A246868 -id:A246868 - cp's OEIS frontend

A089633 Numbers having no more than one 0 in their binary representation.

0, 1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 191, 223, 239, 247, 251, 253, 254, 255, 383, 447, 479, 495, 503, 507, 509, 510, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1023

Offset: 0

Reinhard Zumkeller, Jan 01 2004

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009
Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012
Numbers of the form 2^t - 2^k - 1, 0 <= k < t.
n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018
Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

			From _Tilman Piesk_, May 09 2012: (Start)
This may also be viewed as a triangle:             In binary:
                  0                                         0
               1     2                                 01       10
             3    5    6                          011      101      110
           7   11   13   14                  0111     1011     1101     1110
        15   23   27   29   30          01111    10111    11011    11101    11110
      31  47   55   59   61   62
   63   95  111  119  123  125  126
Left three diagonals are A000225,  A055010, A086224. Right diagonal is A000918. Central column is A129868. Numbers in row n (counted from 0) have n binary 1s. (End)
From _Gus Wiseman_, May 24 2024: (Start)
The terms together with their binary expansions and binary indices begin:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
  11:   1011 ~ {1,2,4}
  13:   1101 ~ {1,3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  23:  10111 ~ {1,2,3,5}
  27:  11011 ~ {1,2,4,5}
  29:  11101 ~ {1,3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  47: 101111 ~ {1,2,3,4,6}
  55: 110111 ~ {1,2,3,5,6}
  59: 111011 ~ {1,2,4,5,6}
  61: 111101 ~ {1,3,4,5,6}
  62: 111110 ~ {2,3,4,5,6}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure, arXiv:0801.0072 [math.CO], 2007-201. See Section 14.
Vladimir Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1.
Index entries for sequences related to binary expansion of n.

Cf. A000120, A007088, A011371, A014132, A023416, A023532, A029931.
Cf. A181741 (primes), union of A081118 and A000918, apart from initial -1.
Cf. A065442, A160502, A265705.
For least binary index (instead of rank) we have A001511.
Applying A019565 (Heinz number of binary indices) gives A077011.
For greatest binary index we have A029837 or A070939, opposite A070940.
Row minima of A118462 (binary ranks of strict partitions).
For sum instead of minimum we have A372888, non-strict A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A277905 groups all positive integers by binary rank of prime indices.
Cf. A000041, A005940, A018819, A147655, A225620, A246868.

a089633 n = a089633_list !! (n-1)
a089633_list = [2 ^ t - 2 ^ k - 1 | t <- [1..], k <- [t-1,t-2..0]]
-- Reinhard Zumkeller, Feb 23 2012

seq(seq(2^a-1-2^b,b=a-1..0,-1),a=1..11); # Robert Israel, Dec 14 2018

fQ[n_] := DigitCount[n, 2, 0] < 2; Select[ Range[0, 2^10], fQ] (* Robert G. Wilson v, Aug 02 2012 *)

{insq(n) = local(dd, hf, v); v=binary(n);hf=length(v);dd=sum(i=1,hf,v[i]);if(dd<=hf-2,-1,1)}
{for(w=0,1536,if(insq(w)>=0,print1(w,", ")))}
\\ Douglas Latimer, May 07 2013

isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018

from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<A089633_list = list(islice(A089633_gen(),30)) # Chai Wah Wu, Feb 10 2023

from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;
A000120(a(u)) <= A000120(a(v)) for uA000120(a(n)) = A003056(n).
a(0)=0, n>0: a(n+1) = Min{m>n: BinOnes(a(n))<=BinOnes(m)} with BinOnes=A000120.
If m = floor((sqrt(8*n+1) - 1) / 2), then a(n) = 2^(m+1) - 2^(m*(m+3)/2 - n) - 1. - Carl R. White, Feb 10 2009
A029931(a(n)) = n and A029931(m) != n for m < a(n). - Reinhard Zumkeller, Feb 28 2014
A265705(a(n),k) = A265705(a(n),a(n)-k), k = 0 .. a(n). - Reinhard Zumkeller, Dec 15 2015
a(A014132(n)-1) = 2*a(n-1)+1 for n >= 1. - Robert Israel, Dec 14 2018
Sum_{n>=1} 1/a(n) = A065442 + A160502 = 3.069285887459... . - Amiram Eldar, Jan 09 2024
A019565(a(n)) = A077011(n). - Gus Wiseman, May 24 2024

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

Offset: 0

Alois P. Heinz, Sep 05 2014

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

			The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3;
   5,  6;
   7, 10;
  11, 14, 15;
  13, 21, 22, 30;
  17, 26, 33, 35, 42;
  19, 34, 39, 55, 66,  70;
  23, 38, 51, 65, 77,  78, 105, 110;
  29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
  ...
Corresponding triangle of strict integer partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (42) (51) (321)
        (7) (61) (52) (43) (421)
     (8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018

Alois P. Heinz, Rows n = 0..42, flattened

Column k=1 gives: A008578(n+1).
Last elements of rows give: A246868.
Row sums give A147655.
Row lengths are: A000009.
Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
      map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A319057 Minimum sum of a strict factorization of n into factors > 1.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 9, 7, 11, 7, 13, 9, 8, 10, 17, 9, 19, 9, 10, 13, 23, 9, 25, 15, 12, 11, 29, 10, 31, 12, 14, 19, 12, 11, 37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 12, 49, 15, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 14, 18, 16, 67, 21, 26

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

a(n) >= A001414(n), with equality iff n is squarefree or four times a squarefree number (i.e., A000188(n) <= 2). - Charlie Neder, Sep 10 2018

			The strict factorizations of 48 are (48), (2*24), (3*16), (4*12), (6*8), (2*3*8), (2*4*6), with sums 48, 26, 19, 16, 14, 13, 12 respectively, so a(48) = 12.

