A005179 -id:A005179 - cp's OEIS frontend

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

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

A072066 Exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m).

Original entry on oeis.org

8, 16, 24, 32, 48, 64, 72, 80, 96, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848

Offset: 1

David Wasserman, Jun 12 2002

nonn

Brown shows that this sequence has density 0 and is a subsequence of A013929. Mei shows that in fact it is a subsequence of A048108. - Charles R Greathouse IV, Jun 07 2013
Not a subsequence of A025487: 80, 108, 112, etc. are not the product of primorials. - Charles R Greathouse IV, Jun 07 2013
The product of any exceptional numbers is an exceptional number. - Thomas Ordowski, Jun 14 2015
Grost proved that p^k is in the sequence if and only if 2^p < prime(k), where p is a prime. - Thomas Ordowski, Jun 15 2015
Only very few of the initial terms, {108, 162, 243, 324, 486, 729, ...} are not multiples of 8. Note that the 2nd to 6th in this list (and certainly more) equal 81*k = (10 + 1/8)*a(n) with n = 2, 3, 4, 5, 7, ... - M. F. Hasler, Jun 15 2022

			m=8 is a term: A005179(8) = 2^3 * 3 = 24 < 30 = 2^1 * 3^1 * 5^1 = A037019(8). - _Jon E. Schoenfield_, Mar 18 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2006), pp. 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
Shu-Yuan Mei, A new class of ordinary integers, video summary of article.
Shu-Yuan Mei, A new class of ordinary integers, Journal of Number Theory, Volume 133, Issue 10, October 2013, Pages 3559-3564.
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.

Cf. A005179, A037019.

select( {is_A072066(n)=A005179(n)<A037019(n)}, [1..9999]) \\ M. F. Hasler, Oct 14 2014, updated Jun 15 2022

Links updated by Michel Marcus and M. F. Hasler, Oct 14 2014

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

Original entry on oeis.org

8, 24, 48, 64, 72, 80, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848, 864, 896

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Where the prime signature of the least number with exactly k divisors differs from the prime signature of the least odd number with exactly k divisors.
Also where A122375(k) differs from A122810(k).
Also where A122376(k) differs from A122811(k).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A005179, A038547, A046523, A122375, A122376, A122810-A122819.

A110821 SuperRefactorable numbers: m=A005179(n) such that k=m/n is an integer.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3600, 5040, 6720, 10080, 15120, 20160, 25200, 32400, 55440, 60480, 100800, 110880, 181440, 221760, 226800, 277200, 665280, 720720, 810000, 907200, 1108800, 1441440, 1587600, 1995840, 2494800, 2882880, 3548160, 3603600

Offset: 1

Thomas Ordowski, Sep 16 2005

nonn

Refactorable numbers, A033950, such that m=A073904(n)=A005179(n).

			k = m/n = 1, 1, 2, 3, 4, 5, 10, 12, 15, 24, 35, 42, ... is an integer.
For instance: 60/12=5, 180/18=10, 240/20=12, 360/24=15.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A033950, A005179, A007416 and A073904.

t = Table[0, {2^10}]; Do[ d = DivisorSigma[0, n]; If[ d < 2^10 && t[[d]] == 0, t[[d]] = n], {n, 2882880}]; Rest[ Union[ t[[ Select[ Range[2^10], IntegerQ[ t[[ # ]]/# ] &]] ]]] (* Robert G. Wilson v, Sep 21 2005 *)

More terms from Robert G. Wilson v, Sep 21 2005

Data corrected by David A. Corneth, Dec 11 2021

A262983 Terms of A005179 divisible by their indices in order of appearance in A005179.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3600, 6720, 5040, 10080, 32400, 15120, 20160, 25200, 60480, 55440, 810000, 100800, 181440, 110880, 226800, 221760, 277200, 907200, 665280, 1587600, 720720, 5670000, 1108800, 3548160, 1995840, 1441440, 2494800, 6350400

Offset: 1

Vladimir Letsko, Oct 06 2015

A005179(n) is in this sequence iff it is divisible by n. Thus this is a subsequence of A005179 indexed by A262981.
Also this sequence is the intersection of A033950 and A005179. Hence this sequence has density zero. - Vladimir Letsko, Dec 16 2016
It seems that this sequence is a subsequence of A262981.
This sequence is not in ascending order as terms of A005179 divisible by their number of divisors do not occur in ascending order. For terms sorted in ascending order see A110821. - David A. Corneth, Dec 10 2021

			12 is a term since it is the smallest positive integer having exactly 6 divisors and divisible by 6.

David A. Corneth, Table of n, a(n) for n = 1..10000
Vladimir Letsko, The table of correspondence between A262983 and A262981

Cf. A000005, A033950, A005179, A110821, A262981, A279373.

Take[#, 33] &@ DeleteCases[#, 0] &@ Function[s, ReplacePart[#, Flatten@ Map[{# -> Function[k, k Boole[Divisible[k, #]]]@ Lookup[s, #]} &, Keys@ s]] &@ ConstantArray[0, Max@ Keys@ s]]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)

a(n) = A005179(A262981(n)).

A000005(a(n)) = A262981(n).

