradiction with demanded values: 2, 3, 4, 5, 6, 7, ... This is why A055079(11)=2048. Larger cases are handled in a similar way.
a(35) = 15239 since A055079(15239) = 2^15239, which has 4588 decimal digits.
A protocol for 15239 is as follows: u=15239; t0=Table[s, {s, 0, 17}]; t1=Table[mr[w], {w, u, u+17}]; t2=t1-t0; g=Table[{w, mr[w]}, {w, u, u+17}]; i1=TimeUsed[]; Write["a(bad)tx1", u, t1, t2, g]; 15239.
Supposed number of A001221(x) which should be larger or equal than A001222(d(x)): {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}.
A001222(d(x)) {3, 6, 1, 2, 2, 4, 2, 6, 2, 5, 4, 5, 2, 5, 2, 3, 5, 4}.
A001222(d(x)) - A001221(x) (negative value means "nasty case") {3, 5, -1, -1, -2, -1, -4, -1, -6, -4, -6, -6, -10, -8, -12, -12, -11, -13} numbers (corresponding d(x) values for some x) together with A001222[d(x)] {{15239, 3}, {15240, 6}, {15241, 1}, {15242, 2}, {15243, 2}, {15244, 4}, {15245, 2}, {15246, 6}, {15247, 2}, {15248, 5}, {15249, 4}, {15250, 5}, {15251, 2}, {15252, 5}, {15253, 2}, {15254, 3}, {15255, 5}, {15256, 4}}.

Ray Chandler, Table of n, a(n) for n = 1..13925

Cf. A055079, A001221, A001222, A000005, A058060, A058061, A057841.

2^a(n) = A057841(n) = A055079(a(n)).

A001221(A055079(a(n))) = 1.

Edited, corrected and extended by Ray Chandler, Aug 14 2010

A338160 Number of ways to represent n as a product of the greatest number of distinct factors.

Original entry on oeis.org

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

Offset: 1

Vladimir Letsko, Oct 14 2020

nonn

a(n) = A058060(n) for 1 < n < 60; a(60) = 3, A058060(60) = 1.
a(n) is the number of factorizations of n into A086435(n) distinct factors > 1.
a(n) depends only on the prime signature of n.

			a(72) = 3 because 72 = 2*3*12 = 2*4*9 = 3*4*6 and 72 cannot be represented as a product of 4 distinct factors each greater than 1 (adding the factor 1 to each product doesn't change anything).

Cf. A086435, A140207, A338159.

a(n)={my(d=divisors(n)); my(F(r,k)=if(r==1, [0,1], my(b=-1,c=0); for(k=2, k, if(r%d[k]==0, my([tb,tc]=self()(r/d[k], k-1)); if(tb>b, b=tb; c=0); if(tb==b, c+=tc))); [b+1, c])); F(n, #d)[2]} \\ Andrew Howroyd, Oct 14 2020

More terms from Andrew Howroyd, Oct 14 2020

A057841 a(n) = 2^A057838(n) corresponding to extremal cases of A055079.

Original entry on oeis.org

4, 8, 2048, 34359738368, 2361183241434822606848, 3138550867693340381917894711603833208051177722232017256448

Offset: 1

Labos Elemer, Nov 24 2000

nonn

			a(7) = 2^419 which has 127 decimal digits.
a(35) = 2^15239 which has 4588 decimal digits.

Ray Chandler, Table of n, a(n) for n=1..17

Cf. A000005, A001221, A001222, A055079, A058060, A058061, A057838.

a(n) = 2^A057838(n) = A055079(A057838(n)).

Edited by Ray Chandler, Aug 14 2010

cp's OEIS Frontend

A058061 Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.

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A055079 Smallest number with exactly n nonprime divisors.

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A057838 Numbers k such that A055079(k) = 2^k.

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A338160 Number of ways to represent n as a product of the greatest number of distinct factors.

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A057841 a(n) = 2^A057838(n) corresponding to extremal cases of A055079.

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