A229099 -id:A229099 - cp's OEIS frontend

A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A322448 and first differs from it at n = 138: A322448(138) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k composite} (1/zeta(k+1) - 1/zeta(k)) = 0.05296279266796920306... . The asymptotic density of this sequence within the nonsquarefree numbers (A013929) is d / (1 - 1/zeta(2)) = 0.13508404411123191108... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002808, A013661, A051903, A229099.
Complement of A074661 within A013929.
Subsequence of A322448 and A322449 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374589, A374590.

filter:= proc(n) local m;
  m:= max(ifactors(n)[2][..,2]);
  m > 1 and not isprime(m)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 14 2024

Select[Range[1200], CompositeQ[Max[FactorInteger[#][[;; , 2]]]] &]

iscomposite(n) = n > 1 && !isprime(n);
is(n) = n > 1 && iscomposite(vecmax(factor(n)[, 2]));

A368541 The number of exponential divisors of the nonsquarefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 29 2023

nonn

The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree (A005117).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A013929, A049419.
Cf. A084936, A174961, A275699, A368038, A368039, A368040.
Cf. A013661, A059956, A229099, A327837.

f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, numdiv(f[i, 2])); if(p > 1, print1(p, ", ")));}

a(n) = A049419(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (A327837 - A059956)/A229099 = 2.53623753427906735929... .

A381391 Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).

Original entry on oeis.org

0, 29, 367, 3866, 39098, 391838, 3920154, 39205902, 392069187, 3920718974, 39207261564, 392072817656, 3920728751139, 39207289143932, 392072896183208, 3920728975677128, 39207289797472001, 392072898095046811, 3920728981307675534, 39207289814141997459, 392072898144605471040

Offset: 1

Michael De Vlieger, Feb 22 2025

nonn

			Let S = A126706.
a(1) = 0 since the smallest term in S is 12.
a(2) = 29 since S(1..29) = {12, 18, 20, 24, ..., 99, 100}, etc.

Cf. A011557, A071172, A126706, A267574, A372403, A229099, A380403.

Table[10^n - Sum[PrimePi@ Floor[10^(n/k)], {k, 2, Floor[Log2[10^n]]}] - Sum[MoebiusMu[k]*Floor[10^n/(k^2)], {k, Floor[Sqrt[10^n]]}], {n, 10}]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381391(n):
    m = 10**n
    return int(-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))-sum(mobius(k)*(m//k**2) for k in range(2, isqrt(m)+1))) # Chai Wah Wu, Feb 23 2025

a(n) = 10^n - Sum_{k = 2..log_2(10^n)} pi(floor(10^(n/k))) - Sum_{k = 1..floor(sqrt(10^n))} mu(k)*floor(10^n/k^2), where pi = A000720 and mu = A008683.

a(n) = A011557(n) - A071172(n) - A267574(n).

A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

Original entry on oeis.org

1, 2, 8, 12, 14, 16, 25, 39, 42, 44, 46, 49, 51, 53, 59, 70, 73, 78, 81, 83, 85, 86, 101, 103, 105, 116, 118, 119, 126, 130, 135, 137, 139, 142, 144, 147, 148, 158, 161, 163, 170, 171, 178, 181, 186, 188, 190, 192, 194, 195, 204, 207, 209, 212, 216, 219, 224, 229

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity. A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.66565771136..., where x=(1/zeta(2)) and m=floor(log(1-x)/log(x))=1.

See A077468 for Mathematica program by Robert G. Wilson v.

			a(3)=8 since (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^8 < 1 and (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^k > 1 for 2<k<8.

Cf. A077468, A080057, A080059, A229099.

a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(1/zeta(2)) and frac(y) = y - floor(y).

A191102 Decimal expansion of (1/3)*arccos(6/Pi^2-1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 12 2011

			.65789338396252593180869437710606678919613274849811661638...

M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227.

Cf. A192930, A229099.

RealDigits[(ArcCos[6/Pi^2-1])/3,10,120][[1]] (* Harvey P. Dale, Nov 29 2015 *)

acos(6/Pi^2-1)/3 \\ Michel Marcus, Sep 19 2017

A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

Original entry on oeis.org

2, 6, 3, 6, 14, 6, 10, 18, 5, 12, 14, 30, 36, 30, 22, 15, 42, 7, 10, 26, 24, 42, 30, 21, 62, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13

Offset: 1

Amiram Eldar, May 21 2024

A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.
The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.

Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }

from math import prod, isqrt
from sympy import mobius, factorint
def A373058(n):
    def f(x): return n+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 prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # Chai Wah Wu, Jul 22 2024

a(n) = A336563(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .

A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 15 2016

			0.1742197830347247005585740721805346916511057518703135572332637051646...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 15.
Randell Heyman, Pairwise non-coprimality of triples, arXiv preprint, arXiv:1309.5578 [math.NT], 2013-2014.
Randell Heyman, Topics in divisibility: pairwise coprimality, the GCD of shifted sets and polynomial irreducibility, PhD thesis, University of New South Wales, 2015.
Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
Eric Weisstein's MathWorld, Carefree Couple.

Cf. A065464, A065473, A229099.

$MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
P = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
Q = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
F3 = 1 - 18/Pi^2 + 3P - Q;
RealDigits[F3, 10, digits][[1]]

1 - 3/zeta(2) + 3 * prodeulerrat(1 - (2*p-1)/p^3) - prodeulerrat(1 - (3*p-2)/p^3) \\ Amiram Eldar, Mar 03 2021

Equals 1 - 18/Pi^2 + 3P - Q, where P is A065464 and Q is A065473.

A372634 Numerators of the reduced fraction of all rational number representations (a/b, with a and b being integers) which can themselves be reduced by at least one common prime factor of at most prime(n).

Original entry on oeis.org

1, 1, 9, 457, 11213, 273347, 79439651, 5761023199, 277886746829, 33449007905699, 32197332748181219, 2322572170370125769, 3907895853135787075289, 657439531892484346088851, 63187594618979703535273733, 13660992716321028635960170769

Offset: 1

Brian Lee Burtner, May 08 2024

The numerators of the fraction F(n) = a(n)/A072044(n) may be generated directly by use of inclusion-exclusion; e.g., 1/4 + 1/9 + 1/25 - 1/225 - 1/100 - 1/36 + 1/900 = 9/25.
Following Euler, they may also be generated via products.
a(n)/A072044(n) -> 1 - 6/Pi^2 (provable via Euler, see references).  This value is the supremal proportion of all rational number representations a/b that are reducible by some common factor (or, more broadly: the proportion of all pairs of integers a,b that are not coprime).
A072045(n)/A072044(n) gives the complementary proportion of all rational number representations that are irreducible by any prime factor of at most A000040(n).  This analogously converges to 6/Pi^2, the infimal proportion of all rational number representations a/b that are simply irreducible.

			For n=3, 1 - (3/4)*(8/9)*(24/25) = 9/25.
Exactly 9/25 of all rational number representations are reducible by at least one prime factor of at most 5.

Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.

Leonhard Euler, Introductio in Analysin Infinitorum, Vol 1.
Wikipedia, Probability of coprimality.

Cf. A072044, A072045, A229099.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*(1-1/ithprime(n)^2)) end:
a:= n-> numer(1-b(n)):
seq(a(n), n=1..16);  # Alois P. Heinz, May 11 2024

a[n_]:=Numerator[1-Product[1-1/Prime[k]^2,{k,n}]]; Array[a,16] (* Stefano Spezia, May 11 2024 *)

a(n) = numerator(1 - prod(k=1, n, (prime(k)^2-1)/prime(k)^2)); \\ Michel Marcus, May 08 2024

a(n) = numerator(1 - Product_{k=1..n} (1 - 1/prime(k)^2)).

cp's OEIS Frontend

A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

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A368541 The number of exponential divisors of the nonsquarefree numbers.

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A381391 Number of k <= 10^n that are neither squarefree nor prime powers (i.e., k is in A126706).

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A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

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A191102 Decimal expansion of (1/3)*arccos(6/Pi^2-1).

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A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

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A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.

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A372634 Numerators of the reduced fraction of all rational number representations (a/b, with a and b being integers) which can themselves be reduced by at least one common prime factor of at most prime(n).

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