A166721 -id:A166721 - cp's OEIS frontend

A166722 a(n) is the number of divisors of A166721(n).

1, 3, 5, 9, 7, 15, 21, 27, 11, 25, 45, 13, 35, 33, 63, 75, 39, 81, 49, 17, 55, 105, 135, 99, 19, 65, 51, 189, 77, 125, 117, 147, 225, 165, 57, 243, 91, 175, 23, 85, 315, 195, 297, 153, 231, 95, 405, 245, 69, 375, 351, 119, 275, 441, 171, 121, 273, 567, 495, 255, 525

Offset: 1

Alexander Isaev (i2357(AT)mail.ru), Oct 20 2009

This is a permutation of the odd numbers A005408. - Alois P. Heinz, Mar 04 2018

			a(8) = A000005(A166721(8)) = A000005(900) = A000005(2^2 * 3^2 * 5^2) = (2+1)*(2+1)*(2+1) = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Alois P. Heinz)

Cf. A000005, A005408, A048691, A136404, A152674, A166721.

a(n) = A000005(A166721(n)).

Proper definition (and removal of obscure Comments entries) by Jon E. Schoenfield, Mar 03 2018

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

A354530 Numbers k such that k^2 is a minimal number; numbers k whose square is in A007416.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 60, 64, 72, 96, 120, 180, 192, 210, 216, 256, 288, 360, 420, 480, 512, 576, 768, 840, 864, 900, 960, 1080, 1260, 1440, 1536, 1680, 1728, 1800, 2048, 2304, 2520, 2880, 3360, 3840, 4320, 4608, 4620, 5400, 6144, 6300, 6720, 6912, 7200, 7560

Offset: 1

Jianing Song, Aug 16 2022

Numbers k such that there is no m < k^2 such that d(m) = d(k^2), d = A000005. Since only squares have an odd number of divisors, also numbers k such that there is no m < k such that d(m^2) = d(k^2).

			8 is a term since 8^2 = 64 has 7 divisors and no smaller number (smaller square) has that many divisors.

David A. Corneth, Table of n, a(n) for n = 1..14008 (first 310 terms from Jianing Song, terms <= 10^20).

Cf. A007416, A000005, A025487.
Square root of A166721. Also A016017 or A071571 sorted.
Cf. also A166722.

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n, ", "); v = Set(concat(v, d))); ); } \\ from Michel Marcus's program for A166721

d(a(n)^2) = A166722(n).

cp's OEIS Frontend

A166722 a(n) is the number of divisors of A166721(n).

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

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A354530 Numbers k such that k^2 is a minimal number; numbers k whose square is in A007416.

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PARI

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