A052485 -id:A052485 - cp's OEIS frontend

A331801 Integers that are sum of two nonsquarefree numbers.

8, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Bernard Schott, Jan 26 2020

nonn

Proposition: All integers > 23 are terms of this sequence (see link Prime Curios!).
Proof by exhaustion:
1) For numbers {4*k} with k>=6, then 4*k = 4*(k-1) + 4 is a term as 4*(k-1) and 4 are nonsquarefree;
2) For numbers {4*k+1} with k>=6, then 4*k+1 = 4*(k-2) + 9 is a term as 4*(k-2) and 9 are nonsquarefree;
3) For numbers {4*k+2} with k>=6, then 4*k+2 = 4*(k-4) + 18 is a term as 4*(k-4) and 18 are nonsquarefree;
4) For numbers {4*k+3}; with k=6, 27 = 9+18 is a term as 9 and 18 are nonsquarefree, and with k>=7, 4*k+3 = 4*(k-6) + 27 is also a term as 4*(k-6) and 27 are nonsquarefree.
Conclusion: every integer > 23 is sum of two nonsquarefree numbers (QED).

			13 = 4 + 9 and 21 = 9 + 12 are terms of this sequence as 4, 9 and 12 are nonsquarefree numbers.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 23 (Rupinski)

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331802 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

max = 85; Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &] (* Amiram Eldar, Feb 04 2020 *)
Join[{8,12,13,16,17,18,20,21,22},Range[24,100]] (* or *) Complement[Range[100],{1,2,3,4,5,6,7,9,10,11,14,15,19,23}] (* Harvey P. Dale, Dec 04 2024 *)

isok(m) = {for (i=1, m-1, if (!issquarefree(i) && !issquarefree(m-i), return (1));); return(0);} \\ Michel Marcus, Jan 31 2020

A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 176

Offset: 1

Amiram Eldar, Mar 29 2024

Subsequence of A283050 and first differs from it at n = 100: A283050(100) = 300 = 2^2 * 3 * 5^2 is not a term of this sequence.
Powerful numbers and nonpowerful numbers k such that 1 < A249740(k) < A277698(k), or equivalently, 1 < A006530(A057521(k)) < A020639(A055231(k)).
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} f(p)/(p^2-p+1) = 0.32131800923..., where f(p) = Product_{primes q <= p} (q^2-q+1)/(q^2-1).

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

Subsequence of A013929 and A283050.

Cf. A001694, A006530, A020639, A052485, A055231, A057521, A059956, A249740, A277698.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[e] > 1 && (Min[e] > 1 || Max[e[[FirstPosition[e, 1][[1]] ;; -1]]] == 1)]; Select[Range[200], q]

is(n) = {my(e = apply(x->if(x > 1, 2, 1), factor(n)[,2])); n > 1 && vecmax(e) > 1 && vecsort(e, , 4) == e;}

A378766 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is powerful (in A001694).

Original entry on oeis.org

1, 3, 4, 8, 6, 9, 10, 16, 25, 27, 12, 32, 14, 36, 17, 49, 64, 19, 72, 21, 81, 23, 100, 26, 108, 121, 125, 29, 128, 31, 144, 169, 34, 196, 37, 200, 216, 39, 225, 41, 243, 43, 256, 45, 288, 47, 289, 50, 324, 343, 52, 361, 54, 392, 56, 400, 58, 432, 60, 441, 62, 484

Offset: 1

Michael De Vlieger, Dec 18 2024

The sequence is a list of indices m of powerful numbers a(m).

See comments in A379051 for more information.

			a(1) = 1 since 1 is powerful, validating the appearance of 1 as an index of a powerful number in the sequence.
a(2) = 3 since self-referential 2 would prove false; 2 is not powerful, but 3 mandates a powerful number a(3).
a(3) = 4 since a(2) = 3, and 4 is the smallest powerful number that has not appeared.
a(4) = 8 since a(3) = 4, and 8 is the smallest powerful number that has not appeared.
a(5) = 6 since m = 5 has not appeared, and 6 is the smallest weak (nonpowerful, in A052485) number k > n.
a(6) = 9 since a(5) = 6, and 9 is the smallest powerful number that has not appeared, etc.

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

Cf. A001694, A052485, A121053, A379051, A379053.

nn = 120;
u = 3; v = {}; w = {}; c = 1;
s = Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^30];
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
{1}~Join~Reap[Do[
  If[MemberQ[w, n], k = s[[c]];
    w = DeleteCases[w, n],
    m = Min[{s[[c]], u, v}];
    If[And[Divisible[m, rad[m]^2], CompositeQ[m], n < m],
      AppendTo[v, n]];
      If[Length[v] > 0,
        If[v[[1]] == m, v = Rest[v] ] ]; k = m];
  AppendTo[w, k]; If[k == s[[c]], c++]; Sow[k];
    If[n + 1 >= u, u++;
      While[And[Divisible[u, rad[u]^2], CompositeQ[u]], u++] ], {n, 2, nn}] ][[-1, 1]]

A382219 Product of the largest and smallest exponents in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 4, 1, 1, 2, 1, 1, 1, 16, 1, 2, 1, 2, 1, 1, 1, 3, 4, 1, 9, 2, 1, 1, 1, 25, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 4, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 36, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 16, 1, 1, 2, 1, 1, 1, 3, 1, 2

Offset: 1

Ilya Gutkovskiy, Mar 19 2025

nonn

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 1/zeta(2) for k = 1 and 1/zeta(k+1) - 1/zeta(k) for k >= 2, and the asymptotic mean of this sequence is A033150, the same densities and mean as in A051903, since a(n) = A051903(n) for nonpowerful numbers n (A052485) whose asymptotic density is 1. - Amiram Eldar, Mar 28 2025

Cf. A005361, A033150, A051903, A051904, A052485, A062977, A066048, A304233, A333352.

