A024619 -id:A024619 - cp's OEIS frontend

A089993 Penultimate prime divisor of numbers that are not powers of primes (A024619).

2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 5, 2, 3, 2, 3, 2, 3, 5, 3, 2, 3, 5, 2, 2, 3, 2, 7, 3, 2, 2, 3, 5, 2, 3, 2, 3, 7, 2, 3, 2, 5, 2, 2, 3, 2, 3, 2, 5, 2, 2, 5, 3, 2, 3, 5, 2, 3, 2, 7, 3, 2, 3, 2, 3, 3, 5, 3, 7, 2, 3, 2, 3, 5, 3, 2, 11, 2, 5, 2, 3, 2, 3, 2, 3, 7, 5

Offset: 1

Cino Hilliard, Jan 14 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A024619, A076820, A089992.

Select[Table[If[Length[(f = FactorInteger[n])] > 1, f[[-2, 1]], 1], {n, 1, 150}], # > 1 &]

f(n) = a=factor(n);v=a[,1];ln=length(v);if(ln>1,return(v[ln-1]));
g(m) = for(x=2,m,if(f(x)>0,print1(f(x)",")))

a(n) = A076820(A024619(n)). - Amiram Eldar, Apr 12 2021

Offset corrected by Amiram Eldar, Apr 12 2021

A109353 a(n) is the sum of the distinct prime divisors of A024619(n).

Original entry on oeis.org

5, 7, 5, 9, 8, 5, 7, 10, 13, 5, 15, 9, 10, 14, 19, 12, 5, 21, 16, 7, 12, 13, 8, 25, 5, 7, 20, 15, 5, 16, 9, 22, 31, 10, 33, 10, 18, 16, 19, 26, 14, 5, 39, 8, 21, 18, 18, 7, 43, 12, 22, 45, 32, 13, 10, 20, 25, 34, 49, 24, 5, 9, 14, 7, 22, 15, 15, 55, 5, 18, 40, 9, 24, 28, 31, 16, 61, 24

Offset: 1

Cino Hilliard, Aug 21 2005

			a(3)=5 because the 3rd non-prime-power is 12 and its prime factors sum to 5.

Cf. A024619.

distinct(n) = \sum the distinct prime factors of n { local(a,x,m,ln,s); for(m=2,n, s=0; a=ifactord(m); ln=length(a); if(ln > 1, for(x=1,ln, s+=a[x]; ); print1(s",") ) ) } ifactord(n,m=0) = \distinct prime factors of n { local(f,j,k,flist); flist=[]; f=Vec(factor(n,m)); for(j=1,length(f[1]), flist = concat(flist,f[1][j]) ); return(flist) }

Edited by Don Reble, Jul 23 2006

A264762 a(n) is the index of A246946(n) in A024619, integers x that satisfy omega(x) >= 2.

Original entry on oeis.org

1, 3, 6, 10, 13, 2, 7, 20, 26, 34, 5, 23, 44, 55, 17, 21, 4, 12, 31, 41, 16, 67, 8, 36, 50, 25, 29, 38, 9, 22, 54, 70, 30, 113, 14, 63, 86, 42, 47, 11, 28, 66, 85, 37, 136, 19, 76, 106, 61, 65, 15, 39, 90, 116, 48, 64, 79, 69, 73, 18, 45, 102, 133, 60, 207, 32, 117, 164

Offset: 1

Michel Marcus, Nov 23 2015

nonn

Is this a permutation of the positive integers?

			The first 6 terms of A246946 are 6, 12, 18, 24, 30 and 10, that is, the 1st, 3rd, 6th, 10th, 13th and 2nd terms of A024619.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024619, A246946.

