A361098 -id:A361098 - cp's OEIS frontend

A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, 363, 375, 378, 384, 396, 405, 416, 448, 450, 468, 480, 486, 490, 504, 507, 525, 528, 540, 550, 560, 567, 578, 588, 594, 600

Offset: 1

Michael De Vlieger, Aug 03 2023

nonn

Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1.
Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098.
Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T.
This sequence gives k that are in S but not in T.

			Let B = A126706.
B(1) = 12 is not in the sequence since 3*6 > 12.
B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18.
B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1).
B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48.
B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc.

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

Cf. A001221, A001222, A001694, A007947, A053669, A119288, A126706, A286708, A361098.

nn = 2^10; 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, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]]

This sequence is A361098 \ A286708.

A364998 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) <= k but rad(k)A053669(k) > k.

Original entry on oeis.org

18, 24, 90, 120, 126, 150, 168, 180, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 630, 666, 696, 738, 744, 774, 840, 846, 888, 954, 984, 990, 1032, 1050, 1062, 1098, 1128, 1170, 1206, 1260, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1470, 1494

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) > 0, A360543(k) = 0. There exist nondivisors m < k such that rad(m) = rad(k); however, m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k do not exist.

			Let b(n) = A126706(n), S = A360768, and T = A363082.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is not in S.
b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is in S but not T.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, thus 40 is neither in S nor T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364999 in blue, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms, using the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates a strong quasiperiodic pattern approximately mod 169.

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360768, A361098, A363082, A364997, A364999.

Select[Select[Range[1500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r <= k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A363082 and A360768.

A364997 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) > k but rad(k)A053669(k) < k.

Original entry on oeis.org

40, 45, 56, 63, 88, 99, 104, 117, 136, 152, 153, 171, 175, 176, 184, 207, 208, 232, 248, 261, 272, 275, 279, 280, 296, 297, 304, 315, 325, 328, 333, 344, 351, 368, 369, 376, 387, 423, 424, 425, 440, 459, 464, 472, 475, 477, 488, 495, 496, 513, 520, 531, 536, 539

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = 0, A360543(k) > 0. There exist m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k, but nondivisors m < k do not exist such that rad(m) = rad(k).

			Let b(n) = A126706(n), S = A360767, and T = A360765.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is in S but not in T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, neither is less than 18, hence 18 is not in S but is in T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is not in S but is in T.
b(7) = a(1) = 40 since p*r = 5*10 = 50 and q*r = 3*10 = 30. We have both 50 > 40 and 30 < 40, thus 40 is in both S and T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus for n = 1..400. Terms in this sequence are colored black, those in A364999 in blue, in A364998 in gold, and in A361098 in red.
Michael De Vlieger, Plot of b(n), n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus for n = 1..14400 using the same color scheme as immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms b(n) in this sequence are colored black, else white.

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360765, A360767, A361098, A364998, A364999.

Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r < k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A360765 and A360767.

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[#]} &]

A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.

Original entry on oeis.org

12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 404, 412, 420, 428, 436, 444, 452, 460, 476, 492, 508, 516, 524, 532, 548

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.
Apparently this is A081770 without the leading 4. - R. J. Mathar, Sep 05 2023
From Peter Munn, Mar 05 2024: (Start)
The preceding observation is true for the whole sequence, for reasons outlined below.
To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).
For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.
(End)

			Let b(n) = A126706(n), S = A360767, and T = A363082.
b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.

Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.

Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A363082 and A360767.
From Peter Munn, Feb 21 2024: (Start)
a(n) = 2*A039956(n+1).
Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)
From Michael De Vlieger, Mar 08 2024: (Start)
{a(n)} = A366825 \ A366460, i.e., even terms in A366825.
A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)

A372972 Numbers k such that A372720(k) is negative.

Original entry on oeis.org

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

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

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 5000}], ?(# < 0 &)][[All, 1]]

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)!.

A372864 Numbers k such that A372720(k) = 0.

Original entry on oeis.org

1, 500, 578, 722, 750, 1058, 1500, 1682, 1922, 2646, 2744, 3430, 3645, 4800, 5202, 5346, 5476, 5488, 5625, 6318, 6400, 6724, 7168, 7396, 8000, 8836, 10092, 10976, 11236, 11532, 11979, 12005, 13068, 13924, 14450, 14884, 15309, 16810, 16875, 16896, 18050, 18225

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since f(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since f(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
Apart from terms in this sequence, all the rest of the terms k in R are such that g(k) is negative.
There are no 3-smooth numbers k > 1 in this sequence, however there are 3 terms {500, 6400, 8000} in A033846 (with s = rad(k) = 10). For s = 2*3*23, there are 6 terms {19044, 25392, 38088, 70656, 536544, 953856}.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 1 since tau(1) - f(1) = 1 - 1 = 0.
a(2) = 500 = 2^2 * 5*3, since tau(500) - f(500)
     = (2+1)*(3+1) - card({10,20,40,50,80,100,160,200,250,320,400,500})
     = 12 - 12 = 0.
a(3) = 578 = 2*17^2, since tau(578) - f(578)
     = (1+1)*(2+1) - card({34,68,136,272,544,578})
     = 6 - 6 = 0, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Cf. A000005, A001221, A007947, A008479, A010846, A372720, A372972.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 20000}], ?(# == 0 &)][[All, 1]]

A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

Original entry on oeis.org

50, 75, 80, 98, 112, 135, 147, 189, 240, 242, 245, 252, 270, 294, 300, 336, 338, 350, 352, 360, 363, 378, 396, 416, 450, 468, 480, 490, 504, 507, 525, 528, 540, 550, 560, 578, 588, 594, 600, 605, 612, 624, 650, 672, 684, 700, 702, 720, 722, 726, 735, 750, 756

Offset: 1

Michael De Vlieger, Feb 09 2024

nonn

Does not contain 3-smooth numbers.
Contains neither A168263 nor A367511.
Conjecture: contains most highly composite numbers.

			Let q = A053669(k) and let p = A119288(k).
For s = 10, we have {50, 80}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 10*{ max(5, 3) <= m < 10 : rad(m) | 10 }
   = 10*{5, 8} = {50, 80}.
For s = 15, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 15*{ max(5, 2) <= m < 15 : rad(m) | 15 }
   = 15*{5, 9} = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
For s = 30, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 30*{ max(3, 7) <= m < 30 : rad(m) | 30 }
   = 30*{8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27}
   = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.

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

Cf. A002182, A003586, A053669, A119288, A120944, A168263, A341645, A361098, A364702, A366250, A367511.

nn = 756;
Select[Select[Range[12, nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &],
  And[Max[#2, #3] <= #1 < #4, ! AllTrue[#5, # > 1 &]] & @@
    {#1/#4, #2, #3, #4, #5} & @@
    {#1, #2[[2, 1]], #3, Times @@ #2[[All, 1]], #2[[All, -1]]} & @@
    {#, FactorInteger[#], If[OddQ[#], 2,
        q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Union of {k = m*s : rad(m) | s, max(p, q) <= m < s}, where s is in A120944.
{a(n)} = A364702 \ A366250.
{a(n)} = A361098 \ A341645.

cp's OEIS Frontend

A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

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A364998 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.

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A364997 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) > k but rad(k)*A053669(k) < k.

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

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A364996 Union of A360767 and A363082.

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A364999 Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.

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A372972 Numbers k such that A372720(k) is negative.

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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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A364998 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) <= k but rad(k)A053669(k) > k.

A364997 Numbers k neither squarefree nor prime power such that rad(k)A119288(k) > k but rad(k)A053669(k) < k.

A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.