Name clarified by David A. Corneth, Dec 10 2021

A140752 Indices of elements of A005179 that belong to A134865 (in order of their appearance in A134865). The sorted version of this sequence is given by A138394.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 16, 20, 24, 30, 48, 60, 64, 72, 80, 84, 100, 108, 126, 192, 162, 224, 189, 384, 448, 512, 576, 672, 11520

Offset: 1

Max Alekseyev, May 26 2008

Cf. A000005, A005179, A134865, A138394, A140753.

a(n) = A000005(A134865(n))

a(31)=11520 from Ray Chandler, Jun 30 2008

A140753 Subsequence of elements of A005179 that appear in A134865.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 2520, 5040, 7560, 10080, 15120, 20160, 45360, 50400, 100800, 352800, 705600, 332640, 665280, 4324320, 8648640, 17297280, 21621600, 43243200, 13492656777600

Offset: 1

Max Alekseyev, May 26 2008

Rearrangement of elements of A134865 in order of their appearance in A005179. Hence A134865 represents a sorted variant of this sequence.

Cf. A000005, A005179, A134865, A138394, A140752.

a(n) = A005179(A138394(n))

a(31)=13492656777600 from Ray Chandler, Jun 30 2008

A328897 Odd numbers k > 1 such that A005179(k-1) > A005179(k) < A005179(k+1).

Original entry on oeis.org

27, 45, 75, 105, 117, 135, 147, 165, 187, 189, 231, 243, 245, 275, 285, 297, 315, 333, 345, 357, 375, 387, 403, 405, 423, 425, 427, 429, 435, 437, 459, 473, 495, 507, 525, 555, 567, 583, 585, 605, 621, 627, 637, 663, 665, 675, 693, 729, 731, 735, 741, 763, 765, 775, 777, 795

Offset: 1

Jianing Song, Oct 30 2019

nonn

As only square numbers have an odd number of divisors, for odd k, A005179(k) is usually larger than either A005179(k-1) or A005179(k+1) (or both). This sequence lists the exceptions. There are 71 terms below 10^3, 963 terms below 10^4, 11179 terms below 10^5. It seems that the density of this sequence over all the odd numbers is > 0.2.

Is there any odd k such that A005179(k) is smaller than A005179(k-3), A005179(k-1), A005179(k+1) and A005179(k+3)? There is no such k < 10^5.

			27 is a term because the smallest number with 27 divisors is 900, which is smaller than both A005179(26) = 12288 and A005179(28) = 960, so 27 is a term.
45 is a term because the smallest number with 45 divisors is 3600, which is smaller than both A005179(44) = 15360 and A005179(46) = 12582912, so 45 is a term.

Jianing Song, Table of n, a(n) for n = 1..11179 (all terms below 10^5)

Cf. A005179, A339863.

A := [seq(A005179(n), n=1..800)];
isA := k -> k::odd and A[k] < A[k-1] and A[k] < A[k+1]:
select(isA, [$3..799]); # Peter Luschny, Oct 30 2019

isA328897(k) = (k%2&&k>1) && A005179(k)<A005179(k-1) && A005179(k)<A005179(k+1) \\ Corrected by Jianing Song, Dec 05 2021

A339863 Odd numbers k > 1 such that A005179(k-1) > A005179(k) < A005179(k+1) > A005179(k+2) < A005179(k+3).

Original entry on oeis.org

187, 243, 403, 423, 425, 427, 435, 583, 663, 729, 763, 775, 845, 891, 1003, 1083, 1125, 1265, 1267, 1375, 1395, 1419, 1545, 1573, 1575, 1615, 1643, 1645, 1755, 1771, 1813, 1843, 1885, 1925, 1953, 2035, 2275, 2385, 2403, 2523, 2525, 2533, 2635, 2673, 2695

Offset: 1

Jianing Song, Dec 19 2020

nonn

Odd numbers k such that k and k+2 are both in A328897.

Despite the fact that only square numbers have an odd number of divisors, there are surprisingly many terms here. The numbers of terms below 10^3, 10^4 and 10^5 are 14, 208 and 3004 respectively.

			The smallest numbers with exactly 582, 583, 584, 585 and 586 divisors are ~3.565*10^30, ~2.659*10^20, ~4.958*10^24, 406425600 and ~2.387*10^88 respectively. We have A005179(582) > A005179(583) < A005179(584) > A005179(585) < A005179(586), hence 583 is a term.

Matthew House, Table of n, a(n) for n = 1..10000 (terms 1..3004 below 10^5 from Jianing Song)

Cf. A005179, A328897.

isA339863(k) = if(k%2&&k>1, my(v=vector(5, n, A005179(k-2+n))); v[2]

A122813 Terms in A005179 where prime signature differs from that of corresponding term in A038547.

Original entry on oeis.org

24, 360, 2520, 7560, 10080, 15120, 50400, 60480, 83160, 110880, 166320, 352800, 967680, 332640, 3870720, 554400, 665280, 2822400, 1081080, 61931520, 1441440, 247726080, 2162160, 3880800, 10644480, 3963617280, 4324320, 42577920, 7207200

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1371

Cf. A005179, A038547, A046523, A122812, A122814.

a(n) = A005179(A122812(n)).

cp's OEIS Frontend

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

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A072066 Exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m).

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A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

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A110821 SuperRefactorable numbers: m=A005179(n) such that k=m/n is an integer.

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A262983 Terms of A005179 divisible by their indices in order of appearance in A005179.

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A140752 Indices of elements of A005179 that belong to A134865 (in order of their appearance in A134865). The sorted version of this sequence is given by A138394.

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A140753 Subsequence of elements of A005179 that appear in A134865.

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A328897 Odd numbers k > 1 such that A005179(k-1) > A005179(k) < A005179(k+1).

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