N:= 2000: # get all terms before the first term > N.
W:= Vector(N,t -> if nops(numtheory:-factorset(t))<=1 then 0 else 1 fi):
WS:= ListTools:-PartialSums(convert(W,list)):
m:= 1:
F:= {2,3}:
A[1]:= WS[6]:
W[6]:= 0:
for n from 2 do
  while W[m] = 0 and m < N do m:= m+1 od;
  for k from m to N do
     if W[k] = 1 and nops(numtheory:-factorset(k) intersect F) = 2 then
        A[n]:= WS[k];
        W[k]:= 0;
        F:= numtheory:-factorset(k);
        break
     fi
  od;
  if k > N then break fi;
od:
seq(A[i],i=1..n-1); # Robert Israel, Nov 23 2015

v246946(nn) = {a = 6; fa = (factor(a)[,1])~; va = [a]; vs = va; k = 0; while (k!= nn, k = 1; while (!((#setintersect(fa, (factor(k)[,1])~) == 2) && (! vecsearch(vs, k))), k++); a = k; fa = (factor(a)[,1])~; va = concat(va, k); vs = vecsort(va);); va;}
v024619(nn) = {va = []; for (n=1, nn, if (omega(n) >= 2, va = concat(va, n));); va;}
lista(nn) = {v = v246946(nn); w = v024619(vecmax(v)); for (k=1, #v, for (j=1, #w, if (w[j] == v[k], print1(j, ", "); break);););}

A335270 Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.

Original entry on oeis.org

228, 1645, 7725, 88473, 20295895122, 22550994580

Offset: 1

Amiram Eldar, May 29 2020

Since 1 is the only proper unitary divisor of powers of prime (A000961), they are trivial terms and hence they are excluded from this sequence.
The corresponding harmonic means are 4, 5, 5, 9, 18, 20.
Equivalently, numbers m such that omega(m) > 1 and (usigma(m)-1) | m*(2^omega(m)-1), where usigma is the sum of unitary divisors (A034448), and 2^omega(m) - 1 = A034444(m) - 1 = A309307(m) is the number of the proper unitary divisors of m.
The squarefree terms of A247077 are also terms of this sequence.
a(7) > 10^12, if it exists. - Giovanni Resta, May 30 2020
Conjecture: all terms are of the form n*(usigma(n)-1) where usigma(n)-1 is prime. - Chai Wah Wu, Dec 17 2020

			228 is a term since the harmonic mean of its proper unitary divisors, {1, 3, 4, 12, 19, 57, 76} is 4 which is an integer.

The unitary version of A247077.

Cf. A006086, A000961, A024619, A034444, A034448, A077610, A309307, A335268, A335269.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10^5], (omega = PrimeNu[#]) > 1 && Divisible[# * (2^omega-1), usigma[#] - 1] &]

a(5)-a(6) from Giovanni Resta, May 30 2020

A360529 a(n) is the smallest k > A024619(n) such that rad(k) = rad(A024619(n)), where rad(n) = A007947(n).

Original entry on oeis.org

12, 20, 18, 28, 45, 24, 40, 63, 44, 36, 52, 56, 60, 99, 68, 175, 48, 76, 117, 50, 84, 88, 75, 92, 54, 80, 153, 104, 72, 275, 98, 171, 116, 90, 124, 147, 325, 132, 136, 207, 140, 96, 148, 135, 152, 539, 156, 100, 164, 126, 425, 172, 261, 176, 120, 637, 184, 279, 188, 475, 108, 112, 297, 160, 204, 208

Offset: 1

Michael De Vlieger, May 01 2023

nonn

Permutation of A126706.

Let m = A024619(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the successor to m in R_m.

			A024619(1) = 6; the smallest k > 6 such that rad(k) = 6 is a(1) = 12.
A024619(2) = 10; the smallest k > 10 such that rad(k) = 10 is a(2) = 20.
A024619(3) = 12; the smallest k > 12 such that rad(k) = rad(12) = 6 is a(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..2^16.

Cf. A001597, A007947, A020639, A024619, A065642, A360719.

rad[x_] := rad[x] = Times@ FactorInteger[x][[All, 1]]; Table[k = m + 1; Function[r, If[SquareFreeQ[m], m*FactorInteger[m][[1, 1]],  While[rad[k] != r, k++]; k]][rad[m]], {m, Select[Range[2, 104], ! PrimePowerQ[#] &]}]

a(n) = A065642 \ A001597.

