A181815 -id:A181815 - cp's OEIS frontend

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

1, 2, 4, 3, 6, 12, 9, 24, 10, 20, 15, 40, 30, 60, 28, 21, 56, 42, 84, 63, 168, 126, 336, 140, 66, 189, 280, 132, 99, 264, 198, 528, 220, 396, 297, 440, 792, 156, 117, 312, 234, 624, 260, 468, 351, 520, 936, 390, 1040, 1872, 780, 585, 306, 1560, 340, 612, 459, 680, 1224, 510, 1360, 2448, 1020, 765, 342, 2040, 1530, 684, 513

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A002182)
Michael De Vlieger, List of a(n) for n = 1..779674 (gzipped text file, prepared from Flammenkamp's data)
Michael De Vlieger, List of a(n) for n = 1..500000 (noncompressed text file, an abridged version of above)

Cf. A002182, A002183, A056239, A108951, A112778, A112779, A306802, A319626, A324381, A324382, A324581, A324582, A324888, A329040, A329605, A329900, A330743, A330748.

Subsequence of A181815.

Map[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 69] ] (* Michael De Vlieger, Jan 13 2020, imports b-file at A002182 *)

a(n) = A329900(A002182(n)) = A319626(A002182(n)).
a(n) = A181815(A306802(n)).
A108951(a(n)) = A002182(n). [Highly composite numbers (undeflated)]
A056239(a(n)) = A112778(n). [Number of prime factors, counted with multiplicity]
A001222(a(n)) = A112779(n). [Largest exponent in the prime factorization]
A329605(a(n)) = A002183(n). [Number of divisors]
A329040(a(n)) = A324381(n).
A324888(a(n)) = A324382(n).
a(A330748(n)) = A330743(n).

More linking formulas added by Antti Karttunen, Jan 13 2020

A181820 a(1) = 1; for n > 1, if A025487(n) = Product p(i)^e(i), then a(n) = Product p(e(i)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 8, 11, 9, 14, 12, 13, 15, 22, 20, 17, 21, 18, 26, 16, 25, 28, 19, 33, 30, 34, 24, 35, 44, 23, 39, 42, 38, 40, 55, 27, 52, 29, 50, 51, 36, 49, 66, 46, 56, 65, 45, 68, 31, 70, 57, 32, 60, 77, 78, 58, 88, 85, 63, 76, 37, 110, 69, 48, 84, 91, 75, 102, 62, 54, 98, 104, 95

Offset: 1

Matthew Vandermast, Dec 07 2010

nonn

A permutation of the positive integers.

The partition given by the prime signature of A025487(n) has Heinz number a(n). - Pontus von Brömssen, Mar 25 2023

			A025487(8) = 24 = 2^3*3 has the exponents (3,1) in its canonical prime factorization. Accordingly, a(8) = prime(3)*prime(1) (i.e., A000040(3)*A000040(1)), which equals 5*2=10.

A181815 is another mapping from the members of A025487 to the positive integers. Also see A181819, A181821.

Cf. A000040, A122111, A361808 (inverse), A361809 (fixed points).

a(n) = A181819(A025487(n)).

a(n) = A122111(A181815(n)).

A329898 a(n) is the position of 2*A025487(n) in A025487.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Numbers k for which A007814(A025487(k)) > A007949(A025487(k)), i.e., numbers k for which the 2-adic valuation of A025487(k) is larger than its 3-adic valuation.

Numbers k for which A181815(k) is even.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A329897 (complement), A330683 (and its permutation).
Cf. A007814, A007949, A025487, A329904 (a left inverse), A329906.
Positions of even terms in A181815, zeros in A330682.

(* First, load the function f at A025487, then: *)
With[{s = Union@ Flatten@ f@ 6}, Map[If[2 # > Max@ s, Nothing, FirstPosition[s, 2 #][[1]] ] &, s]] (* Michael De Vlieger, Jan 11 2020 *)

upto_e = 64; \\ 64 -> 43608 terms.
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329898list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t =
A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,2*(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
v329898 = A329898list(upto_e);
A329898(n) = v329898[n];

For all n >= 1, A329904(a(n)) = n.

