A120944 -id:A120944 - cp's OEIS frontend

A362045 a(n) = smallest k such that k > m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

48, 125, 224, 243, 567, 512, 832, 960, 1331, 2048, 1715, 2048, 2187, 1792, 2944, 4131, 3125, 4617, 3712, 3968, 8125, 4374, 5589, 5000, 8192, 9317, 6144, 8192, 10625, 8192, 19683, 15379, 19683, 12032, 11875, 11016, 11907, 13568, 12500, 19683, 13122, 14375, 15104, 16807, 15616, 19683, 19683, 17576, 45619

Offset: 1

Michael De Vlieger, Apr 05 2023

nonn

The smallest k such that k > p^2 such that p is prime and rad(k) | p is p^3.

			a(1) = 48 since m = 6 and the smallest k > m^2 such that rad(k) | 6 is 48. This is to say, the number that follows 6^2 in A003586 is 48.
a(2) = 80 since m = 10 and the smallest k > m^2 such that rad(k) | 10 is 125. This is to say, the number that precedes 10^2 in A003592 is 125.
Table of n = 1..12, m = A120944(n), m^2, and a(n).
   n    m    m^2   a(n)
  ---------------------
   1    6     36     48
   2   10    100    125
   3   14    196    224
   4   15    225    243
   5   21    441    567
   6   22    484    512
   7   26    676    832
   8   30    900    960
   9   33   1089   1331
  10   34   1156   2048
  11   35   1225   1715
  12   38   1444   2048

Michael De Vlieger, Table of n, a(n) for n = 1..32768
Michael De Vlieger, Scatterplot of a(n), m^2, and b(n), n = 1..2^14, where b(n) = A362044(n) is shown in blue, m^2 in black, and a(n) in red.

Cf. A007947, A120944, A289280, A362044.

Table[m = k^2 + 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m++]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]

A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

Original entry on oeis.org

32, 80, 128, 135, 343, 352, 512, 864, 891, 1088, 875, 1216, 1053, 1728, 2048, 2187, 1375, 2187, 2048, 2048, 3125, 4224, 2187, 4802, 4736, 3773, 5832, 5248, 4913, 5504, 7047, 4459, 7533, 8192, 6859, 10368, 10935, 8192, 11264, 8991, 12312, 12167, 8192, 5831, 8192, 9963, 10449, 16640, 16807, 17152, 18432

Offset: 1

Michael De Vlieger, Apr 05 2023

nonn

The largest k such that k < p^2 such that p is prime and rad(k) | p is p itself.

			a(1) = 32 since m = 6 and the largest k < m^2 such that rad(k) | 6 is 32. This is to say, the number that precedes 6^2 in A003586 is 32.
a(2) = 80 since m = 10 and the largest k < m^2 such that rad(k) | 10 is 80. This is to say, the number that precedes 10^2 in A003592 is 80.
Table of n = 1..12, m = A120944(n), a(n), and m^2.
   n    m    a(n)   m^2
  ---------------------
   1    6     32     36
   2   10     80    100
   3   14    128    196
   4   15    135    225
   5   21    343    441
   6   22    352    484
   7   26    512    676
   8   30    864    900
   9   33    891   1089
  10   34   1088   1156
  11   35    875   1225
  12   38   1216   1444

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), m^2, and b(n), n = 1..2^14, where b(n) = A362045(n) is shown in red, m^2 in black, and a(n) in blue.

Cf. A007947, A120944, A289280, A362045.