Squarefree m implies a(n) = lpf(m)*m = A020639(m)*m.

A381864 Numbers k in A024619 such that p^(m+1) == r (mod k) where r is also in A024619 for all p | n.

Original entry on oeis.org

15, 33, 35, 44, 45, 51, 63, 65, 66, 69, 70, 75, 76, 77, 80, 85, 87, 88, 90, 91, 92, 95, 99, 102, 104, 105, 115, 119, 123, 130, 133, 135, 138, 140, 141, 143, 144, 145, 152, 153, 154, 159, 160, 161, 170, 172, 174, 175, 176, 177, 180, 184, 185, 187, 188, 189, 190

Offset: 1

Michael De Vlieger, Apr 06 2025

nonn

This sequence intersects neither A381750 nor A382120.

			Table of a(n) for n = 1..12, showing prime decomposition (facs(a(n))), p_x^(m+1) mod n, where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc. Brackets appear around residues that are not prime powers.
                       p_x^(m+1) mod n
 n  a(n)  facs(a(n))   p_1   p_2   p_3
-----------------------------------------
 1   15   3 * 5        12    10
 2   33   3 * 11       15    22
 3   35   5 * 7        20    14
 4   44   2^2 * 11     20    33
 5   45   3^2 * 5      36    35
 6   51   3 * 17       30    34
 7   63   3^2 * 7      18    28
 8   65   5 * 13       60    39
 9   66   2 * 3 * 11   62    15    55
10   69   3 * 23       12    46
11   70   2 * 5 * 7    58    55    63
12   75   3 * 5^2       6    50

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

Cf. A000961, A024619, A381750, A382120.

nn = 190, Reap[Do[If[! PrimePowerQ[n], If[NoneTrue[Map[PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ], PrimePowerQ], Sow[n]]], {n, 2, nn}] ][[-1, 1]]

A382120 Numbers k in A024619 such that there exists a prime p | k for which p^(m+1) == r (mod k), where r is also in A024619, and a prime q | k for which q^(m+1) == r (mod k), where r is a prime power.

Original entry on oeis.org

10, 18, 20, 21, 22, 26, 28, 30, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 55, 57, 58, 60, 68, 72, 74, 78, 82, 84, 86, 93, 94, 96, 98, 100, 106, 108, 110, 111, 114, 116, 117, 118, 122, 124, 126, 129, 132, 134, 136, 142, 146, 147, 148, 150, 156, 158, 162, 164, 165

Offset: 1

Michael De Vlieger, Apr 06 2025

nonn

This sequence intersects neither A381750 nor A381864.

			Table of a(n) for select n, showing prime decomposition (facs(a(n))), p_x^(m+1) mod n, where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc. Brackets appear around residues that are not prime powers.
                          p_x^(m+1) mod n
 n  a(n)  facs(a(n))      p_1   p_2   p_3
-----------------------------------------
 1   10   2 * 5           [6]    5
 2   18   2 * 3^2        [14]    9
 3   20   2^2 * 5        [12]    5
 4   21   3 * 7           [6]    7
 5   22   2 * 11         [10]   11
 6   26   2 * 13          [6]   13
 7   28   2^2 * 7          4   [21]
 8   30   2 * 3 * 5        2   [21]    5
 9   34   2 * 17         [30]   17
10   36   2^2 * 3^2      [28]    9
11   38   2 * 19         [26]   19
22   60   2^2 * 3 * 5      4   [21]    5

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

Cf. A000961, A024619, A381750, A381864.

nn = 165, Reap[Do[If[! PrimePowerQ[n], If[CountDistinct@ Map[Boole@ PrimePowerQ@ PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ] == 2, Sow[n]]], {n, 2, nn}] ][[-1, 1]]

A138296 Table T(k,n) read along antidiagonals: sum of the k-th powers of the distinct prime factors of A024619(n).

