A007416 -id:A007416 - cp's OEIS frontend

A099312 Exponent of greatest power of 2 dividing the n-th minimal number.

0, 1, 2, 1, 2, 4, 3, 2, 4, 2, 6, 3, 4, 2, 6, 4, 3, 6, 4, 3, 2, 6, 10, 2, 4, 4, 3, 6, 10, 4, 12, 4, 6, 2, 4, 6, 3, 10, 5, 12, 6, 4, 10, 6, 4, 6, 3, 4, 12, 2, 4, 10, 6, 5, 4, 6, 12, 16, 10, 3, 6, 10, 5, 6, 4, 4, 6, 12, 16, 6, 4, 10, 6, 18, 4, 10, 12, 5, 5, 10, 12, 4, 5, 16

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416.

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

Cf. A007416, A007814, A099311, A099314, A099316.

a(n) = A007814(A007416(n)).

A099314 Exponent of greatest power of 3 dividing the n-th minimal number.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 0, 2, 4, 1, 2, 2, 1, 2, 0, 2, 4, 2, 4, 1, 3, 2, 2, 1, 2, 3, 1, 2, 2, 4, 2, 4, 2, 2, 4, 2, 6, 2, 2, 3, 1, 0, 4, 3, 2, 1, 2, 4, 3, 2, 4, 2, 1, 2, 4, 2, 6, 0, 2, 2, 4, 3, 2, 4, 1, 4, 2, 2

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416.

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

Cf. A007416, A007949, A099313, A099312, A099316.

a(n) = A007949(A007416(n)).

A099383 Number of partitions of n into minimal numbers.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 16, 16, 24, 24, 33, 33, 46, 46, 62, 62, 82, 82, 106, 106, 138, 138, 174, 174, 220, 220, 274, 274, 339, 339, 414, 414, 507, 507, 611, 611, 737, 737, 881, 881, 1049, 1049, 1239, 1239, 1466, 1466, 1717, 1717, 2012, 2012, 2344, 2344

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416;

a(2*n) = a(2*n+1) = A099385(n).

			a(8) = #{6+2, 6+1+1, 4+4, 4+2+2, 4+2+1+1, 4+1+1+1+1, 2+2+2+2,
2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1} = 11.

Cf. A099384, A000041, A007416, A005179.

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

A233284 a(n) = largest m such that 1, 2, ..., m divide n-th Fibonacci number; a(n) = A055874(A000045(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 6, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 12

Offset: 1

Antti Karttunen, Dec 12 2013

nonn

It seems that the records occur at the positions given by A233283: 1, 3, 12, 60, 120, 840, 2520, 12600, ...

The corresponding record values begin as 1, 2, 4, 6, 12, 16, 24, 36, ... (maybe A007416?).

Antti Karttunen, Table of n, a(n) for n = 1..12600

Differs from A233285 for the first time at n=120, where a(120)=12, while A233285(120)=7.

Cf. A055874, A000045, A001175-A001177, A233283, A007416.

A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

Original entry on oeis.org

1, 2, 3, 7, 13, 53, 37, 311, 89, 151, 223, 2045, 281, 3241, 1163, 827, 659, 9037, 1069, 17611, 1511, 4211, 28181, 122119, 2423, 10627, 88483, 6997, 7561, 98965, 5443, 88099, 6473, 95603, 309073, 50543, 10271, 192709, 508051, 438979, 14323, 305107, 26203

Offset: 0

Gus Wiseman, Mar 20 2024

nonn

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.
Every nonnegative integer belongs to A370820, so this sequence is infinite.
Are there any terms with more than two prime factors?

			The terms together with their prime indices begin:
       1: {}
       2: {1}
       3: {2}
       7: {4}
      13: {6}
      53: {16}
      37: {12}
     311: {64}
      89: {24}
     151: {36}
     223: {48}
    2045: {3,80}
     281: {60}
    3241: {4,90}
    1163: {192}
     827: {144}
     659: {120}
    9037: {4,210}
    1069: {180}
   17611: {5,252}

Counting prime factors instead of divisors (see A303975) gives A062447(>0).
The sorted version is A371181.
A000005 counts divisors.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000720, A000792, A005179, A007416, A355739, A370348, A370802, A370808, A371130, A371165, A371177.

rnnm[q_]:=Max@@Select[Range[Min@@q,Max@@q],SubsetQ[q,Range[#]]&];
posfirsts[q_]:=Table[Position[q,n][[1,1]],{n,Min@@q,rnnm[q]}];
posfirsts[Table[Length[Union @@ Divisors/@PrimePi/@First/@If[n==1, {},FactorInteger[n]]],{n,1000}]]

f(n) = my(list=List(), f=factor(n)); for (i=1, #f~, fordiv(primepi(f[i,1]), d, listput(list, d))); #Set(list); \\ A370820
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, May 02 2024

A134865 Numbers k meeting the following criterion: if k is a multiple of d, then it is also a multiple of the smallest number with same number of divisors as d.

