A303606 -id:A303606 - cp's OEIS frontend

A361098 Intersection of A360765 and A360768.

36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525

Offset: 1

Michael De Vlieger, Mar 15 2023

nonn

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
Subset of A126706. All terms are neither prime powers nor squarefree.
From Michael De Vlieger, Aug 03 2023: (Start)
Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
This sequence contains {A002182 \ A168263}. (End)

			For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
For n = 18, A360480(n) = | {10, 14, 15} | = 3,
            A360543(n) = | {} | = 0,
            A361235(n) = | {4, 8, 16} | = 3,
            A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
            A360543(n) = | {30} | = 1,
            A361235(n) = | {8, 16, 27, 32} | = 4,
            A355432(n) = | {24} | = 1.
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
    n |  a + b =  c | d + e = f | g + tau + phi - 1 =  n
  ------------------------------------------------------
   36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 +  9 + 12 - 1 =  36
   48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 =  48
   50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 +  6 + 20 - 1 =  50
   54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 +  8 + 18 - 1 =  54
   72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 =  72
   75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 +  6 + 40 - 1 =  75
   80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 =  80
   96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 =  96
   98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 +  6 + 42 - 1 =  98
  100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 +  9 + 40 - 1 = 100
  108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
  112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

nn = 2^16;
a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Reap[ Do[n = s[[j]];
    If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
      FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]

A365308 Powers of primorials P(k)^m, k > 1, m > 1, where P(k) = A002110(k).

Original entry on oeis.org

36, 216, 900, 1296, 7776, 27000, 44100, 46656, 279936, 810000, 1679616, 5336100, 9261000, 10077696, 24300000, 60466176, 362797056, 729000000, 901800900, 1944810000, 2176782336, 12326391000, 13060694016, 21870000000, 78364164096, 260620460100, 408410100000, 470184984576

Offset: 1

Michael De Vlieger, Oct 02 2023

Proper subset of A303606, in turn a proper subset of A286708, in turn a proper subset of A126706.

Numbers in A322793 that are not powers of 2.

			Terms less than 10^4 include P(2)^2 = 36, P(2)^3 = 216, P(2)^4 = 1296, P(2)^5 = 7776, and P(3)^2 = 900.

Michael De Vlieger, Table of n, a(n) for n = 1..4935
Michael De Vlieger, 1024 X 1024 Bitmap showing A322793(n) in black if a power of 2 (i.e., in A000079) else white if in this sequence, n = 1..2^20, arranged from left to right in rows, then from top to bottom.

Cf. A000079, A002110, A100778, A126706, A286708, A303606, A322793, A325374.

nn = 2^39; k = 2; P = 6; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]] ][[-1, 1]]

Intersection of A100778 and A303606.
This sequence is {A325374 \ {A002110 \ {1,2}}} = {A322793 \ {A000079 \ {1,2}}}.
Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/(P(k)*(P(k)-1)) = 0.03450573145072369022... . - Amiram Eldar, Mar 10 2024

A359280 Powerful numbers that are neither prime powers nor powers of squarefree composites.

Original entry on oeis.org

72, 108, 144, 200, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2888, 2916, 3087, 3136, 3200, 3267, 3456, 3528, 3600, 3872, 3888, 3969

Offset: 1

Michael De Vlieger, Aug 01 2023

nonn

Numbers k such that omega(k) > 1 and for prime power factors p^e | k, multiplicities e > 1, yet the multiplicities are not equal.
Subset of A286708, which in turn is a subset of A361098, itself a subset of A126706, the sequence of numbers neither squarefree nor prime powers.
Since A001694 = Union({1}, A246547, A286708), this sequence is a subset of A001694.

			Let b(n) = A286708(n).
b(1) = 36 is not in the sequence since rad(36) = A007947(36) = 6, and 36 = 6^2.
b(2) = a(1) = 72 since 72 is not a perfect power of rad(72).
b(3) = 100 = rad(100)^2 = 10^2, so it is not in the sequence.
b(4) = a(2) = 108, since 108 is not a perfect power of rad(108) = 6.
b(5) = a(3) = 144, since 144 is not a perfect power of rad(144) = 6.
b(6) = 196 is not in the sequence since 196 = rad(196)^2 = 14^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot A001694(ym + x) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and both prime powers and those in A303606 in white.
Michael De Vlieger, Plot A286708(n) at (x, y) for n = ym + x, m = 1024 and x = 1..m, y = 0..m-1, showing terms in this sequence in black, and those in A303606 in white.