Table[Max @@ (#[[2]] & /@ FactorInteger[n]) Min @@ (#[[2]] & /@ FactorInteger[n]), {n, 90}]

a(n) = if(n == 1, 1, my(e = factor(n)[,2]); vecmin(e) * vecmax(e)); \\ Amiram Eldar, Mar 28 2025

If n = Product (p_j^k_j) then a(n) = min{k_j} * max{k_j}.

a(n) = A051903(n) * A051904(n) for n > 1.

A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

Original entry on oeis.org

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 84, 88, 92, 98, 99, 104, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 184, 188, 189, 198, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 242, 244

Offset: 1

Michael De Vlieger, Jul 19 2025

nonn

			Table of n, a(n) and prime decomposition for n = 1..12:
 n  a(n)
------------------
 1   20 = 2^2 * 5
 2   28 = 2^2 * 7
 3   40 = 2^3 * 5
 4   44 = 2^2 * 11
 5   45 = 3^2 * 5
 6   50 = 2 * 5^2
 7   52 = 2^2 * 13
 8   56 = 2^3 * 7
 9   63 = 3^2 * 7
10   68 = 2^2 * 17
11   75 = 3 * 5^2
12   76 = 2^2 * 19
Let q = A053669 and let gpf = A006530.
The number 12 = 2^2*3 is not in the sequence since q(12) > gpf(12), i.e., 5 > 3.
The number 18 = 2*3^2 is not in the sequence since q(18) > gpf(18), i.e., 5 > 3.
a(1) = 20 = 2^2*5 since q(20) < gpf(20), i.e., 3 < 5.
The number 60 = 2^2*3*5 is not a term since q(60) > gpf(60), i.e., 7 > 5, etc.

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

Cf. A001694, A002110, A006939, A007947, A013929, A052485, A053669, A055932, A080259, A126706, A286708, A332785, A380543.

f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q]; Select[Range[256], Nor[Length[#2] == 1, Max[#2[[All, -1]]] == 1, Divisible[#1, Apply[Times, #2[[All, 1]]]^2], f[#1] > #2[[-1, 1]]] & @@ {#, FactorInteger[#]} &]

Intersection of A332785 and A080259 = A332785 \ A055932 = A126706 \ A286708 \ A380543.

A339553 Number of ordered factorizations of n into perfect powers > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 8

Offset: 1

Ilya Gutkovskiy, Dec 08 2020

nonn

Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A050363, A052485 (positions of 0's), A075802, A294068.

a[n_] := If[n == 1, n, Sum[If[d < n, Boole[GCD @@ FactorInteger[n/d][[All, 2]] > 1] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 128}]

a(1) = 1; a(n) = Sum_{d|n, dA075802(n/d) * a(d).

A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 126, 128, 130, 132, 135, 136, 138, 140, 147, 148, 150, 152

Offset: 1

Luca Onnis, Jul 07 2022

nonn

There are no primes in the sequence, since A007956(p^2) = p for all primes p.
There are infinitely many terms, in fact p^2 is a term for all primes p.
If 8k+1 is not a perfect square, then p^k is a term for all primes p.
Depends only on the prime signature: n is in this sequence if and only if A046523(n) is in this sequence. - Charles R Greathouse IV, Jul 08 2022
Contains all the weak numbers (A052485) aside from the primes (A000040) and squarefree semiprimes (A006881). - Charles R Greathouse IV, Jul 08 2022

			4 is a term of this sequence because there are no numbers k such that A007956(k) = 4.
2^10 is not a term of this sequence because A007956(32) = 1024 (Note that 8*10+1=81=9^2 is a perfect square).
p^4 belongs to this sequence for all primes p, in fact 8*4+1=33 is not a perfect square, so there are no numbers h such that A007956(h) = p^4.

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Subsequences by prime signature: A001248 (p^2), A054753 (p^2*q), A030514 (p^4), A065036 (p^3*q), A007304 (p*q*r), A050997 (p^5), A085986 (p^2*q^2).

Cf. A007956, A052485, A106543.

Complement[Complement[Table[n, {n, 2, 1000}], Select[NumericalSort[Table[Times @@ Most[Divisors[n]], {n, 1000000}]], # != 1 && # < 1000 &]], Select[Table[Prime[n], {n, 1, 1000}], # < 1000 &]]

a(n) = n + O(n log log n/log n). - Charles R Greathouse IV, Jul 08 2022

cp's OEIS Frontend

A331801 Integers that are sum of two nonsquarefree numbers.

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A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

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A378766 Lexicographically earliest infinite sequence of distinct positive numbers with the property that n is a member of the sequence iff a(n) is powerful (in A001694).

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A382219 Product of the largest and smallest exponents in the prime factorization of n.

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A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

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A339553 Number of ordered factorizations of n into perfect powers > 1.

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A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

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