A033847 -id:A033847 - cp's OEIS frontend

A033848 Numbers whose prime factors are 2 and 11.

22, 44, 88, 176, 242, 352, 484, 704, 968, 1408, 1936, 2662, 2816, 3872, 5324, 5632, 7744, 10648, 11264, 15488, 21296, 22528, 29282, 30976, 42592, 45056, 58564, 61952, 85184, 90112, 117128, 123904, 170368, 180224, 234256, 247808, 322102

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k = 5/11. - Michel Marcus, Sep 22 2012

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

Cf. A033845, A033846, A033847, A033849, A033850, A033851, A143201.

import Data.Set (singleton, deleteFindMin, insert)
a033848 n = a033848_list !! (n-1)
a033848_list = f (singleton (2*11)) where
   f s = m : f (insert (2*m) $ insert (11*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

N:= 10^6: # to get all terms <= N
S:= {seq(seq(2^i*11^j, i=1..ilog2(floor(N/11^j))),j=1..floor(log[11](N/2)))}:
sort(convert(S,list)); # Robert Israel, Oct 26 2017

Select[Range[10^6], FactorInteger[#][[All, 1]] == {2, 11} &] (* Michael De Vlieger, Oct 26 2017 *)
Sort[Flatten[Table[Table[2^j 11^k, {j, 1, 8}], {k, 1, 8}]]] (* Vincenzo Librandi, Oct 27 2017 *)

A143201(a(n)) = 10. - Reinhard Zumkeller, Sep 13 2011

Sum_{n>=1} 1/a(n) = 1/10. - Amiram Eldar, Dec 22 2020

A033851 Numbers whose prime factors are 5 and 7.

Original entry on oeis.org

35, 175, 245, 875, 1225, 1715, 4375, 6125, 8575, 12005, 21875, 30625, 42875, 60025, 84035, 109375, 153125, 214375, 300125, 420175, 546875, 588245, 765625, 1071875, 1500625, 2100875, 2734375, 2941225, 3828125, 4117715, 5359375, 7503125

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k)/k == 24/35. - Artur Jasinski, Nov 09 2008

Subsequence of A143202. - Reinhard Zumkeller, Sep 13 2011

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

Cf. A003595, A147571-A147575, A147576-A147580. [Artur Jasinski, Nov 09 2008]
Cf. A033845, A033846, A033847, A033848, A033849, A033850, A143201, A143202.
Subsequence of A256617.

import Data.Set (singleton, deleteFindMin, insert)
a033851 n = a033851_list !! (n-1)
a033851_list = f (singleton (5*7)) where
   f s = m : f (insert (5*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

a = {}; Do[If[EulerPhi[x]/x == 24/35, AppendTo[a, x]], {x, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
Take[With[{nn=10},Sort[Flatten[Table[5^i 7^j,{i,nn},{j,nn}]]]],40] (* Harvey P. Dale, Feb 09 2013 *)

a(n) = 35*A003595(n). - Artur Jasinski, Nov 09 2008
A143201(a(n)) = 3. - Reinhard Zumkeller, Sep 13 2011
Sum_{n>=1} 1/a(n) = 1/24. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Bob Selcoe, Mar 24 2017

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017
See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017
Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
Cf. A065642.
Cf. A008479 (index of the row where n is located), A285329 (of the column).

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

From Antti Karttunen, Apr 17 2017: (Start)
A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
A(A008479(n), A285329(n)) = n for all n >= 2.
(End)

A288162 Numbers whose prime factors are 2 and 13.

Original entry on oeis.org

26, 52, 104, 208, 338, 416, 676, 832, 1352, 1664, 2704, 3328, 4394, 5408, 6656, 8788, 10816, 13312, 17576, 21632, 26624, 35152, 43264, 53248, 57122, 70304, 86528, 106496, 114244, 140608, 173056, 212992, 228488, 281216, 346112, 425984, 456976, 562432, 692224, 742586, 851968, 913952

Offset: 1

Bernard Schott, Jun 06 2017

nonn

Numbers k such that phi(k)/k = 6/13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033847, A033848, A033849, A033850, A033851, A086780, A107326, A143207.

