A053212 -id:A053212 - cp's OEIS frontend

A064787 Inverse permutation to A053212.

1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 23, 10, 31, 15, 13, 12, 58, 14, 74, 16, 18, 29, 122, 17, 25, 40, 21, 22, 224, 19, 267, 20, 38, 69, 33, 24, 453, 89, 49, 26, 636, 28, 737, 43, 30, 141, 995, 27, 53, 35, 84, 57, 1523, 34, 59, 36, 108, 257, 2244, 32, 2528, 310, 41, 37, 77, 52

Offset: 1

N. J. A. Sloane, Oct 20 2001

a(n) is the index of A005179(n) in A007416; equivalently, a(n) is the number of minimal numbers (numbers in A007416) that are <= A005179(n). - Jianing Song, Aug 16 2022

			a(23) = 122 because d(n) (A000005(n)) takes 121 different values before it first reaches 23 (at n = 2^22).

Cf. A053212, A005179, A007416, A000005.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a064787 = (+ 1) . fromJust . (`elemIndex` a053212_list)
-- Reinhard Zumkeller, Apr 18 2015

More terms from Naohiro Nomoto, Oct 31 2001

More terms from David Wasserman, Aug 14 2002

A007416 The minimal numbers: sequence A005179 arranged in increasing order.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 24, 36, 48, 60, 64, 120, 144, 180, 192, 240, 360, 576, 720, 840, 900, 960, 1024, 1260, 1296, 1680, 2520, 2880, 3072, 3600, 4096, 5040, 5184, 6300, 6480, 6720, 7560, 9216, 10080, 12288, 14400, 15120, 15360, 20160, 25200, 25920, 27720, 32400, 36864, 44100

Offset: 1

N. J. A. Sloane

Numbers k such that there is no x < k such that A000005(x) = A000005(k). - Benoit Cloitre, Apr 28 2002
A047983(a(n)) = 0. - Reinhard Zumkeller, Nov 03 2015
Subsequence of A025487. If some m in A025487 is the first term in that sequence having its number of divisors, m is in this sequence. - David A. Corneth, Aug 31 2019

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (first 1000 from T. D. Noe and to 10000 from David A. Corneth)
Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2005), pp. 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

Subsequence of A025487; A002182 is a subsequence.
Cf. A005179, A099317, A099312, A099314, A099316, A099318.
Cf. A000005, A047983, A166721 (subsequence of squares).
Cf. A053212 and A064787 (the sequence {A000005(a(n))} and its inverse permutation).

a007416 n = a007416_list !! (n-1)
a007416_list = f 1 [] where
   f x ts = if tau `elem` ts then f (x + 1) ts else x : f (x + 1) (tau:ts)
            where tau = a000005' x
-- Reinhard Zumkeller, Apr 18 2015

for n from 1 to 10^5 do
  t:= numtheory:-tau(n);
  if not assigned(B[t]) then B[t]:= n fi;
od:
sort(map(op,[entries(B)]));# Robert Israel, Nov 11 2015

A007416 = Reap[ For[ s = 1, s <= 10^5, s++, If[ Abs[ Product[ DivisorSigma[0, i] - DivisorSigma[0, s], {i, 1, s-1}]] > 0, Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Nov 19 2012, after Pari *)

for(s=1,10^6,if(abs(prod(i=1,s-1,numdiv(i)-numdiv(s)))>0,print1(s,",")))

is(n)=my(d=numdiv(n));for(i=1,n-1,if(numdiv(i)==d, return(0))); 1 \\ Charles R Greathouse IV, Feb 20 2013

A283980(n,f=factor(n))=prod(i=1, #f~, my(p=f[i, 1]); if(p==2, 6, nextprime(p+1))^f[i, 2])
A025487do(e) = my(v=List([1, 2]), i=2, u = 2^e, t); while(v[i] != u, if(2*v[i] <= u, listput(v, 2*v[i]); t = A283980(v[i]); if(t <= u, listput(v, t))); i++); Set(v)
winnow(v,lim=v[#v])=my(m=Map(),u=List()); for(i=1,#v, if(v[i]>lim, break); my(t=numdiv(v[i])); if(!mapisdefined(m,t), mapput(m,t,0); listput(u,v[i]))); m=0; Vec(u)
list(lim)=winnow(A025487do(logint(lim\1-1,2)+1),lim) \\ Charles R Greathouse IV, Nov 17 2022

A090128 Distinct values of sigma(k), the sum of divisors, in order of appearance as k grows.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 28, 14, 24, 31, 39, 20, 42, 32, 36, 60, 40, 56, 30, 72, 63, 48, 54, 91, 38, 90, 96, 44, 84, 78, 124, 57, 93, 98, 120, 80, 168, 62, 104, 127, 144, 68, 126, 195, 74, 114, 140, 186, 121, 224, 108, 132, 180, 234, 112, 128, 252, 171, 156, 217

Offset: 1

Labos Elemer, Jan 16 2004

Constructed by reading A000203 and deleting values that already appeared earlier: A000203(15)=24 is dropped because equal to A000203(14). A000203(17)=18 is dropped because equal to A000203(10) etc. - R. J. Mathar, May 27 2024

Giovanni Resta, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A002191 (terms in ascending order).

Cf. A053212, A090127.

t = Table[DivisorSigma[1, w], {w, 100}]; u = Union[t]; uu = Union@ Table[ Min[ Flatten[ Position[t, u[[j]]]]], {j, Length[u]}]; Table[ t[[uu[[j]]]], {j, Length[uu]}]
DeleteDuplicates[DivisorSigma[1,Range[100]]] (* Harvey P. Dale, Dec 01 2018 *)

A256259 Sum of divisors of the minimal numbers (A007416).

Original entry on oeis.org

1, 3, 7, 12, 28, 31, 60, 91, 124, 168, 127, 360, 403, 546, 508, 744, 1170, 1651, 2418, 2880, 2821, 3048, 2047, 4368, 3751, 5952, 9360, 9906, 8188, 12493, 8191, 19344, 15367, 22568, 22506, 24384, 28800, 26611, 39312, 32764, 51181, 59520, 49128, 79248, 99944, 92202, 112320, 116281, 106483, 160797

Offset: 1

Omar E. Pol, Apr 20 2015

nonn

Has a symmetric representation in the same way as A000203 and all its subsequences.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated from the b-file at A007416)

Cf. A000203, A005179, A007416, A007626, A053212, A063072, A081533, A237271, A237593.

(* The d-th element in list minDiv[n, b] is the smallest numbers k<=n with exactly d<=b divisors, otherwise it is zero. Computation stops as soon as either inequality fails. *)
minDiv[n_, b_] :=
Module[{list = Array[0 &, b], k = 1, d},
  While[k <= n, d = DivisorSigma[0, k];
   If[d <= b && list[[d]] == 0, list[[d]] = k];
   If[d <= b, k++, k = n + 2]]; list]
a256259[n_, b_] :=
Map[DivisorSigma[1, #] &, Sort[Select[minDiv[n, b], # != 0 &]]]
a256259[100000, 300] (* computes the first 60 elements of the sequence *)
(* Hartmut F. W. Hoft, Apr 27 2015 *)

a(n) = A000203(A007416(n)).

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A064787 Inverse permutation to A053212.

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A007416 The minimal numbers: sequence A005179 arranged in increasing order.

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A090128 Distinct values of sigma(k), the sum of divisors, in order of appearance as k grows.

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A256259 Sum of divisors of the minimal numbers (A007416).

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A053213 Differences between the minimal numbers (A007416).

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