Cf. A001694, A007947, A082695, A126706, A286708, A303606, A361098.

nn = 5000; s = Rest@ Select[Union@ Flatten@Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Not@*PrimePowerQ]; Select[s, !SameQ @@ FactorInteger[#][[All, -1]] &]

from math import isqrt
from sympy import mobius, integer_nthroot
def A359280(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        j = isqrt(x)
        c, l = n+x+3-(y:=x.bit_length())+squarefreepi(j)+sum(squarefreepi(integer_nthroot(x, k)[0]) for k in range(4, y)), 0
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Feb 09 2025

This sequence is A286708 \ A303606.

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) - 1 = 0.094962568855... . - Amiram Eldar, Dec 09 2023

A364996 Union of A360767 and A363082.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 90, 92, 99, 104, 116, 117, 120, 124, 126, 132, 136, 140, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 198, 204, 207, 208, 212, 220, 228, 232, 234, 236, 244, 248, 260, 261

Offset: 1

Michael De Vlieger, Aug 26 2023

nonn

			This sequence is A126706 \ A361098.
Union of A364997, A364998, A364999.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, List of A126706(1..400), 20 in a row. Terms in this sequence are circled, while those in small colored circles appear in A361098. Blue represents numbers in A364702, purple A359280, and magenta A303606.
Michael De Vlieger, Plot b(n) at (x,y) = (n mod 1024, -floor(n/1024)), where terms in this sequence are shown in black, and those in A361098 appear in white.

Cf. A013929, A024619, A126706, A360767, A361098, A363082, A364997, A364998, A364999.

Select[Select[Range[261], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, Or[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

A303661 Powers of squarefree semiprimes that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10648, 11236, 12321, 13225, 13924, 14161, 14884

Offset: 1

Ilya Gutkovskiy, Apr 28 2018

nonn

			1089 is in the sequence because 1089 = 3^2*11^2.
1296 is in the sequence because 1296 = 2^4*3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A006881, A013929, A072774, A072777, A085155, A123711, A200511, A246547, A303606.

Select[Range[15000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && PrimeNu[#] == 2 && ! SquareFreeQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] == 2 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10000] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A303661(n):
    def g(x): return int(-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
    def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A006881(n)-1)*A006881(n)) = Sum_{k>=2} (P(k)^2 - P(2*k))/2 = 0.07160601536406295068..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A363596 a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 12, 3, 10, 2, 12, 2, 420, 60, 24, 3, 90, 10, 420, 42, 660, 60, 360, 30, 3276, 252, 56, 4, 120, 8, 3696, 231, 3570, 210, 36, 2, 103740, 5460, 840, 42, 13860, 660, 27720, 1260, 19320, 840, 5040, 210, 198900, 7956, 10296, 396, 11880, 440, 6384, 228

Offset: 0

Michael De Vlieger, Aug 03 2023

Motivated by Proposition 3.2, p. 10 of the Bedhouche-Farhi paper.
Observations regarding prime power decomposition of terms in a(0..20737):
For n > 300, most terms are in A361098 but not in A286708. A303606 is a subset of A286708, which is a subset of A361098, which in turn is a subset of A126706, numbers that are neither prime powers nor squarefree.
a(34) = 36 is the only term in A286708 (more specifically, in A303606).
a(35) = 2 is the last prime term.
a(29) = 8 is the only composite prime power.
a(190) = 221760 is the last term in A002182, but a(61) = a(102) = 720720 is the largest.
a(191) = 2310 is the last primorial term.
a(1055) = 2207550414530882190 is the last squarefree term. If there are further squarefree terms a(n), n is likely to belong to -1 (mod 24).
a(7055) = 1733187515208453605856007490304335826298500960 is the last term that is not in A361098. a(n) not in A361098 is likely to belong to -1 (mod 24).

			The table below relates b(n) = A091137(n) to a(n), with (n+1)!*a(n) = k!*m = b(n), where k! is the largest factorial that divides b(n).
 n  A067255(b(n)) (n+1)!*a(n)   k! * m
---------------------------------------
 0  0                1! * 1     1! * 1
 1  1                2! * 1     2! * 1
 2  2.1              3! * 2     3! * 2
 3  3.1              4! * 1     4! * 1
 4  4.2.1            5! * 6     6! * 1
 5  5.2.1            6! * 2     6! * 2
 6  6.3.1.1          7! * 12    7! * 12
 7  7.3.1.1          8! * 3     8! * 3
 8  8.4.2.1          9! * 10   10! * 1
 9  9.4.2.1         10! * 2    10! * 2
10  10.5.2.1.1      11! * 12   12! * 1
11  11.5.2.1.1      12! * 2    12! * 2
12  12.6.3.2.1.1    13! * 420  15! * 2
13  13.6.3.2.1.1    14! * 60   15! * 4
14  14.7.3.2.1.1    15! * 24   15! * 24
15  15.7.3.2.1.1    16! * 3    16! * 3
16  16.8.4.2.1.1.1  17! * 90   18! * 5
...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See p. 10.
Michael De Vlieger, Log log scatterplot of a(n+1), n = 0..10^4.
Michael De Vlieger, Plot p(k)^e(k) | a(n) at (x, y) = (n, k), n = 0..2^11, with a color function representing e(k), where black = 1, red = 2, and the largest exponent in the dataset shown in magenta. The bar at bottom shows the number 1 in black, primes in red, composite prime powers in gold, squarefree terms in green, and terms that are neither squarefree nor prime powers in blue.