[n:n in [1..100000] | Set(PrimeDivisors(n)) eq {2,13}];  // Marius A. Burtea, May 10 2019

Select[Range[920000],FactorInteger[#][[All,1]]=={2,13}&] (* Harvey P. Dale, Jun 18 2021 *)

is(n) = factor(n)[, 1]~==[2, 13] \\ Felix Fröhlich, Jun 06 2017

list(lim)=my(v=List(),t); for(n=1,logint(lim\2,13), t=13^n; while((t<<=1)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 11 2017

a(n) = 26 * A107326(n). - David A. Corneth, Jun 06 2017

Sum_{n>=1} 1/a(n) = 1/12. - Amiram Eldar, Dec 22 2020

A343300 a(n) is p1^1 + p2^2 + ... + pk^k where {p1,p2,...,pk} are the distinct prime factors in ascending order in the prime factorization of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 11, 7, 2, 3, 27, 11, 11, 13, 51, 28, 2, 17, 11, 19, 27, 52, 123, 23, 11, 5, 171, 3, 51, 29, 136, 31, 2, 124, 291, 54, 11, 37, 363, 172, 27, 41, 354, 43, 123, 28, 531, 47, 11, 7, 27, 292, 171, 53, 11, 126, 51, 364, 843, 59, 136, 61, 963, 52, 2, 174, 1342, 67, 291, 532, 370, 71, 11, 73

Offset: 1

Giorgos Kalogeropoulos, Apr 11 2021

nonn

From Bernard Schott, May 07 2021: (Start)
a(n) depends only on prime factors of n (see formulas).
Primes are fixed points of this sequence.
Terms are in increasing order in A344023. (End)

			a(60) = 136 because the distinct prime factors of 60 are {2, 3, 5} and 2^1 + 3^2 + 5^3 = 136.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A027748, A344023 (terms ordered).

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

Original entry on oeis.org

4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241

Offset: 1

Michel Marcus, Dec 17 2020

nonn

Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)

			   n  a(n) prime factor set
   1    4  [2]           A000079
   2    9  [3]           A000244
   3   25  [5]           A000351
   4   18  [2, 3]        A033845
   5   49  [7]           A000420
   6   80  [2, 5]        A033846
   7  121  [11]          A001020
   8  169  [13]          A001022
   9  112  [2, 7]        A033847
  10  135  [3, 5]        A033849
  11  289  [17]          A001026
  12  361  [19]          A001029
  13  441  [3, 7]        A033850
  14  352  [2, 11]       A033848
  15  529  [23]          A009967
  16  416  [2, 13]       A288162
  17  841  [29]          A009973
  18  360  [2, 3, 5]     A143207

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).

u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}

a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020

A370266 Numbers k that are not prime powers, such that k/rad(k) >= rad(k), where rad(k) = A007947(k).

Original entry on oeis.org

36, 48, 54, 72, 96, 100, 108, 144, 160, 162, 192, 196, 200, 216, 224, 225, 250, 288, 320, 324, 375, 384, 392, 400, 405, 432, 441, 448, 484, 486, 500, 567, 576, 640, 648, 675, 676, 686, 704, 768, 784, 800, 832, 864, 896, 900, 960, 968, 972, 1000, 1029, 1080, 1089

Offset: 1

Michael De Vlieger, Feb 18 2024

Numbers k = m * s, where s is composite and squarefree, rad(m) | s, and m >= s.

A177492 is a proper subset.

			For s = 6, this sequence contains {36, 48, 54, 72, 96, ...}, i.e., A033845(n) for n >= A010846(6).
For s = 10, this sequence contains {100, 160, 200, 250, 320, ...}, i.e., A033846(n) for n >= A010846(10).
For s = 14, this sequence contains {196, 224, 392, 448, 686, ...}, i.e., A033847(n) for n >= A010846(14).
For s = 15, this sequence contains {225, 375, 405, 675, 1125, ...}, i.e., A033849(n) for n >= A010846(15), etc.

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

Cf. A007947, A010846, A024619, A120944, A126706, A177492, A246547, A286708, A341645, A366250.

Select[Select[Range[2, 1100], Not@*PrimePowerQ], #1/#2 >= #2 & @@ {#, Times @@ FactorInteger[#][[All, 1]]} &]

Set difference of A341645 and A246547.
Intersection of A341645 and A126706.
Union of A286708 and A366250.

A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 210, 240, 252, 270, 300, 330, 360, 390