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, 332640, 352800, 665280, 705600, 4324320, 8648640, 17297280, 21621600, 43243200, 13492656777600

Offset: 1

J. Lowell, Jan 29 2008

Note that this is not a subsequence of A002182: 100800 is in this sequence but not in A002182. - J. Lowell, Feb 22 2008
A subset of A005179. - Max Alekseyev, May 19 2008
A number k is in this sequence iff for every divisor d of k, A005179(A000005(d)) (= A140635(d)) is also a divisor of k. So the question of the finiteness of this sequence is closely related to the form of the elements of A005179. - Max Alekseyev, May 19 2008, May 20 2008
Rearrangement of this sequence, forming a subsequence of A005179, is given by A140753. Corresponding indices of elements of A005179 are given by A138394 and A140752. - Max Alekseyev, May 26 2008
A subsequence of A007416 which is a subsequence of A025487, so every term is primally tight and even (after the first term). Thus if d is a divisor of a term, then the least integer with the same prime signature as d (=A046523(d)) is also a divisor. So only the divisors that are in A025487 need be tested. - Ray Chandler
a(32) > 8*10^25 if it exists. - David A. Corneth, Dec 10 2021

			60 is a multiple of 30 with 8 divisors, but not of 24 (the smallest number with 8 divisors) so 60 is not a term of this sequence.

Cf. A000005, A005179, A007416, A138394, A140752, A140753.

a = {}; For[n = 1, n < 10000, n++, b = Divisors[n]; c = 1; For[i = 1, i < Length[b] + 1, i++, j = 1; While[Length[Divisors[j]] < Length[Divisors[b[[i]]]], j++ ]; If[ ! Mod[n, j] == 0, c = 0]]; If[c == 1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Feb 05 2008 *)

isA134865(n)={ n%2 & return(n==1); fordiv(n, d, bigomega(d)>1 || next; nd=numdiv(d); for(k=4, d, numdiv(k)==nd || next; n%k & return; break)); 1 }
for(n=1,10^7,if(isA134865(n),print1(n,", "))); \\ R. J. Mathar, May 17 2008

a(n) = A005179(A140752(n)). - Max Alekseyev, May 26 2008

More terms from Stefan Steinerberger, Feb 05 2008
More terms from J. Lowell, Feb 22 2008
a(22)-a(30) from Don Reble, May 17 2008
a(31)=13492656777600 from Ray Chandler, Jun 30 2008

A166721 Squares for which no smaller square has the same number of divisors.

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 576, 900, 1024, 1296, 3600, 4096, 5184, 9216, 14400, 32400, 36864, 44100, 46656, 65536, 82944, 129600, 176400, 230400, 262144, 331776, 589824, 705600, 746496, 810000, 921600, 1166400, 1587600, 2073600, 2359296, 2822400, 2985984, 3240000

Offset: 1

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

From Jon E. Schoenfield, Mar 03 2018: (Start)
Numbers k^2 such there is no positive m < k such that A000005(m^2) = A000005(k^2).
Square terms in A007416. (End)

			The positive squares begin 1, 4, 9, 16, 25, 36, 49, 64, ..., and their corresponding numbers of divisors are 1, 3, 3, 5, 3, 9, 3, 7, ...; thus, a(1)=1, a(2)=4, 9 is not a term (it has the same number of divisors as does 4; the same is true of 25, 49, etc.), a(3)=16, a(4)=36, a(5)=64, ... - _Jon E. Schoenfield_, Mar 03 2018

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

Cf. A000005, A005179, A007416, A025487, A048691, A136404, A166722.

 Sort[Module[{nn=2000,tbl},tbl=Table[{n^2,DivisorSigma[0,n^2]},{n,nn}];Table[ SelectFirst[ tbl,#[[2]]==k&],{k,nn}]][[All,1]]/."NotFound"->Nothing] (* Harvey P. Dale, Jun 06 2022 *)

lista(nn) = {v = []; for (n=1, nn, d = numdiv(n^2); if (! vecsearch(v, d), print1(n^2, ", "); v = Set(concat(v, d))););} \\ Michel Marcus, Mar 04 2018

Proper definition and substantial editing by Jon E. Schoenfield, Mar 03 2018

A064787 Inverse permutation to A053212.

Original entry on oeis.org

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

A099384 Number of partitions of n into distinct minimal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 5, 3, 3, 4, 4, 5, 5, 3, 3, 5, 5, 6, 6, 4, 4, 6, 6, 7, 7, 5, 5, 8, 8, 9, 9, 6, 6, 9, 9, 9, 9, 6, 6, 10, 10, 10, 10, 8, 8, 12, 12, 11, 11, 9, 9, 12, 12, 11, 11, 9, 9, 12, 12, 11

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416;

a(2*n) = a(2*n+1) = A099386(n).

			a(18) = #{16+2, 12+6, 12+4+2} = 3.

Cf. A099383, A000009, A007416, A005179.

cp's OEIS Frontend

A099312 Exponent of greatest power of 2 dividing the n-th minimal number.

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A099314 Exponent of greatest power of 3 dividing the n-th minimal number.

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A099383 Number of partitions of n into minimal numbers.

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

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A233284 a(n) = largest m such that 1, 2, ..., m divide n-th Fibonacci number; a(n) = A055874(A000045(n)).

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A371131 Least number with exactly n distinct divisors of prime indices. Position of first appearance of n in A370820.

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A134865 Numbers k meeting the following criterion: if k is a multiple of d, then it is also a multiple of the smallest number with same number of divisors as d.

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A166721 Squares for which no smaller square has the same number of divisors.

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A064787 Inverse permutation to A053212.

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