Cf. A000142, A000720, A091137, A361098.

Table[j = 1; ( Times @@ Reap[While[Sow[#^Floor[n/(# - 1)]] &[Prime[j]] > 1, j++]][[-1, 1]] )/Factorial[n + 1], {n, 0, 60}]

from math import prod, factorial
from sympy import sieve
def A363596(n: int) -> int:
    numer = prod(p ** (n // (p - 1)) for p in sieve.primerange(2, n + 2))
    return numer // factorial(n + 1)
print([A363596(n) for n in range(56)])  # Peter Luschny, Aug 17 2025

a(n) = A091137(n)/(n+1)!.

A365436 a(2^k) = 2^k for all k >= 0. let 2^r be the smallest power of 2 which exceeds n, then a(n) = the least novel m*a(k), where k = 2^r-n, and m is not a prior term.

Original entry on oeis.org

1, 2, 3, 4, 15, 10, 5, 8, 30, 60, 90, 24, 18, 12, 6, 16, 42, 84, 126, 168, 630, 420, 210, 56, 35, 70, 105, 28, 21, 14, 7, 32, 63, 154, 189, 252, 945, 770, 315, 504, 1890, 3780, 5670, 1512, 1134, 756, 378, 144, 54, 108, 162, 216, 810, 540, 270, 72, 45, 110, 135

Offset: 1

David James Sycamore, Nov 09 2023

nonn

Based on a recursion similar to that which produces the Doudna sequence, A005940, (using the least power of 2 exceeding n rather than the greatest power of 2 not exceeding n). All 2^(n-1) terms between between fixed points 2^n and 2^(n+1) are multiples m*a(k) of m, the least unused term, and m is a(2^(k+1)-1).
Conjectured to be a permutation of the positive integers.
From David A. Corneth, Nov 11 2023: (Start)
This is a permutation of the positive integers.
To prove this we'll show that each integer occurs at most once and at least once hence exactly once.
By definition (...a(n) = the least novel...) each positive integer occurs at most once.
Now suppose t is the smallest term not in the sequence. Then there exists u such that a(1)..a(u) contain the positive integers from 1 through t-1. Then a(i) = t for some 1 <= i <= 2^e - 1 where 2^e - 1 >= u. If a(i) != t for 1 <= i <= 2^e-2 then a(2^e - 1) = t as then k = 1, a(k) = 1 and m is not a prior term (t did not occur earlier).
Hence t occurs at least once. As it also occurs at most once every positive integer occurs exactly once and this sequence is a permutation of the positive integers. (End)

			a(3) = 3 since k = 1, a(1) = 1  and 3 is the smallest number which is not already a term.
a(5) = 15 since k = 8-5 = 3, a(3) = 3 and 5 is the smallest number which is not already a term.
a(31) = 7, the least unused term at this point in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
David A. Corneth, PARI program.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
Michael De Vlieger, Fan style binary tree of a(n), n = 1..8191, showing primes in red, squares of primes in orange, other composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, purple (also in A286708), and pink (also in A303606).
Index entries for sequences that are permutations of the natural numbers

Cf. A005940, A356886.

nn = 120;  c[] := False; c[1] = True; m[] := 1; a[1] = 1; c[1] = True;
 Do[If[IntegerQ[#],
      Set[k, i],
       While[Or[c[m[#]], c[Set[k, # m[#]]]], m[#]++] &[
         a[2^Floor[# + 1] - i]]] &@ Log2[i];
      Set[{a[i], c[k]}, {k, True}], {i, nn}];
  Array[a, nn] (* Michael De Vlieger, Nov 13 2023 *)

PARI
```
\\ See PARI link
```

A366854 Powers k^m such that k is neither squarefree nor prime powers, and m > 1.