Original entry on oeis.org

5, 13, 7, 35, 29, 5, 97, 133, 13, 9, 275, 641, 35, 53, 8, 793, 3157, 97, 351, 34, 5, 2315, 15689, 275, 2417, 152, 13, 7, 6817, 78253, 793, 16839, 706, 35, 29, 10, 20195, 390881, 2315, 117713, 3368, 97, 133, 58, 13, 60073, 1953637, 6817, 823671, 16354, 275, 641

Offset: 1

R. J. Mathar, May 07 2008

Row k=1 is A109353. Rows k=2,3 and 4 are subsequences of A005063-A005065.

			Upper left corner of the table starting at row k=1, column n=1:
1|......5.......7.......5.......9.......8.......5.......7.
2|.....13......29......13......53......34......13......29.
3|.....35.....133......35.....351.....152......35.....133.
4|.....97.....641......97....2417.....706......97.....641.
5|....275....3157.....275...16839....3368.....275....3157.
6|....793...15689.....793..117713...16354.....793...15689.
7|...2315...78253....2315..823671...80312....2315...78253.
8|...6817..390881....6817.5765057..397186....6817..390881.

J.-M de Koninck, F. Luca, Integers divisible by sums of powers of their prime factors, J. Num. Theory vol 128 (2008) 557-563.

A024619 := proc(n)
    local a;
    if n = 1 then
        RETURN(6);
    else
        for a from A024619(n-1)+1 do
            if A001221(a) > 1 then
               RETURN(a) ;
            fi ;
        od:
    fi ;
end:
A138296 := proc(n,j)
    local f,beta ;
    beta := 0 ;
    for f in ifactors( A024619(n) )[2] do
        beta := beta+op(1,f)^j ;
    od:
    RETURN(beta) ;
end:
for d from 1 to 10 do for n from 1 to d do printf("%d,",A138296(n,d-n+1)) ; od: od: # R. J. Mathar, May 07 2008

T(k,n) = sum_{d in A000040, d| A024619(n)} d^k.

A365789 Position of A365787(n) in A024619.

Original entry on oeis.org

1, 3, 2, 6, 10, 4, 7, 17, 5, 25, 29, 12, 20, 42, 8, 9, 26, 61, 69, 23, 11, 31, 48, 96, 13, 22, 111, 64, 14, 44, 134, 15, 16, 154, 36, 28, 62, 18, 19, 72, 109, 210, 21, 34, 54, 240, 139, 89, 24, 288, 39, 181, 329, 27, 55, 66, 137, 45, 374, 30, 99, 161, 236, 32

Offset: 1

Michael De Vlieger, Sep 19 2023

nonn

A365787(n) = A286708(n)/A007947(A286708(n)).

Permutation of natural numbers.

			Let b(n) = A286708(n), rad(n) = A007947(n), and c(n) = A024619(n).
a(1) = 1 since b(1)/rad(b(1)) = 36/6 = 6 = c(1).
a(2) = 3 since b(2)/rad(b(2)) = 72/6 = 12 = c(3).
a(3) = 2 since b(3)/rad(b(3)) = 100/10 = 10 = c(2).
a(4) = 6 since b(4)/rad(b(4)) = 108/6 = 18 = c(6).
a(5) = 10 since b(5)/rad(b(5)) = 144/6 = 24 = c(10).
a(6) = 4 since b(6)/rad(b(6)) = 196/14 = 14 = c(4), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..4742
Index entries for sequences that are permutations of the natural numbers

Cf. A007947, A024619, A120944, A286708, A365786, A365787.

nn = 3600;
s = Rest@
  Select[Union@ Flatten@
   Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
   Not @* PrimePowerQ];
t = Select[Range[2, nn], Not @* PrimePowerQ];
Map[FirstPosition[t, #/(Times @@ FactorInteger[#][[All, 1]])][[1]] &, s]

A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.