Table[m = k^2 - 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m--]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]

A365785 a(n) = k such that A120944(k) is the squarefree kernel of A126706(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 6, 4, 1, 2, 7, 1, 3, 8, 5, 10, 1, 4, 12, 2, 14, 6, 8, 15, 1, 3, 9, 2, 7, 1, 3, 19, 13, 8, 20, 14, 22, 4, 10, 24, 1, 5, 25, 8, 12, 16, 27, 2, 1, 28, 14, 18, 30, 11, 6, 8, 15, 34, 5, 1, 3, 22, 2, 36, 23, 7, 38, 1, 39, 3, 4, 41, 19, 27, 43, 8

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

			Let b(n) = A126706(n), c(n) = A120944(n), and squarefree kernel rad(n) = A007947(n).
a(1) = 1 since c(1) = rad(b(1)) = rad(12) = 6.
a(2) = 1 since c(1) = rad(b(2)) = rad(18) = 6.
a(3) = 2 since c(2) = rad(b(3)) = rad(20) = 10.
a(4) = 1 since c(1) = rad(b(4)) = rad(24) = 6.
a(5) = 3 since c(3) = rad(b(5)) = rad(28) = 14, etc.

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

Cf. A007947, A120944, A126706, A365783.

nn = 240;
s = Select[Range[12, nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
t = Select[Range[nn/2], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

A120944(a(n)) = A007947(A126706(n)) = A365783(n).

A366807 a(n) = A020639(A120944(n))*A120944(n).

Original entry on oeis.org

12, 20, 28, 45, 63, 44, 52, 60, 99, 68, 175, 76, 117, 84, 92, 153, 275, 171, 116, 124, 325, 132, 207, 140, 148, 539, 156, 164, 425, 172, 261, 637, 279, 188, 475, 204, 315, 212, 220, 333, 228, 575, 236, 833, 244, 369, 387, 260, 931, 268, 276, 423, 284, 1573, 725

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
Proper subset of A364996, which itself is contained in A126706.

			Let b(n) = A120944(n).
a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

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

Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.

nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A366807(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m*min(primefactors(m)) # Chai Wah Wu, Aug 02 2024

a(n) = A065642(A120944(n)), n > 1.

a(n) = A285109(A120944(n)).

A240592 Number of primes between successive composite squarefree numbers (A120944).

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Antonio Roldán, Apr 08 2014

nonn

First positions where the numbers n=2..5 occur are at: 2, 142, 2180230, 256857823. - Vaclav Kotesovec, Apr 17 2014

			a(4) is 2 because A120944(4)=15, A120944(5)=21, 2 primes between them: 17 and 19.

Cf. A120944, A240593.

freesqrcomp(n)=issquarefree(n)&&!isprime(n)
nextfqc(n)={local(k);k=n+1;while(!freesqrcomp(k),k+=1);return(k)}
primesin(a,b)={local(p=a,q=0);while(p

A240593 The smaller of a pair of consecutive composite squarefree numbers (A120944) without any prime number between them.

Original entry on oeis.org

14, 21, 33, 34, 38, 55, 57, 62, 65, 69, 74, 77, 85, 86, 91, 93, 94, 105, 110, 114, 115, 118, 119, 122, 129, 133, 141, 142, 143, 145, 154, 158, 159, 165, 174, 177, 182, 183, 185, 186, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 230, 235, 237, 246, 247, 253, 254, 258, 259, 265, 266, 273, 285, 286, 287, 290, 295, 298, 299

Offset: 1

Antonio Roldán, Apr 08 2014

nonn

Supersequence of A121495.

			62 is in the sequence because A120944(20)=62, A120944(21)=65, without primes between them.

Cf. A120944, A240592.

Select[Partition[Select[Range[400],CompositeQ[#]&&SquareFreeQ[#]&],2,1], PrimePi[ #[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* Harvey P. Dale, Apr 12 2020 *)

freesqrcomp(n)=issquarefree(n)&&!isprime(n)
nextfqc(n)={local(k); k=n+1; while(!freesqrcomp(k), k+=1); return(k)}
{for(i=2, 1000, if(freesqrcomp(i) && (nextfqc(i)

A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).