A329897 Numbers k for which the 2-adic and 3-adic valuations of A025487(k) are equal, where A025487(k) is the k-th number which is a product of primorial numbers.

Original entry on oeis.org

1, 4, 9, 11, 20, 22, 23, 38, 41, 43, 44, 54, 69, 72, 73, 77, 79, 93, 110, 114, 118, 123, 124, 128, 129, 131, 147, 154, 181, 186, 190, 191, 199, 201, 208, 209, 212, 232, 242, 245, 246, 272, 279, 286, 294, 299, 300, 307, 310, 312, 321, 324, 327, 345, 359, 371, 374, 376, 416, 424, 425, 430, 434, 442, 446, 451, 454, 466, 469

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Numbers k for which A007814(A025487(k)) = A007949(A025487(k)).

Numbers k for which A181815(k) is odd.

Antti Karttunen, Table of n, a(n) for n = 1..7806

Cf. A007814, A007949, A025487, A329898 (complement), A330682 (characteristic function).
Sequence A330683 sorted into ascending order.
Positions of odd terms in A181815.

s = {1}; k = 1; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], k++; If[Equal @@ IntegerExponent[n, {2, 3}], AppendTo[s, k]]], {n, 2, 10^5}]; s (* Amiram Eldar, Jul 28 2023 *)

A330683 a(n) is the position of A283980(A025487(n)) in A025487.

Original entry on oeis.org

1, 4, 11, 9, 23, 20, 44, 41, 22, 79, 38, 73, 43, 131, 69, 124, 77, 212, 118, 72, 201, 54, 110, 129, 327, 191, 123, 312, 93, 181, 209, 493, 300, 199, 474, 154, 286, 128, 324, 725, 190, 454, 147, 272, 310, 697, 245, 434, 208, 490, 1044, 299, 671, 114, 232, 416, 469, 1008, 374, 646, 321, 721, 1481, 451, 974, 186, 359

Offset: 1

Antti Karttunen, Dec 26 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12868

Permutation of A329897.

Cf. A025487, A085089, A101296, A181815, A283980, A329898 (positive integers not in this sequence), A329904 (a left inverse), A329906, A330681.

(* First, load the function f at A025487, then: *)
With[{s = Union@ Flatten@ f@ 10}, TakeWhile[#, # != 0 &] &@ Map[If[# > Max@ s, 0, FirstPosition[s, #][[1]] ] &[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2]] &, s]] (* Michael De Vlieger, Jan 11 2020 *)

upto_e = 101;
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A330683list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,A283980(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
v330683 = A330683list(upto_e);
A330683(n) = v330683[n];

a(n) = A085089(A330681(n)) = A101296(A330681(n)) = A101296(A283980(A025487(n))).

For all n >= 1, A329904(a(n)) = n.

A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 28 2020

nonn

a(n) is the least k for which A337473(k, n) = 1.

Cf. A003961, A108951, A181815, A336835, A337473, A337475.

Cf. A337476 (position of the first occurrence of each n), A337478.

A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337474(n) = for(i=0,oo,if(1==A337473sq(i,n),return(i)));

\\ This version uses binary search, which is faster in certain cases:
isA337473sq1(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(s<2)); prevpid = pid; e += f[i,2]); (s<2));
A337474(n) = if(!bitand(n,n-1),0,my(imin=0,imax=n,imid); for(i=0,oo, imid=(imax+imin)\2; if(1!=isA337473sq1(imid,n), imin = imid+1, if(1!=isA337473sq1(imid-1,n),return(imid), imax = imid-1))));

a(n) = A336835(A108951(n)).
a(A181815(n)) = A337475(n).
For all n >= 0, a(A337476(n)) = n.
For all n >= 0, a(A337478(n)) >= n.

