A007947 -id:A007947 - cp's OEIS frontend

A172420 Numbers k that have measure of smoothness J larger than 5, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 15625, 16384, 17496, 18432, 19683, 20736, 23328, 24576, 26244, 27648, 31104, 32768, 34992, 36864, 39366, 41472, 46656, 49152

Offset: 1

Artur Jasinski, Feb 02 2010

nonn

This sequence is a subsequence of A049094, A172418, and A172419.

Amiram Eldar, Table of n, a(n) for n = 1..3000 (terms 1..368 from Harvey P. Dale)

Cf. A007947, A049094, A059172, A172418, A172419, A172421, A172422.

aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 5, AppendTo[aa, c]], {c, 2, 10000}]; aa
Select[Range[2,50000],Log[Times@@FactorInteger[#][[All,1]],#]>5&] (* Harvey P. Dale, Apr 30 2018 *)

A172421 Numbers k that have measure of smoothness J larger than 6, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

128, 256, 512, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 49152, 52488, 55296, 59049, 62208, 65536, 69984, 73728, 78125, 78732, 82944, 93312, 98304, 104976, 110592, 118098, 124416, 131072, 139968, 147456, 157464, 165888, 177147, 186624, 196608

Offset: 1

Artur Jasinski, Feb 02 2010

nonn

This sequence is a subsequence of A049094, A172418, A172419, and A172420.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007947, A049094, A059172, A172418, A172419, A172420, A172422.

aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 6, AppendTo[aa, c]], {c, 2, 10000}]; aa

More terms from Amiram Eldar, Mar 10 2020

A172422 Numbers k that have measure of smoothness J larger than 7, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 294912, 314928, 331776, 354294, 373248, 390625, 393216, 419904, 442368, 472392, 497664, 524288, 531441, 559872, 589824, 629856, 663552, 708588, 746496, 786432

Offset: 1

Artur Jasinski, Feb 02 2010

nonn

This sequence is a subsequence of A049094, A172418, A172419, A172420, and A172421.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A007947, A049094, A059172, A172418, A172419, A172420, A172421.

aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 7, AppendTo[aa, c]], {c, 2, 10000}]; aa

More terms from Amiram Eldar, Mar 10 2020

A224866 Numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 9, 10, 17, 26, 28, 33, 37, 50, 65, 73, 82, 101, 109, 122, 126, 129, 145, 170, 197, 201, 217, 226, 244, 257, 289, 290, 325, 344, 362, 393, 401, 433, 442, 485, 501, 513, 530, 577, 626, 649, 676, 677, 730, 785, 801, 842, 865, 901, 962, 969, 973, 1001

Offset: 1

Reinhard Zumkeller, Jul 23 2013

nonn

A078310 in natural order.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequences: A078325, A078324.

Cf. A001694, A078310, A007947.

a224866 n = a224866_list !! (n-1)
a224866_list = [x | x <- [2..] , let x' = x - 1, let k = a007947 x',
                    let (y,m) = divMod x' k, m == 0, a007947 y == k]

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[1001], powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)

is(n) = n>1 && ispowerful(n-1) \\ Charles R Greathouse IV, Aug 08 2013, corrected by Amiram Eldar, Jul 31 2022

a(n) = A001694(n) + 1.

A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 11, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 5, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 3, 23, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 0, 3, 1, 4

Offset: 1

Michael De Vlieger, Mar 06 2023

nonn

			a(4) = 0 since k = 1..3 are prime powers.
a(8) = 1 since only k = 6 is such that p = 3, q = 5, but gcd(6, 10) = 2.
a(9) = 1 since the following satisfies definition: {6},
a(16) = 4, i.e., {6, 10, 12, 14},
a(25) = 3, i.e., {10, 15, 20},
a(27) = 6, i.e., {6, 12, 15, 18, 21, 24},
a(32) = 11, i.e., {6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30},
a(36) = 1, i.e., {30},
a(40) = 1, i.e., {30},
a(45) = 1, i.e., {30}, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k <= n, n = 1..36, where a(n) is the number of numbers k in row n shown in blue. Numbers k in green in row n are counted by A360480(n). Together, blue and green numbers are counted by A243823(n) and appear in row n of A272619. Dots at (k, n) in red are divisors, and in yellow and magenta in row n are counted by A243822(n).
Michael De Vlieger, Plot k < n at (x, y) = (k, -n) for n = 1..2^10, where black represents k such that gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k).

