A332785 -id:A332785 - cp's OEIS frontend

A378700 Number of k in A126706 between powerful numbers that are not prime powers.

5, 11, 10, 1, 10, 19, 1, 4, 2, 22, 12, 27, 1, 11, 2, 14, 6, 28, 26, 9, 0, 41, 3, 26, 13, 25, 0, 10, 35, 11, 10, 0, 26, 26, 8, 10, 5, 26, 30, 17, 11, 52, 13, 12, 56, 1, 20, 9, 34, 69, 1, 69, 37, 3, 38, 0, 14, 57, 11, 39, 23, 15, 26, 18, 6, 36, 3, 30, 27, 27, 97

Offset: 1

Michael De Vlieger, Dec 04 2024

Within the sequence S = A126706 of powerful numbers, we have numbers k that are powerful (in A286708) and numbers m that are not powerful (in A332785). This sequence is the number of k between m.

			We partition S = A126706 by numbers k in A286708 (in brackets) and derive the following irregular table:
    12,   18,  20,  24,  28, [36];                                hence a(1) = 5,
    40,   44,  45,  48,  50,  52,  54,  56,  60,  63,   68, [72];       a(2) = 11,
    75,   76,  80,  84,  88,  90,  92,  96,  98,  99, [100];            a(3) = 10,
   104, [108];                                                          a(4) = 1,
   112,  116, 117, 120, 124, 126, 132, 135, 136, 140, [144];            a(5) = 10, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..11210, rendering a(n) = 0 instead as 1/2 for visibility.

Cf. A001694, A126706, A286708, A332785.

s = Select[Range[2^16], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; -1 + Length /@ TakeList[s, Prepend[Differences[#], First[#]] &@ Position[s, _Integer?(Divisible[#, Apply[Times, FactorInteger[#][[All, 1]] ]^2] &)][[All, 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.

A343295 a(n) is the smallest k such that A008477(k) = a(n-1) with a(1) = 144.

Original entry on oeis.org

144, 4096, 1225, 34359738368, 549081

Offset: 1

Bernard Schott, May 10 2021

nonn

The next term is a(6) = 2^741 with 224 digits.
Equivalently, when g is the reciprocal map of f = A008477 as defined in the Name, the terms of this sequence are the successive terms of the infinite iterated sequence {m, g(m), g(g(m)), g(g(g(m))), ...} that begins with m = a(1) = 144, hence f(a(n)) = a(n-1).
Why choose 144? Because it is the third integer, after 36 and 100, for which there exists a new infinite iterated sequence that begins with g(144) = 4096; then f(144) = 128 with the periodic sequence (128, 49, 128, 49, ...) (see A062307). Explanation: 144 is the 5th nonsquarefree number in A342973 that is also squareful; the 3 such first integers 36, 64, 81 are terms of the infinite iterated sequence A343293, while 100 is a term of the infinite iterated sequence A343294.
Remember that the nonsquarefree terms in A342973 that are not squareful (A332785) have no preimage by f.
All the terms are nonsquarefree but also powerful, hence they are in A001694.
a(n) < a(n+2) (last comment in A008477) but a(n) < a(n+1) or a(n) > a(n+1).
Prime factorizations from a(1) to a(6): 2^4*3^2, 2^12, 5^2*7^2, 2^35, 3^2*13^2*19^2, 2^741.
It appears that a(2m) = 2^q for some q > 1, and a(2m+1) = r^2 for some r > 1.

			a(1) = 144; 4096 = 2^12 so f(4096) = 12^2 = 144: also 12288 = 2^12*3^1 and f(12288) = 12^2*1^3 = 144; we have f(4096) = f(12288) = 144, but as 4096 < 12288, hence g(144) = 4096 and a(2) = 4096.
a(2) = 4096 = f(1225) = f(2450), but as 1225 < 2450, g(4096) = 1225 and a(3) = 1225.

Cf. A001694, A008477, A062307, A332785, A342511, A342973, A343293, A343294.

A386822 Irregular table T(n,k) = Product_{j = 1..k} prime(j)^(n-j+1), n >= 0, k = 1..n.

Original entry on oeis.org

1, 2, 4, 12, 8, 72, 360, 16, 432, 10800, 75600, 32, 2592, 324000, 15876000, 174636000, 64, 15552, 9720000, 3333960000, 403409160000, 5244319080000, 128, 93312, 291600000, 700131600000, 931875159600000, 157486901972400000, 2677277333530800000

Offset: 0

Michael De Vlieger, Aug 31 2025

Proper subset of A025487, in turn a proper subset of A055932.
For n > 1, T(n,n) is in A332785.
For 1 < k < n, T(n,k) is in A286708, where A286708 is the sequence of powerful numbers (i.e., in A001694) that are not prime powers.
For n > 1 and k > 1, T(n,k) is in A126706.

			Table begins:
  n\k   1      2        3          4           5
  ----------------------------------------------
  0:    1;
  1:    2;
  2:    4,    12;
  3:    8,    72,     360;
  4:   16,   432,   10800,     75600;
  5:   32,  2592,  324000,  15876000,  174636000;
Table of n, a(n) = P(k)^m * Q(k), for n < 12, illustrating prime power factor exponents, where k = omega(a(n)) = A001221(a(n)), P = A002110, and Q = A006939:
                                     Exponents of
 n     a(n)                  k   m   2.3.5.7
---------------------------------------------------
 1       1                           .
 2       2 = P(1)^0 * Q(1)   1   0   1
 3       4 = P(1)^1 * Q(1)   1   1   2
 4      12 = P(2)^0 * Q(2)   2   0   2.1
 5       8 = P(1)^2 * Q(1)   1   2   3
 6      72 = P(2)^1 * Q(2)   2   1   3.2
 7     360 = P(3)^0 * Q(3)   3   0   3.2.1
 8      16 = P(1)^3 * Q(1)   1   3   4
 9     432 = P(2)^2 * Q(2)   2   2   4.3
10   10800 = P(3)^1 * Q(3)   3   1   4.3.2
11   75600 = P(4)^0 * Q(4)   4   0   4.3.2.1

Michael De Vlieger, Table of n, a(n) for n = 0..703 (rows n = 0..37, flattened.)

Cf. A000079, A001694, A006939, A025487, A055932, A126706, A286708, A332785, A387491.

Table[Product[Prime[j]^(n - j + 1), {j, k}], {n, 8}, {k, n}] // Flatten

T(0,1) = 1 by convention.
T(n,1) = A000079(n) = 2^n.
T(n,n) = A006939(n).

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A378700 Number of k in A126706 between powerful numbers that are not prime powers.

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

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A343295 a(n) is the smallest k such that A008477(k) = a(n-1) with a(1) = 144.

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A386822 Irregular table T(n,k) = Product_{j = 1..k} prime(j)^(n-j+1), n >= 0, k = 1..n.

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