A182763 Numbers that set records for number of ordered factorizations as A025487(j)*A025487(k).

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 48, 96, 192, 384, 576, 768, 1152, 2304, 4608, 9216, 11520, 17280, 23040, 34560, 69120, 138240, 276480, 414720, 552960, 829440, 1105920, 1658880, 3317760, 6635520, 9953280, 13271040, 19906560, 26542080, 39813120, 53084160, 79626240

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

Subsequence of A025487: it includes A025487(n) iff A182762(n) > A182762(m) for all m < n.

			24 has 6 factorizations into two members of A025487 (24 = 1*24 = 2*12 = 4*6 = 6*4 = 12*2 = 24*1). No smaller number has more than 5 such factorizations. Hence, 24 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..344
Amiram Eldar, Table of n, a(n) = A025487(i), i, A182762(i) = A182764(n) for n = 1..344

Subsequence of A181818.

Cf. A025487, A181815, A182762, A182764.

More terms from Max Alekseyev, Jan 22 2012

A304886 Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, May 21 2018

Row n consists of terms k such that A025487(n) = the product of primorials p_k#, the k in row n written least to greatest k.
For m = A025487(n) in A000079 (i.e., m is an integer power of 2), row n contains A000079(m) 1s.
For m = A025487(n) in A002110 (i.e., m is a primorial) row n contains a single term k that is the index of m in A002110.

			Triangle begins as in rightmost column, which lists the terms that occur on row n. Maximum value of each row is given by A061394(n).
   n  A025487(n)   Row n
--------------------------------
   1        1      0
   2        2      1
   3        4      1,1
   4        6      2
   5        8      1,1,1
   6       12      1,2
   7       16      1,1,1,1
   8       24      1,1,2
   9       30      3
  10       32      1,1,1,1,1
  11       36      2,2
  12       48      1,1,1,2
  13       60      1,3
  14       64      1,1,1,1,1,1
  15       72      1,2,2
  16       96      1,1,1,1,2
  17      120      1,1,3
  18      128      1,1,1,1,1,1,1
  19      144      1,1,2,2
  20      180      2,3
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..8600
Michael De Vlieger, Concordance of A025487, A051282, A061394, and A304886
Michael De Vlieger, Indices of primorials whose product is highly composite
Michael De Vlieger, Indices of primorials whose product is superabundant

Cf. A025487, A051282 (row lengths), A061394 (row maximum), A124832, A181815.

Cf. also A307056.

(* Simple (A025487(n) < 10^5): *)
{{0}}~Join~Map[With[{w = #}, Reverse@ Array[Function[k, Count[w, _?(# >= k &)] ], Max@ w]] &, Select[Array[{#, FactorInteger[#][[All, -1]]} &, 400], Times @@ Boole@ {#1 == Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3], #2 == #3} == 1 & @@ {#1, #2, Sort[#2, Greater]} & @@ # &][[All, -1]] ]
(* Efficient (A025487(n) < 10^23): *)
f[n_] := Block[{ww, g, h},
  g[x_] := Apply[Times,
    MapIndexed[Prime[First@ #2]^#1 &, x]];
  h[x_] := Reverse@
    Array[Function[k, Count[x, _?(# >= k &)] ], Max@ x];
  ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 +
     Length@ NestWhileList[NextPrime@ # &, 1,
     Times @@ {##} <= n &, All] ];
  Map[h, SortBy[Flatten[#, 1], g]] &@
   Map[Block[{w = #, k = 1},
      Apply[
         Join, {{ConstantArray[1, Length@ w]},
           If[Length@ # == 0, #, #[[1]]] }] &@ Reap[
         Do[
          If[# < n,
            Sow[w]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@
               g@ Set[w,
               If[k == 1,
                 MapAt[# + 1 &, w, k],
                 PadLeft[#, Length@ w, First@ #] &@
                   Drop[MapAt[# + Boole[i > 1] &, w, k],
                    k - 1] ]], {i, Infinity}] ][[-1]] ] &, ww]]; {{0}}~Join~f@ 400

