A322448 -id:A322448 - cp's OEIS frontend

A366992 The sum of divisors of n that are not terms of A322448.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 47, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 47, 84, 144

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365682 at n = 64.
The sum of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and the largest of them is A366994(n).

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

Cf. A000203, A046100, A365682, A366991, A366994.

f[p_, e_] := 1 + p + Total[p^Select[Range[e], PrimeQ]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1] + sum(j = 1, f[i, 2], if(isprime(j), f[i, 1]^j)));}

Multiplicative with a(p^e) = 1 + p + Sum_{primes q <= e} p^q.
a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.77864544487983775708..., where f(x) = (1-x) * (1 + Sum_{k>=1} (1 + 1/x + Sum_{primes q <= k} 1/x^q) * x^(2*k)).

A366991 The number of divisors of n that are not terms of A322448.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365680 at n = 64.
The number of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The sum of these divisors is A366992(n) and the largest of them is A366994(n).

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

Cf. A000005, A000720, A046100, A365680, A366989, A366992, A366994.

f[p_, e_] := PrimePi[e] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, primepi(f[i, 2]) + 2);}

Multiplicative with a(p^e) = A000720(e) + 2.

a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

A366994 The largest divisor of n that is not a term of A322448.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365683 at n = 64.
The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and their sum is A366992(n).

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

Cf. A007917, A322448, A365683, A366989, A366991, A366992.

f[p_, e_] := p^If[e == 1, 1, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}

Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.
a(n) <= n, with equality if and only if n is not in A322448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.

A322449 Numbers whose prime factorization contains only composite exponents.

Original entry on oeis.org

1, 16, 64, 81, 256, 512, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 19683, 20736, 28561, 32768, 38416, 40000, 41472, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256

Offset: 1

Alois P. Heinz, Dec 08 2018

nonn

Differs from A117453 first at n = 13: a(13) = 5184 = 2^6 * 3^4, A117453(13) = 6561 = 3^8.

			5184 = 2^6 * 3^4 is a term because all exponents are composite numbers.
1 is a term, because it has no prime factorization, and "the empty set has every property". - _N. J. A. Sloane_, Aug 25 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002808, A117453, A322448.

Join[{1},Select[Range[250000],AllTrue[FactorInteger[#][[;;,2]],CompositeQ]&]] (* Harvey P. Dale, Aug 25 2024 *)

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k in A002808} 1/p^k) = 1.1028952548... . - Amiram Eldar, Jul 02 2022

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

Original entry on oeis.org

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]));

A339328 Integers m such that A240471(m) > A115588(m).

Original entry on oeis.org

16, 24, 27, 28, 32, 36, 44, 48, 50, 52, 54, 55, 60, 64, 65, 68, 72, 76, 77, 80, 81, 84, 85, 90, 91, 92, 95, 96, 98, 100, 105, 108, 110, 112, 115, 116, 119, 120, 124, 125, 126, 128, 130, 132, 133, 135, 136, 140, 143, 144, 145, 148, 150, 152, 154, 155, 156, 160

Offset: 1

Thomas Scheuerle, Nov 30 2020

nonn

Integers m such that integer part of the harmonic mean of divisors of m is greater than the number of distinct prime numbers necessary to represent m.
For all m not in this sequence this integer part is equal to the number of distinct prime numbers necessary to represent m.
This correlation between A240471 and A115588 contains some apparently random component.
If the integer part of the harmonic mean of divisors of m equals 1 we will find an 1 in A115588(m) too, for all m. If the integer part of the harmonic mean of divisors of m equals 2 we will find 2 in A115588(m) too, with probability of ~0.9877 for m in range 2-1000.
For m until 10000 the only exceptions are 16 and 27. If the integer part of the harmonic mean of divisors of m equals 3 we will find 3 in A115588(m) too, with probability of ~0.1983 for m in range 2-1000. For integer parts greater than 3 the probability gets fast smaller.
If m is a square of a prime it is not in this sequence.
Let m be a semiprime with two distinct prime factors p1 and p2. If m >= 3(1+p1+p2) then m is in this sequence. Example: 55 > 3(1+5+11). This can be generalized for k-almostprimes if all factors are distinct: If m(2^k) >= (1+k)sigma(m) then m is in this sequence. Example: 105*8 > 4*192.
Let p be a prime greater than 2. Let o be a natural number >0 without divisor p, then if m = o*p^p, m is in this sequence. This can be generalized for a set of distinct primes >2 {p_1,p_2,...,p_n} and any permutation of this set {p_a,p_b,...,p_z}, then if m = o*p_1^p_a*p_2^p_b*...*p_n^p_z, m is in this sequence. Example: 3960 = 55*2^3*3^2.
The sequence includes all numbers whose prime factorization contains at least one composite exponent (A322448).

Cf. A240471, A115588, A322448.

listf(f, list) = {for (k=1, #f~, listput(list, f[k,1]); if (isprime(f[k,2]), listput(list, f[k,2]), if (f[k,2] > 1, my(vexp = Vec(listf(factor(f[k,2]), list))); for (i=1, #vexp, listput(list, vexp[i]););););); list;}
a8(n) = {my(f=factor(n), list=List()); #select(isprime, Set(Vec(listf(f, list))));}
a1(n) = n*numdiv(n)\sigma(n);
isok(m) = a1(m) > a8(m); \\ Michel Marcus, Dec 02 2020

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A366992 The sum of divisors of n that are not terms of A322448.

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A366991 The number of divisors of n that are not terms of A322448.

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A366994 The largest divisor of n that is not a term of A322448.

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A322449 Numbers whose prime factorization contains only composite exponents.

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

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A339328 Integers m such that A240471(m) > A115588(m).

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