Original entry on oeis.org

3, 4, 4, 3, 3, 5, 5, 3, 4, 6, 3, 6, 4, 3, 6, 4, 3, 4, 6, 6, 3, 3, 4, 4, 7, 3, 3, 7, 3, 7, 5, 3, 5, 7, 3, 3, 3, 7, 4, 5, 3, 3, 7, 3, 7, 5, 5, 4, 3, 8, 3, 5, 8, 3, 4, 8, 4, 4, 8, 5, 3, 3, 8, 4, 3, 5, 8, 4, 5, 4, 3, 3, 4, 8, 3, 5, 8, 3, 4, 8, 3, 3, 5, 8, 4, 3, 8

Offset: 1

Michael De Vlieger, Feb 18 2024

			Let b(n) = A120944(n).
a(1) = 3 since b(1) = 6, p = 2, and q = 3; 1 + Ceiling(log 3/log 2) = 3.
  For s = 6, { k = m*s : rad(m) | s } = A003586 begins {1, 2, 3, 4, 6, ...};
  there are 2 powers of 2 before q = 3 so c(6) = 2 = a(1) - 1.
a(2) = 4 since b(2) = 10, p = 2, and q = 5; 1 + Ceiling(log 5/log 2) = 4.
  For s = 10, { k = m*s : rad(m) | s } = A003592 begins {1, 2, 4, 5, 8, 10, ...};
  there are 3 powers of 2 before q = 5 so c(10) = 3 = a(2) - 1.
a(6) = 5 since b(6) = 22, p = 2, and q = 11; 1 + Ceiling(log 11/log 2) = 5.
  For s = 22, { k = m*s : rad(m) | s } = A003596 begins {1, 2, 4, 8, 11, ...};
  there are 4 powers of 2 before q = 11 so c(22) = 4 = a(6) - 1, etc

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

Cf. A007947, A020639, A119288, A120944.

Map[1 + Ceiling[Log[##]] & @@ FactorInteger[#][[1 ;; 2, 1]] &, Select[Range[300], And[CompositeQ[#], SquareFreeQ[#]] &]]

Let c(s) be the number of powers p^m of p = lpf(s) = A020639(s) that precede q = A119288(s) in the sequence { k = m*s : rad(m) | s }, where rad(n) = A007947(n).

a(n) = 1 + c(A120944(n)).

A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

Original entry on oeis.org

162, 250, 686, 1875, 7203, 2662, 4394, 750, 3993, 578, 12005, 722, 6591, 2058, 1058, 14739, 73205, 20577, 1682, 1922, 142805, 5346, 36501, 3430, 2738, 102487, 6318, 3362, 417605, 3698, 73167, 199927, 89373, 4418, 651605, 5202, 25725, 5618, 13310, 151959, 6498

Offset: 1

Michael De Vlieger, Jun 24 2024

Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.
For k = p^m, m > 0, in S(p), p prime, tau(p^m) > A008479(p^m) since tau(p^m) = m + 1 and A008479(p^m) = m. Therefore we consider only composite squarefree q in this sequence.
a(n) is in A126706.
Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.

			a(1) = 162 since the 12th term in S(6) = A033845 = {6, 12, 18, 24, 36, 48, 54, ..., 162, ...} is the smallest k = S(6, i) such that tau(S(6, i)) <= i: tau(162) = 10 while i = 12.
a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
Table of first and some notable terms:
       n        q     i         a(n) a(n)/q  A372720(a(n))
  --------------------------------------------------------
       1        6    12         162   3^3         -2
       2       10     9         250   5^2         -1
       3       14    10         686   7^2         -2
       4       15    11        1875   5^3         -1
       5       21    13        7203   7^3         -3
       6       22    12        2662   11^2        -4
       7       26    13        4394   13^2        -5
       8       30    16         750   5^2          0
      82      210    51       26250   5^3        -11
    1061     2310    99      635250   5^2 * 11    -3
   15013    30030   222    25375350   5 * 13^2   -30
  268015   510510   338   679488810   11^3       -18

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..268015 in blue, and A120944(n)^3 in red.
Michael De Vlieger, Diagram of S(6) = A033845 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function relates to A372720(k), with positive values from largest in light yellow grading to A372720(k) = 1 in orange, and negative values with smallest absolute value in dark blue to greatest in light blue. a(1) = 162 appears at right.
Michael De Vlieger, Diagram of S(30) = A143207 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function is as above, but with A372720(k) = 0 in red. a(2) = 750 appears at left in red.