For row n > 1, Product_{k=1..A051282(n)} A000040(T(n,k)) = A181815(n). [Product of primes indexed by nonzero terms of row n is equal to A181815(n)] - Antti Karttunen, Dec 28 2019

A329906 a(0) = 1; a(1) = 2; after which a(2n) = A329898(a(n)), a(2n+1) = A330683(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 6, 9, 7, 23, 15, 38, 8, 20, 13, 22, 10, 44, 30, 110, 19, 69, 49, 128, 12, 41, 27, 72, 17, 43, 29, 54, 14, 79, 56, 272, 37, 181, 136, 482, 26, 118, 86, 307, 61, 208, 156, 424, 16, 73, 52, 190, 34, 123, 89, 242, 24, 77, 55, 147, 36, 93, 66, 114, 18, 131, 97, 596, 68, 416, 323, 1448, 48, 286, 218, 990, 164, 711

Offset: 0

Antti Karttunen, Dec 24 2019

nonn

Note the indexing: domain begins from zero, but the range does not include it.

			This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A329898 the parent, and each child to the right is obtained by applying A330683 to the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        5......../ \........11                  6......../ \........9
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    7       23         15       38          8       20         13       22
  10 44   30  110    19  69    49 128     12 41   27  72     17  43   29  54
etc.

Index entries for sequences that are permutations of the natural numbers

Cf. A329905 (inverse permutation).

Cf. A163511, A181815, A329897, A329898, A329901, A330683.

A329906(n) = if(n<2,1+n,if(!(n%2),A329898(A329906(n/2)),A330683(A329906(n\2))));

a(0) = 1; a(1) = 2; after which a(2n) = A329898(a(n)), a(2n+1) = A330683(a(n)).

a(n) = A329901(A163511(n)).

A329905 a(1) = 0; a(2) = 1; and for n > 2, a(n) = A330682(n) + 2*a(A329904(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 12, 7, 16, 5, 24, 14, 32, 10, 48, 28, 64, 20, 13, 96, 15, 9, 56, 128, 40, 26, 192, 30, 18, 112, 256, 80, 52, 384, 60, 36, 11, 224, 512, 25, 160, 29, 17, 104, 768, 120, 72, 22, 448, 1024, 50, 320, 31, 58, 34, 208, 1536, 240, 144, 44, 896, 2048, 100, 640, 62, 116, 68, 21, 416, 3072, 27, 49, 480, 288, 88, 57

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Note the indexing: domain begins from one, but the range contains also zero.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A329906 (inverse permutation).

Cf. A000035, A025487, A061394, A181815, A243071, A329897, A329898, A329904, A329907, A330682.

A329905(n) = if(n<=2,n-1,(A181815(n)%2)+2*(A329905(A329904(n))));

a(1) = 0; a(2) = 1; and for n > 2, if A181815(n) is odd, a(n) = 1 + 2*a(A329904(n)), otherwise a(n) = 2*a(A329904(n)).
a(n) = A243071(A181815(n)).
For all n >= 1, A000120(a(n)) = A061394(n).
For all n >= 2, A070939(a(n)) = A329907(n).

cp's OEIS Frontend

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

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A181820 a(1) = 1; for n > 1, if A025487(n) = Product p(i)^e(i), then a(n) = Product p(e(i)).

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A329898 a(n) is the position of 2*A025487(n) in A025487.

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A329897 Numbers k for which the 2-adic and 3-adic valuations of A025487(k) are equal, where A025487(k) is the k-th number which is a product of primorial numbers.

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A330683 a(n) is the position of A283980(A025487(n)) in A025487.

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A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

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A182763 Numbers that set records for number of ordered factorizations as A025487(j)*A025487(k).

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A304886 Irregular triangle where row n contains indices k where the product of A002110(k) = A025487(n).

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