Original entry on oeis.org

6, 12, 14, 24, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643

Offset: 1

Michael De Vlieger, Mar 27 2025

nonn

Numbers k in A024619 such that A381804(k) = 0.
Let S(n,p) be the set of distinct power residues r (mod n) beginning with empty product and recursively multiplying by prime p | n. For example, S(10,2) = {1,2,4,8,6}.
This sequence builds on A381750, taking the tensor product T(k) (mod k) of S(k,p), p | k. If all products r (mod k) are such that rad(r) | k, then k is in this sequence. Distinct residues r (mod k) in T(k) are listed in row k of A381801.
Proper subset of A381750.
A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
Conjecture: 12 and 24 are the only nonsquarefree numbers in this sequence, i.e., in A126706.

			Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), row a(n) of A381801:
                        Row a(n) of A381801
 n    a(n)  facs(a(n))  k (mod a(n)) such that rad(k) | a(n).
-------------------------------------------------------------
 1      6   2 * 3       {0, 1, 2, 3, 4}
 2     12   2^2 * 3     {0, 1, 2, 3, 4, 6, 8, 9}
 3     14   2 * 7       {0, 1, 2, 4, 7, 8}
 4     24   2^3 * 3     {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
 5     39   3 * 13      {0, 1, 3, 9, 13, 27}
 6     62   2 * 31      {0, 1, 2, 4, 8, 16, 31, 32}
 7    155   5 * 31      {0, 1, 5, 25, 31, 125}
 8    254   2 * 127     {0, 1, 2, 4, 8, 16, 32, 64, 127, 128}
 9   3279   3 * 1093    {0, 1, 3, 9, 27, 81, 243, 729, 1093, 2187}
10   5219   17 * 307    {0, 1, 17, 289, 307, 4913}
Let b = A381750.
a(1) = 6 since T(6) (mod 6) = {1,2,4} X {1,3} = {{1,2,4},{3,0,0}}; all residues r (mod 6) in T(6) (i.e., in row 6 of A381801) are such that rad(r) | 6.
a(2) = 12 since T(12) (mod 12) = {1,2,4,8} X {1,3,9} = {{1,2,4,8},{3,6,0,0},{9,6,0,0}}; all residues r (mod 12) in T(12) are such that rad(r) | 12.
a(3) = 14 since T(14) (mod 14) = {1,2,4,8} X {1,7} = {{1,2,4,8},{7,0,0,0}}; all residues r (mod 14) in T(14) are such that rad(r) | 14.
a(4) = 24 since T(24) (mod 24) = {1,2,4,8,16} X {1,3,9} = {{1,2,4,8,16},{3,6,12,0,0},{9,18,0,0,0}}; all residues r (mod 24) in T(24) are such that rad(r) | 24.
b(6) = 56 is not in the sequence since 49*2 = 98 = 42 (mod 56), rad(42) does not divide 56.
b(8) = 112 is not in the sequence since 49*4 = 196 = 84 (mod 112), rad(84) does not divide 112, etc.

Michael De Vlieger, Efficient Mathematica code for this sequence (2025).

Cf. A007947, A024619, A381750, A381801.

cp's OEIS Frontend

A089993 Penultimate prime divisor of numbers that are not powers of primes (A024619).

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A109353 a(n) is the sum of the distinct prime divisors of A024619(n).

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A264762 a(n) is the index of A246946(n) in A024619, integers x that satisfy omega(x) >= 2.

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A335270 Numbers that are not powers of primes (A024619) whose harmonic mean of their proper unitary divisors is an integer.

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A360529 a(n) is the smallest k > A024619(n) such that rad(k) = rad(A024619(n)), where rad(n) = A007947(n).

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A381864 Numbers k in A024619 such that p^(m+1) == r (mod k) where r is also in A024619 for all p | n.

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A382120 Numbers k in A024619 such that there exists a prime p | k for which p^(m+1) == r (mod k), where r is also in A024619, and a prime q | k for which q^(m+1) == r (mod k), where r is a prime power.

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A138296 Table T(k,n) read along antidiagonals: sum of the k-th powers of the distinct prime factors of A024619(n).

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