Cf. A000005, A000010, A001221, A007947, A010846, A045763, A051953, A243822, A243823, A246547, A272619, A360480, A360765.

nn = 120; rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; c = Select[Range[4, nn], CompositeQ]; Table[Function[{q, r}, Count[TakeWhile[c, # <= n &], _?(And[PrimeNu[#] > q, Divisible[rad[#], r]] &)]] @@ {PrimeNu[n], rad[n]}, {n, nn}]

a(n) = A243823(n) - A360480(n).
a(n) = A045763(n) - A243822(n) - A360480(n).
a(n) = A051953(n) - A000005(n) - A243822(n) - A360480(n).
a(n) = A051953(n) - A010846(n) - A360480(n).
a(n) = A243823(n) = A045763(n) for n in A246547.
For prime power n = p^e, n > 1, a(n) = p^(e-1) - e.
For n in A360765, a(n) > 0.

A377070 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) = bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 9, 7, 8, 9, 4, 10, 25, 11, 8, 12, 18, 27, 13, 4, 14, 49, 9, 15, 25, 16, 17, 8, 12, 18, 27, 19, 8, 20, 50, 125, 9, 21, 49, 4, 22, 121, 23, 16, 24, 36, 54, 81, 25, 4, 26, 169, 27, 8, 28, 98, 343, 29, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 31

Offset: 1

Michael De Vlieger, Oct 25 2024

Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k = bigomega(n), that is, numbers m such that rad(m) | n and m has the same number of prime factors with repetition than does n.

			Triangle begins:
    n    row n of this sequence:
   -------------------------------------------
    1:   {1}
    2:   {2}
    3:   {3}
    4:   {4}
    5:   {5}
    6:   {4, 6, 9}
    7:   {7}
    8:   {8}
    9:   {9}
   10:   {4, 10, 25}
   ...                       (Select rows appear below)
   12:   {8, 12, 18, 27}
   14:   {4, 14, 49}
   15:   {9, 15, 25}
   18:   {8, 12, 18, 27}
   20:   {8, 20, 50, 125}
   24:   {16, 24, 36, 54, 81}
   30:   {8, 12, 18, 20, 27, 30, 45, 50, 75, 125}
   42:   {8, 12, 18, 27, 28, 42, 63, 98, 147, 343}
   60:   {16, 24, 36, 40, 54, 60, 81, 90, 100, 135, 150, 225, 250, 375, 625}.
.
Diagrams of the rank omega(n)-1 simplexes created by row n of this sequence for select n, ordering k in row n by prime decomposition. The number k = n appears in brackets:
Rank 3:
   n = 30:                    n = 42:
             8                         8
           /  \                      /  \
         12 -- 20                  12 -- 28
        /  \  /  \                /  \  /  \
      18 --[30]-- 50            18 --[42]-- 98
     /  \  /  \  /  \          /  \  /  \  /  \
   27 -- 45 -- 75 -- 125     27 -- 63 --147 -- 343
.
   n = 60:     16
              /  \
            24 -- 40
           /  \  /  \
         36 --[60]-- 50
        /  \  /  \  /  \
      54 -- 90 -- 75 --125
     /  \  /  \  /  \  /  \
   81 --150 --135 --375 --625
.
Rank 4:
   n = 210:
   16
        40
   24   56
             100
        60   140
   36   84   196
                   250
             150   350
        90  [210]  490
   54  126   294   686
                            625
                     375    875
              225    525   1225
        135   315    735   1715
   81   189   441   1029   2401

Michael De Vlieger, Table of n, a(n) for n = 1..12021, (rows n = 1..1500, flattened)
Michael De Vlieger, Diagrams of select a(n) illustrating rank omega(n)-1 simplexes formed by the arrangement of terms in row n by prime power decomposition.
Michael De Vlieger, Log log scatterplot of a(n), rows n = 1..65536 (1278755 terms).

Cf. A001221, A001222, A007947, A024619, A376248, A377071.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; Select[Range[n^PrimeNu[n]], Divisible[n, rad[#]] && PrimeOmega[#] == k &], {n, 30}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) = bigomega(n) }.
For prime p, row p of this sequence is {p}, generally for prime power p^k, row p^k of this sequence is {p^k}.
For n in A024619, row n of this sequence has more than 1 term.
A377071(n) = length of row n of this sequence.

A379368 Denominators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

Original entry on oeis.org

1, 2, 6, 3, 15, 10, 70, 35, 105, 210, 2310, 385, 5005, 10010, 30030, 15015, 255255, 510510, 9699690, 4849845, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 44618574, 3187041, 92424189, 308080630, 9550499530, 4775249765, 1302340845, 2604681690, 18232771830

Offset: 1

Amiram Eldar, Dec 21 2024

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.

Amiram Eldar, Table of n, a(n) for n = 1..1000
N. G. de Bruijn, On the number of integers <= x whose prime factors divide n, Illinois Journal of Mathematics, Vol. 6, No. 1 (1962), pp. 137-141.
Olivier Robert and Gérald Tenenbaum, Sur la répartition du noyau d'un entier, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 802-914.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.6, pp. 24-26.

Cf. A007947, A073355, A370896, A379367 (numerators), A379370.

rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Denominator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]

rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} 1/A007947(k)).

