A055079 -id:A055079 - cp's OEIS frontend

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

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

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

A033273 Number of nonprime divisors of n.

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, 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

Offset: 1

N. J. A. Sloane

nonn

Ray Chandler, Table of n, a(n) for n=1..100000

Cf. A000005, A001221, A010051, A027750.
Cf. A055079, A059992, A180040.
Cf. A001620, A077761.

a033273 = length . filter ((== 0) . a010051) . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[NumberOfDivisors(n) - #PrimeDivisors(n): n in [1..150]]; // Vincenzo Librandi, May 17 2017

Table[Length[Select[Divisors[n], ! PrimeQ[#] &]], {n, 104}] (* Jayanta Basu, May 23 2013 *)
Table[DivisorSigma[0, n] - PrimeNu[n], {n, 100}] (* Vincenzo Librandi, May 17 2017 *)
Table[Count[Divisors[n],?CompositeQ],{n,110}]+1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 11 2019 *)

a(n) = numdiv(n) - omega(n); \\ Michel Marcus, Apr 28 2016

a(n) = A000005(n) - A001221(n).
a((n)) = n and a(m) <> n for m < A055079(n). - Reinhard Zumkeller, Dec 16 2013
G.f.: Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Jan 17 2017
Dirichlet g.f.: zeta(s)*(zeta(s)-primezeta(s)). - Benedict W. J. Irwin, Jul 11 2018
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 1 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Nov 27 2022

More terms from Reinhard Zumkeller, Sep 02 2003
Corrected error in offset. - Jaroslav Krizek, May 04 2009
Extended by Ray Chandler, Aug 07 2010

A303555 Triangle read by rows: T(n,k) = 2^(n-k)*prime(k)#, 1 <= k <= n, where prime(k)# is the product of first k primes.

Original entry on oeis.org

2, 4, 6, 8, 12, 30, 16, 24, 60, 210, 32, 48, 120, 420, 2310, 64, 96, 240, 840, 4620, 30030, 128, 192, 480, 1680, 9240, 60060, 510510, 256, 384, 960, 3360, 18480, 120120, 1021020, 9699690, 512, 768, 1920, 6720, 36960, 240240, 2042040, 19399380, 223092870, 1024, 1536, 3840, 13440, 73920, 480480, 4084080, 38798760, 446185740, 6469693230

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

T(n,k) = the smallest number m having exactly n prime divisors counted with multiplicity and exactly k distinct prime divisors.

			T(5,4) = 420 = 2^2*3*5*7, hence 420 is the smallest number m such that bigomega(m) = 5 and omega(m) = 4 (see A189982).
Triangle begins:
    2;
    4,   6;
    8,  12,  30;
   16,  24,  60,  210;
   32,  48, 120,  420, 2310;
   64,  96, 240,  840, 4620, 30030;
  128, 192, 480, 1680, 9240, 60060, 510510;
  ...

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers
Index to sequences related to prime signature

Cf. A000079, A001221, A001222, A002110, A005179, A038547, A055079, A070175, A303557 (central terms).

Flatten[Table[2^(n - k) Product[Prime[j], {j, k}], {n, 10}, {k, n}]]

A061283 Smallest number with exactly 2n-1 divisors.

Original entry on oeis.org

1, 4, 16, 64, 36, 1024, 4096, 144, 65536, 262144, 576, 4194304, 1296, 900, 268435456, 1073741824, 9216, 5184, 68719476736, 36864, 1099511627776, 4398046511104, 3600, 70368744177664, 46656, 589824, 4503599627370496, 82944

Offset: 1

Labos Elemer, May 22 2001

nonn

The terms are always squares (because the divisors of a nonsquare N come in pairs, d and N/d, and so their number is always even - N. J. A. Sloane, Dec 26 2018).

			For n=15, a(15)=144 with 15 divisors: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.

Amiram Eldar, Table of n, a(n) for n = 1..1661 (terms 1..1000 from T. D. Noe)

Cf. A000005, A000290, A005408, A003680, A016017, A037992, A055079, A048691.

Second bisection of A005179.

mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n-1] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)

a(n) = Min{k | A000005(k)=2n-1}.
a((p+1)/2) = 2^(p-1) for odd prime p. [Corrected by Jianing Song, Aug 30 2021]
From Jianing Song, Aug 30 2021: (Start)
a(n) = A016017(n)^2.
a(n) <= 2^(2n-2), where the equality holds if and only if n=1 or 2n-1 is prime. (End)

A016017 Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways.