Cf. A000005, A008479, A120944, A126706, A372720.

(* First, load function f from A162306 *)
Table[k = 1; s = f[n, n^3]; While[DivisorSigma[0, n*s[[k]]] - k > 0, k++]; s[[k]], {n, Select[Range[6, 120], And[SquareFreeQ[#], CompositeQ[#]] &]}]

A361102 1 together with numbers having at least two distinct prime factors.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

nonn

This is the union of 1 and A024619. It is the sequence C used in the definition of A360519. Since C is central to the analysis of A360519 it deserves its own entry.

This has the same relationship to A024619 as A000469 does to A120944 for squarefree numbers.

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

Cf. A000469, A024619, A120944, A126706, A246655, A360519.

isa := n -> is(irem(ilcm(seq(1..n-1)), n) = 0):
aList := upto -> select(isa, [seq(1..upto)]):
aList(112); # Peter Luschny, May 17 2023

Select[Range[120], Not@*PrimePowerQ] (* Michael De Vlieger, May 17 2023 *)

from sympy import primepi, integer_nthroot
def A361102(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

def A361102List(upto: int) -> list[int]:
    return sorted(Set(1..upto).difference(prime_powers(upto)))
print(A361102List(112))  # Peter Luschny, May 17 2023

From Peter Luschny and Michael De Vlieger, May 17 2023: (Start)
The sequence is the complement of the prime powers in the positive integers, a = A000027 \ A246655.
k is in this sequence <=> k divides lcm(1, 2, ..., k-1). (End)
This sequence is {1} U { A120944 U A126706 } = {1} U A024619. - Michael De Vlieger, May 17 2023

Offset set to 1 by Peter Luschny, May 17 2023

A048146 Sum of non-unitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 24, 5, 0, 12, 16, 0, 0, 0, 30, 0, 0, 0, 41, 0, 0, 0, 36, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 36, 0, 48, 0, 0, 0, 48, 0, 0, 24, 62, 0, 0, 0, 36, 0, 0, 0, 105, 0, 0, 20, 40, 0, 0, 0, 84, 39, 0, 0, 64, 0, 0, 0, 72, 0, 54, 0

Offset: 1

Labos Elemer

nonn

			If n = 1000, the 12 non-unitary divisors are {2, 4, 5, 10, 20, 25, 40, 50, 100, 200, 250, 500} and their sum is a(n) = a(1000) = 1206. a(16) = a(2^4) = (2^4 - 2) / (2 - 1)= 14.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034444, A000203, A006881, A048105, A048106, A048107, A048109, A048111, A005117, A120944.

Cf. A002117, A072691.

us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Table[ DivisorSigma[1, n] - us[n], {n, 1, 100} ]
(* Second program: *)
Table[DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 91}] (* Michael De Vlieger, Nov 20 2017 *)

a(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1) \\ Charles R Greathouse IV, Jun 17 2015

from sympy.ntheory.factor_ import divisor_sigma, udivisor_sigma
def A048146(n): return divisor_sigma(n)-udivisor_sigma(n) # Chai Wah Wu, Aug 22 2024

a(n) = A000203(n) - A034448(n) = sigma(n) - usigma(n). a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = (p^k - p) / (p - 1), for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k >=2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * (1 - 1/zeta(3)) = 0.1382506... . - Amiram Eldar, Dec 09 2022

Edited by Jaroslav Krizek, Mar 01 2009

cp's OEIS Frontend

A362045 a(n) = smallest k such that k > m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

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A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

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A365785 a(n) = k such that A120944(k) is the squarefree kernel of A126706(n).

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A366807 a(n) = A020639(A120944(n))*A120944(n).

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A240592 Number of primes between successive composite squarefree numbers (A120944).

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A240593 The smaller of a pair of consecutive composite squarefree numbers (A120944) without any prime number between them.

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PARI

A370454 a(n) = 1 + ceiling((log q)/(log p)), where p = A020639(s) and q = A119288(s) is the second smallest distinct prime factor of squarefree composite s = A120944(n).

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A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

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