Charlie Neder, Table of n, a(n) for n = 1..1000

Cf. A001055, A007716, A045778, A056239, A162247, A215366, A246868, A318871, A318953, A318954.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[strfacs[n/d],Min@@#>d&],{d,Rest[Divisors[n]]}]];
Table[Min[Total/@strfacs[n]],{n,100}]

A318871 Minimum Heinz number of a factorization of n into factors > 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 49, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(1) = 1 = the empty product.
a(12) = 35 = 5 * 7 = prime(3) * prime(4).
a(16) = 49 = 7^2 = prime(4)^2.
a(23) = 83 = prime(23).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001055, A007716, A056239, A064988, A162247, A215366, A246868.

a:= proc(n) option remember; `if`(n=1, 1, min(seq(a(d)*
      ithprime(n/d), d=numtheory[divisors](n) minus {n})))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 05 2018

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@#&/@facs[n]],{n,100}]

A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 15, 20, 27, 36, 48, 64, 30, 40, 54, 72, 96, 128, 7, 25, 45, 60, 80, 81, 108, 144, 192, 256, 14, 50, 90, 120, 160, 162, 216, 288, 384, 512, 21, 28, 75, 100, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024, 42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048, 35, 63, 84, 112, 125, 225, 300, 400

Offset: 1

Antti Karttunen, Nov 14 2016

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.
Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.
Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024
Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - Flávio V. Fernandes, Jul 19 2025

			The irregular table begins as:
  row terms
   0   1;
   1   2;
   2   3,  4;
   3   6,  8;
   4   5,  9,  12,  16;
   5  10, 18,  24,  32;
   6  15, 20,  27,  36,  48,  64;
   7  30, 40,  54,  72,  96, 128;
   8   7, 25,  45,  60,  80,  81, 108, 144, 192, 256;
   9  14, 50,  90, 120, 160, 162, 216, 288, 384, 512;
  10  21, 28,  75, 100, 135, 180, 240, 243, 320, 324, 432,  576,  768, 1024;
  11  42, 56, 150, 200, 270, 360, 480, 486, 640, 648, 864, 1152, 1536, 2048;
...

Antti Karttunen, Table of n, a(n) for n = 1..166
Index entries for sequences that are permutations of the natural numbers

Cf. A019565 (the left edge, the only terms that are squarefree).
Cf. A000079 (the trailing edge).
Cf. A048675, A260443, A277886, A277896, A277903, A277904.
Row lengths are A018819 (number of partitions of binary rank n).
A000009 counts strict partitions, ranks A005117.
A029837 stc_sum or A070939 bin_len, opposite A070940 binexp_lastpos_1.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, cf. A001222, A003963, A056239, A296150.
A372890 adds up binary ranks of partitions, strict A372888.
Cf. A000040, A000041, A001511, A005940, A118462, A215366, A225620, A246868.
Cf. A097248, A048675.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Select[Range[0,2^k],Total[2^(prix[#]-1)]==k&],{k,0,10}] (* Gus Wiseman, May 25 2024 *)

(definec (A277905 n) (A277905bi (A277903 n) (A277904 n)))
(define (A277905bi row col) (let outloop ((k (A019565 row)) (col col)) (if (zero? col) k (let inloop ((j (+ 1 k))) (if (= (A048675 j) row) (outloop j (- col 1)) (inloop (+ 1 j))))))) ;; Very slow implementation.
;; Implementation based on a naive recurrence:
(definec (A277905 n) (if (= 1 n) n (let ((maybe_next (A277896 (A277905 (- n 1))))) (if (not (zero? maybe_next)) maybe_next (A019565 (A277903 n))))))

a(1) = 1; for n > 1, if A277896(a(n-1)) > 0, then a(n) = A277896(a(n-1)), otherwise a(n) = A019565(A277903(n)). [A naive recurrence for a one-dimensional version.]
Other identities. For all n >= 1:
A048675(a(n)) = A277903(n).

A318953 Maximum Heinz number of a strict factorization of n into factors > 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 21, 23, 33, 31, 39, 41, 51, 55, 57, 59, 69, 67, 87, 85, 93, 83, 111, 97, 123, 115, 129, 109, 165, 127, 159, 155, 177, 187, 195, 157, 201, 205, 231, 179, 255, 191, 237, 253, 249, 211, 285, 227, 319, 295, 303, 241, 345, 341, 357, 335, 327

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The Heinz number of a factorization (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The strict factorizations of 80 are (2*4*10), (2*5*8), (2*40), (4*20), (5*16), (8*10), (80), with Heinz numbers 609, 627, 519, 497, 583, 551, 409 respectively, so a(80) = 627.