Original entry on oeis.org

1, 2, 4, 8, 6, 32, 64, 12, 256, 512, 24, 2048, 36, 30, 16384, 32768, 96, 72, 262144, 192, 1048576, 2097152, 60, 8388608, 216, 768, 67108864, 288, 1536, 536870912, 1073741824, 120, 576, 8589934592, 6144, 34359738368, 68719476736, 180, 864

Offset: 1

Robert G. Wilson v

nonn

From Jianing Song, Aug 30 2021: (Start)
a(n) is the smallest number whose square has exactly 2n-1 divisors.
a(n) is the earliest occurrence of 2n-1 in A048691. (End)

			a(1)=1 and a(2)=2 because 1/2 = 1/3 + 1/6 = 1/4 + 1/4.
a(3)=4 because 1/4 = 1/5 + 1/20 = 1/6 + 1/12 = 1/8 + 1/8.
a(4)=8 because 1/8 = 1/9 + 1/72 = 1/10 + 1/40 = 1/12 + 1/24 = 1/16 + 1/16.
a(5)=6 because 1/6 = 1/7 + 1/42 = 1/8 + 1/24 = 1/9 + 1/18 = 1/10 + 1/15 = 1/12 + 1/12.

David W. Wilson, Table of n, a(n) for n = 1..1000 (terms 122, 365 and 608 corrected by _Amiram Eldar_, Nov 08 2024)

Identical to A071571 shifted right.

Cf. A000005, A000290, A005408, A005179, A003680, A037992, A055079, A048691.

f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; t = Table[0, {50}]; Do[a = f[2, n]; If[a < 51 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 2^30}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {k = 1; while (numdiv(k^2) != (2*n-1), k++); return (k); }; \\ Amiram Eldar, Jan 07 2019 after Michel Marcus at A071571

a(n+1) <= 2^n.
From Labos Elemer, May 22 2001: (Start)
a(n) = sqrt(A061283(n)).
a(n) = sqrt(Min{k| A000005(k)=2n-1}).
a((p+1)/2) = 2^((p-1)/2) = 2^A005097(i) if p is the i-th odd prime. [Corrected by Jianing Song, Aug 30 2021] (End)
a(n) is the least k such that (tau(k^2) + 1)/2 = n. - Vladeta Jovovic, Aug 01 2001

Entry revised by N. J. A. Sloane, Aug 14 2005

Offset corrected by David W. Wilson, Dec 27 2018

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

A287661 Smallest odd number with exactly n nonprime divisors.

Original entry on oeis.org

1, 9, 27, 45, 105, 135, 225, 405, 315, 675, 177147, 1155, 945, 3375, 1575, 6075, 2835, 10125, 18225, 3465, 4725, 30375, 50625, 11025, 25515, 91125, 14175, 10395, 23625, 273375, 1476225, 17325, 33075, 759375, 50031545098999707, 31185, 70875, 1366875, 127575

Offset: 1

Ilya Gutkovskiy, May 29 2017

nonn

			a(5) = 105 because 105 has 8 divisors {1, 3, 5, 7, 15, 21, 35, 105} among which 5 are nonprime {1, 15, 21, 35, 105} and 105 is the smallest odd with exactly 5 nonprime divisors.

Giovanni Resta, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor

Cf. A001221, A002110, A003680, A005179, A033273, A038547, A055079, A070826.

A033273(a(n)) = n.

a(35)-a(39) from Giovanni Resta, May 31 2017

cp's OEIS Frontend

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

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

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A033273 Number of nonprime divisors of n.

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A303555 Triangle read by rows: T(n,k) = 2^(n-k)*prime(k)#, 1 <= k <= n, where prime(k)# is the product of first k primes.

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Mathematica

A061283 Smallest number with exactly 2n-1 divisors.

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Mathematica

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A016017 Smallest k such that 1/k can be written as a sum of exactly 2 unit fractions in n ways.

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Mathematica

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A059992 Numbers with an increasing number of nonprime divisors.

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Mathematica

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A287661 Smallest odd number with exactly n nonprime divisors.

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