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system (RRS) does not intersect A126706 (i.e., the sequence of numbers that are neither squarefree nor prime powers). Alternatively, numbers k whose RRS is a subset of A303554 (i.e., the union of powers of primes and squarefree numbers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q = A382248(k). Then this sequence is that of k such that k < m.
There are 72 terms in this sequence.
Sequences A048597 and A051250 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = 2^2 * 3 = 12.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 16, 32}.
  11 is in the sequence since 11 < m, m = 2^2 * 3 = 12, but 13 is not, since 13 > 12.
  9 is in the sequence since 9 < m, m = 2^2 * 5 = 20.
  25 is not a term since 25 > 12, and 27 is not a term since 27 > 20.
For omega = 2, we have the subset {6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 48, 50, 54, 72, 96, 108, 144, 162}.
  38 = 2*19 is a term since 38 < 45, 45 = 3^2 * 5, but 46 = 2*23 is not, since 46 > 45.
  15 = 3*5 is a term since 15 < 20, but 21 is not, since 21 > 20 and 35 is not, since 35 > 12.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162}, since m = 5^2 * 7 = 175.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, 50}, since m = 3^2 * 7 = 63.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28}, since m = 3^2 * 5 = 45, etc.
For omega = 3, we have the subset {30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 240, 252, 270, 300, 360, 450, 480}, of which {30, 42, 66, 70, 78, 102, 114, 138, 174} are squarefree.
  Intersection with A143207 = {k : rad(k) = 30} is {30, 60, 90, .., 480} because m = 7^2 * 11 = 539.
  Intersection with 42*A108319 = {k : rad(k) = 42} is {42, 84, 126, 168}, since m = 5^2 * 11 = 275, etc.
For omega = 4, we have the subset {210, 330, 390, 420, 510, 630, 840, 1050, 1260, 1470}, of which {210, 330, 390, 510} are squarefree.
  Intersection with A147571 = {k : rad(k) = 210} is {210, 420, 630, 840, 1050, 1260, 1470} since m = 11^2 * 13 = 1573, etc.
For omega = 5, we have 2310 = 2*3*5*7*11, a term since 2310 < 13*17 = 2873; 2730 = 2*3*5*7*13 is not a term.
There are no terms larger than 2310, since the intersection with A147572 = {2310}, 2730 is not a term, and k = Product_{i=1..j} prime(i), k > prime(j+1)^2 * prime(j+2) for j > 5. Therefore the sequence is finite like A051250.

Michael De Vlieger, Table of n, a(n) for n = 1..72

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A126706, A303554, A380539, A382248, A382960.

Select[Range[30030], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system does not intersect A286708 (i.e., powerful numbers that are not prime powers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q^2 = A382767(k). Then this sequence is that of k such that k < m.
This sequence is finite following arguments akin to those in A051250 and A382659, with 626 terms.
Sequences A048597, A051250, and A382659 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = (2*3)^2 = 36.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 64, 81, 128}.
  31 is in the sequence since 31 < m, m = (2*3)^2 = 36, but 37 is not a term since 37 > 36.
  25 is in the sequence since 25 < m, m = 36.
  49 is not a term since 49 > 36, and 243 is not a term since 243 > 100, 100 = (2*5)^2, etc.
For omega = 2, we have the squarefree numbers {6, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218}.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, .., 1152}, since m = (5*7)^2 = 1225.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, ..., 400}, since m = (3*7)^2 = 441.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28, 56, ..., 224}, since m = (3*5)^2 = 225.
  Intersection with A033848 = {k : rad(k) = 15} is {15, 45, 75, 135}, since m = (2*7)^2 = 196, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..626

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A286708, A380539, A382659 (k such that k < p^2*q), A382767.

Select[Range[510510], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A370409 Numbers k = m * s, where s is composite and squarefree, rad(m) divides s, and 1 < m <= s, where rad() = A007947().

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 50, 52, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 196, 198, 204, 207, 208

Offset: 1

Michael De Vlieger, Feb 22 2024

nonn

A177492 is a proper subset.

Proper subset of A126706.

			Let T(j,k) = row j of A162306 and let s = A120944(n), n > 1.
This sequence contains finite sequences R(s) = s * T(s, 2..A010846(s)). The cardinality of R(s) is A010846(s)-1.
For s = 6, this sequence contains {12, 18, 24, 36},
  i.e., A033845(2..A010846(6)).
For s = 10, this sequence contains {20, 40, 50, 80, 100},
  i.e., A033846(2..A010846(10)).
For s = 14, this sequence contains {28, 56, 98, 112, 196},
  i.e., A033847(2..A010846(14)).
For s = 15, this sequence contains {45, 75, 135, 225},
  i.e., A033849(2..A010846(15)), etc.

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

Cf. A005117, A007947, A120944, A126706, A162306, A177492, A370266.

cp's OEIS Frontend

A033848 Numbers whose prime factors are 2 and 11.

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A033851 Numbers whose prime factors are 5 and 7.

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A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

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A288162 Numbers whose prime factors are 2 and 13.

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A343300 a(n) is p1^1 + p2^2 + ... + pk^k where {p1,p2,...,pk} are the distinct prime factors in ascending order in the prime factorization of n.

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A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

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A370266 Numbers k that are not prime powers, such that k/rad(k) >= rad(k), where rad(k) = A007947(k).

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