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318954.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Max[Times@@Prime/@#&/@Select[facs[n],UnsameQ@@#&]],{n,100}]

A064554 a(n) = Min {k | A064553(k) = n}.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 13, 8, 9, 14, 29, 12, 37, 26, 21, 16, 53, 18, 61, 28, 39, 58, 79, 24, 49, 74, 27, 52, 107, 42, 113, 32, 87, 106, 91, 36, 151, 122, 111, 56, 173, 78, 181, 116, 63, 158, 199, 48, 169, 98, 159, 148, 239, 54, 203, 104, 183, 214, 271, 84, 281, 226, 117, 64

Offset: 1

Reinhard Zumkeller, Sep 21 2001

nonn

A064553(a(n)) = n and A064553(a(k)) <> k for k < a(n). For prime p, a(p)=prime(p-1), which is sequence A055003. - T. D. Noe, Dec 12 2004
a(n) is not multiplicative because a(7*13) = a(91) = 463, but a(7)*a(13) = 13*37 = 481 and 91 is the smallest possible such n. - Christian G. Bower, May 19 2005
a(n) = A080688(n,1). - Reinhard Zumkeller, Oct 01 2012
Minimal shifted Heinz number of a factorization of n, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - Gus Wiseman, Sep 05 2018

T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064554

Cf. A064555, A001055, A064553.
Cf. A055003 (prime(prime(n)-1)).
Cf. A007716, A056239, A162247, A215366, A246868, A318871.

a064554 = head . a080688_row  -- Reinhard Zumkeller, Oct 01 2012

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@facs[n]],{n,100}] (* Gus Wiseman, Sep 05 2018 *)

A318954 Minimum shifted Heinz number of a strict factorization of n into factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 6, 13, 10, 19, 14, 29, 15, 37, 26, 21, 34, 53, 33, 61, 35, 39, 58, 79, 30, 89, 74, 57, 65, 107, 42, 113, 85, 87, 106, 91, 66, 151, 122, 111, 70, 173, 78, 181, 145, 129, 158, 199, 102, 223, 161, 159, 185, 239, 114, 203, 130, 183, 214, 271, 105

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1).

			The strict factorizations of 60 are (2*3*10), (2*5*6), (2*30), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60), with shifted Heinz numbers 138, 154, 218, 105, 201, 215, 217, 253, 277 respectively, so a(60) = 105.

Cf. A001055, A007716, A045778, A056239, A080688, A162247, A215366, A246868, A318871, A318953.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@Select[facs[n],UnsameQ@@#&]],{n,100}]

A347047 Smallest squarefree semiprime whose prime indices sum to n.

Original entry on oeis.org

6, 10, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 3

Gus Wiseman, Aug 22 2021

nonn

Compared to A001747, we have 21 instead of 22 and lack 2 and 4.
Compared to A100484 (shifted) we have 21 instead of 22 and lack 4.
Compared to A161344, we have 21 instead of 22 and lack 4 and 8.
Compared to A339114, we have 11 instead of 9 and lack 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The initial terms and their prime indices:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   21: {2,4}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}

The opposite version (greatest instead of smallest) is A332765.
These are the minima of rows of A338905.
The nonsquarefree version is A339114 (opposite: A339115).
A001358 lists semiprimes (squarefree: A006881).
A024697 adds up semiprimes by weight (squarefree: A025129).
A056239 adds up prime indices, row sums of A112798.
A246868 gives the greatest squarefree number whose prime indices sum to n.
A320655 counts factorizations into semiprimes (squarefree: A320656).
A338898, A338912, A338913 give the prime indices of semiprimes.
A338899, A270650, A270652 give the prime indices of squarefree semiprimes.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
A339362 adds up prime indices of squarefree semiprimes.
Cf. A001221, A087112, A089994, A098350, A176504, A338900, A338901, A338904, A338907/A338908, A339005, A339191.

Table[Min@@Select[Table[Times@@Prime/@y,{y,IntegerPartitions[n,{2}]}],SquareFreeQ],{n,3,50}]

from sympy import prime, sieve
def a(n):
    p = [0] + list(sieve.primerange(1, prime(n)+1))
    return min(p[i]*p[n-i] for i in range(1, (n+1)//2))
print([a(n) for n in range(3, 58)]) # Michael S. Branicky, Sep 05 2021

cp's OEIS Frontend

A089633 Numbers having no more than one 0 in their binary representation.

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A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

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A319057 Minimum sum of a strict factorization of n into factors > 1.

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A318871 Minimum Heinz number of a factorization of n into factors > 1.

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A277905 Irregular table: Each row n (n >= 0) lists in ascending order all A018819(n) numbers k for which A048675(k) = n.

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A318953 Maximum Heinz number of a strict factorization of n into factors > 1.

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Mathematica

A064554 a(n) = Min {k | A064553(k) = n}.

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Haskell