Original entry on oeis.org

144, 324, 400, 576, 784, 1296, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225

Offset: 1

Michael De Vlieger, Jan 01 2024

Analogous to A303606 = { k^m : Omega(k) = omega(k) > 1, m > 1 }, i.e., squarefree composite k (in A120944) raised to m > 1. Proper subset of A131605, itself a proper subset of A286708, which is in turn a proper subset of A126706. This sequence does not intersect Achilles numbers A052486.

			Let b(n) = A126706(n).
a(1) = b(1)^2 = 12^2 = 144. Since 144 = 2^4*3^2, it is both powerful and a perfect power.
a(2) = b(2)^2 = 18^2 = 324.
a(3) = b(3)^2 = 20^2 = 400.
a(8) = b(1)^3 = 12^3 = 1728, etc.

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

Cf. A001597, A001694, A052486, A120944, A126706, A131605, A246547, A286708, A303606.

nn = 20000; i = 1; k = 2;
MapIndexed[Set[S[First[#2]], #1] &,
  Select[Range@ Sqrt[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &] ];
Union@ Reap[
  While[j = 2;
    While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,
    k++; i++] ][[-1, 1]]

This sequence is A126706(i)^m, m > 1.
A131605 = union of {1}, A303606, and {a(n)}.
A286708 = union of A303606, {a(n)}, and A052486.
A001597 = union of {1}, A246547, A303606, and {a(n)}.
A001694 = union of A246547, A303606, {a(n)}, and A052486.

A383018 Numbers k such that A008683(k) = 0 and A382883(k) = -1.

Original entry on oeis.org

36, 64, 100, 196, 216, 225, 441, 484, 676, 729, 1000, 1024, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10648, 11236, 12321, 13225, 13924, 14161

Offset: 1

Peter Luschny, Apr 14 2025

nonn

See the comments in A382883.

Cf. A382883, A008683, A303606, A383105.

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; Select[Range[15000], !SquareFreeQ[#] && f[#] == -1 &] (* Amiram Eldar, Apr 29 2025 *)

print([n for n in range(1, 14444) if moebius(n) == 0 and A382883(n) == -1])

Subsequence of A383105.

A383394 Perfect powers of Achilles numbers.

Original entry on oeis.org

5184, 11664, 40000, 82944, 153664, 186624, 250000, 373248, 419904, 455625, 640000, 746496, 937024, 944784, 1259712, 1265625, 1327104, 1750329, 1827904, 1882384, 2458624, 3240000, 3779136, 4000000, 5345344, 6718464, 7290000, 8000000, 8340544, 9529569, 10240000

Offset: 1

Michael De Vlieger, Aug 01 2025

Proper subset of A131605, where A286708 is the union of A131605 and A052486. Therefore these are both powerful numbers and perfect powers, unlike Achilles numbers themselves.
Proper subset of A366854.
This sequence does not intersect A303606, also a proper subset of A131605.

			Table of n, a(n) for n = 1..12:
 n      a(n)
--------------------------------
 1     5184 =  72^2 = 2^6  * 3^4
 2    11664 = 108^2 = 2^4  * 3^6
 3    40000 = 200^2 = 2^6  * 5^4
 4    82944 = 288^2 = 2^10 * 3^4
 5   153664 = 392^2 = 2^6  * 7^4
 6   186624 = 432^2 = 2^8  * 3^6
 7   250000 = 500^2 = 2^4  * 5^6
 8   373248 =  72^3 = 2^9  * 3^6
 9   419904 = 648^2 = 2^6  * 3^8
10   455625 = 675^2 = 3^6  * 5^4
11   640000 = 800^2 = 2^10 * 5^4
12   746496 = 864^2 = 2^10 * 3^6

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

Cf. A001597, A052486, A126708, A131605, A286708, A303606, A366854, A386762.

nn = 2^24; mm = Sqrt[nn]; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], GCD @@ FactorInteger[#][[;; , -1]] == 1 &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

from math import isqrt
from sympy import integer_nthroot, mobius
def A383394(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while f(kmin) < kmin: kmin >>= 1		
        kmin = max(kmin,kmax >> 1)
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x):
        c, l = squarefreepi(integer_nthroot(x,3)[0])+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))-1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Aug 11 2025

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A303661 Powers of squarefree semiprimes that are not squarefree.

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A363596 a(n) = (Product_{k=1..pi(n+1)} prime(k)^floor(n/(prime(k)-1) ) )/(n+1)!.

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A365436 a(2^k) = 2^k for all k >= 0. let 2^r be the smallest power of 2 which exceeds n, then a(n) = the least novel m*a(k), where k = 2^r-n, and m is not a prior term.

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A366854 Powers k^m such that k is neither squarefree nor prime powers, and m > 1.

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