A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 1, 1, 5, 0, 6, 0, 7, 6, 4, 0, 10, 0, 10, 8, 11, 0, 13, 3, 13, 6, 14, 0, 21, 0, 11, 12, 17, 10, 20, 0, 19, 14, 21, 0, 29, 0, 22, 19, 23, 0, 28, 5, 28, 18, 26, 0, 33, 14, 29, 20, 29, 0, 42, 0, 31, 25, 26, 16, 45, 0, 34, 24, 45, 0, 42, 0, 37

Offset: 1

Michael De Vlieger, Feb 14 2025

Number of k <= n in the cototient of n that do not share the same squarefree kernel as n.
Define a number k "neutral" to n to be such that 1 < gcd(k,n) < k, that is, k neither divides n nor is coprime to n. A045763(n) is the number of k < n such that k is neutral to n.
Define quality Q(k) to be true if k is such that 1 < gcd(k,n) and rad(k) != rad(n).
Then for k <= n and n > 1, a(n) = A045763(n), but admitting divisors k | n such that rad(k) != rad(n), and eliminating occasional nondivisors k such that rad(k) = rad(n), i.e., k listed in row n of A359929 for n = A360768(i).

			a(6) = 3 since {2, 3, 4} are neither coprime to 6 and do not have the squarefree kernel 6.
a(8) = 1 since only 6 is neither coprime to 8 and does not share the squarefree kernel 2 with 8.
a(10) = 5 since {2, 4, 5, 6, 8} are neither coprime to 10 nor have the squarefree kernel 10.
a(12) = 6 since {2, 3, 4, 8, 9, 10} are neither coprime to 12 nor have the squarefree kernel 6.
a(14) = 7 since {2, 4, 6, 7, 8, 10, 12} are neither coprime to 14 nor have the squarefree kernel 14, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A000010, A005361, A008479, A355432, A359929, A381094.

{0}~Join~Table[n - EulerPhi[n] - DivisorSigma[0, n/rad[n]], {n, 2, 120}]

a(1) = 0, a(p) = a(4) = 0.
a(n) = A045763(n) - A005361(n).
For n > 1, a(n) = n - phi(n) - tau(n/rad(n)) = A000010(n) - A005361(n).
For n > 1, a(n) = n - A000010(n) - A008479(n) + A355432(n).

A066301 a(n) = 0 if n is squarefree, otherwise 1 + a(n/rad(n)) where rad = A007947 (squarefree kernel).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1

Offset: 1

Reinhard Zumkeller, Jan 01 2002

This sequence is not the same as A046660.

			a(24) = 1 + a(24/rad(24)) = 1 + a(24/6) = 1 + a(4) = 1 + (1+a(4/rad(4))) = 1 + (1+a(4/2)) = 2 + a(2) = 2 + 0 = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A007947, A003557, A008966, A033150, A046660, A051903.

a066301 1 = 0
a066301 n = a051903 n - 1  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Max[FactorInteger[n][[;;, 2]]] - 1; Array[a, 100] (* Amiram Eldar, Jan 05 2024 *)

a(n)=if(n>1, vecmax(factor(n)[,2])-1, 0) \\ Charles R Greathouse IV, Jul 15 2013

a(n) = A051903(n)-1 for n > 1, a(1) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024

A075423 rad(n) - 1, where rad(n) is the squarefree kernel of n (A007947).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 1, 2, 9, 10, 5, 12, 13, 14, 1, 16, 5, 18, 9, 20, 21, 22, 5, 4, 25, 2, 13, 28, 29, 30, 1, 32, 33, 34, 5, 36, 37, 38, 9, 40, 41, 42, 21, 14, 45, 46, 5, 6, 9, 50, 25, 52, 5, 54, 13, 56, 57, 58, 29, 60, 61, 20, 1, 64, 65, 66, 33, 68, 69, 70, 5, 72, 73, 14, 37, 76, 77

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

a(n) < n for all n, see A075425.

A075424(n) = a(a(n)) for n>1.

a(n)=factorback(factor(n)[,1])-1 \\ Charles R Greathouse IV, Jul 09 2013

cp's OEIS Frontend

A172420 Numbers k that have measure of smoothness J larger than 5, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

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A172421 Numbers k that have measure of smoothness J larger than 6, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

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A172422 Numbers k that have measure of smoothness J larger than 7, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

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A224866 Numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

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A360543 a(n) = number of numbers k < n, gcd(k, n) > 1, such that omega(k) > omega(n) and rad(n) | rad(k), where omega(n) = A001221(n) and rad(n) = A007947(n).

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A377070 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) = bigomega(n), where rad = A007947 and bigomega = A001222.

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A379368 Denominators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

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A381096 Number of k <= n such that k is neither coprime to n and rad(k) != rad(n), where rad = A007947.

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