A007416 -id:A007416 - cp's OEIS frontend

A140635 Smallest positive integer having the same number of divisors as n.

1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 36, 2, 6, 6, 24, 2, 24, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6, 24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2

Offset: 1

Max Alekseyev, May 19 2008

nonn

a(n) <= n for all n. Moreover, a(n) = n if and only if n belongs to A005179 or A007416.

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

Cf. A000005, A005179, A007416, A139770.

Cf. A019505, A138113, A061300 (sequences that can be defined in terms of this sequence).

a140635[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]!=DivisorSigma[0, #]&]
a140635[{m_, n_}] := Map[a140635, Range[m, n]]
a140635[{1, 89}] (* Hartmut F. W. Hoft, Jun 13 2023 *)

A140635(n) = { my(nd = numdiv(n)); for (i=1, n, if (numdiv(i) == nd, return (i))); }; \\ After A139770, Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n):
    x=d(n)
    m=1
    while True:
        if d(m)==x: return m
        else: m+=1 # Indranil Ghosh, May 27 2017

a(n) = A005179(A000005(n)).

A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241

Offset: 1

Gus Wiseman, Mar 18 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.

			The terms together with their prime indices begin:
      3: {2}
      5: {3}
      6: {1,2}
      9: {2,2}
     10: {1,3}
     11: {5}
     12: {1,1,2}
     17: {7}
     18: {1,2,2}
     20: {1,1,3}
     22: {1,5}
     24: {1,1,1,2}
     25: {3,3}
     27: {2,2,2}
     31: {11}
     34: {1,7}
     36: {1,1,2,2}

Subset of A302540.
Subset of A336101 = powers of 2 times powers of primes.
Positions of 2's in A370820.
Counting prime factors instead of divisors gives A371287.
A000005 counts divisors.
A000961 lists powers of primes, A302596 of prime index.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A005179, A007416, A303975, A319899, A355739, A370348, A370802, A371165-A371170, A371178.

Select[Range[100],Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==2&]

A275700 a(n) = Product_{d|n} prime(d).

Original entry on oeis.org

2, 6, 10, 42, 22, 390, 34, 798, 230, 1914, 62, 101010, 82, 4386, 5170, 42294, 118, 547170, 134, 951258, 12410, 14694, 166, 170807910, 2134, 24846, 23690, 3285114, 218, 660741510, 254, 5540514, 42470, 49206, 55726, 21399271530, 314, 65526, 68470, 3126785046, 358

Offset: 1

Jaroslav Krizek, Aug 05 2016

nonn

a(n) mod n = 0 for n: 1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, ...

			a(4) = 42 because the divisors of 4 are: 1, 2 and 4; and prime(1) * prime(2) * prime(4) = 2 * 3 * 7 = 42.

Cf. A007445 (Sum_{d|n} prime(d)).
A version for binary indices is A034729.
Partitions of this type are counted by A054973, strict case of A371284.
The sorted version is A371283, squarefree case of A371288.
These numbers have products A371286, unsorted version A371285.
A000005 counts divisors, row-lengths of A027750.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A001221, A005179, A007416.

[(&*[NthPrime(d): d in Divisors(n)]): n in [1..100]]

Table[Times@@(Prime[#]&/@Divisors[n]),{n,50}] (* Harvey P. Dale, Jun 16 2017 *)

a(n) = my(d=divisors(n)); prod(i=1, #d, prime(d[i])) \\ Felix Fröhlich, Aug 05 2016

use ntheory ":all"; sub a275700 { vecprod(map { nth_prime($) } divisors($[0])); } # Dana Jacobsen, Aug 09 2016

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: A101296(i) = A101296(j) => a(i) = a(j).

For all i, j: a(i) = a(j) <=> A000005(i) = A000005(j).

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

Cf. A000005, A007416 (positions of records, and also the first occurrence of each n).

Cf. also A101296, A286603, A286610, A286619, A286621, A286622, A286626, A286378.

With[{nn = 119}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[0, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000005(n) = numdiv(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000005(n))),"b286605.txt");

A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288

Offset: 1

Gus Wiseman, Mar 22 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.

			The prime indices of 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Joseph Likar, Table of n, a(n) for n = 1..10000

The squarefree case is A371283, unsorted version A275700.
Partitions of this type are counted by A371284, strict A054973.
Products of squarefree terms are A371286, unsorted version A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Union[prix[#]]==Divisors[Max@@prix[#]]&]

A099316 Greatest 3-smooth number dividing the n-th minimal number.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 24, 36, 48, 12, 64, 24, 144, 36, 192, 48, 72, 576, 144, 24, 36, 192, 1024, 36, 1296, 48, 72, 576, 3072, 144, 4096, 144, 5184, 36, 1296, 192, 216, 9216, 288, 12288, 576, 432, 3072, 576, 144, 5184, 72, 1296, 36864, 36, 1296, 9216, 46656, 288

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, Table of n, a(n) for n = 1..20000 (first 1000 terms from Amiram Eldar)

Cf. A007416, A065331, A099315, A003586, A099312, A099314.

a(n) = A065331(A007416(n)).

A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

Original entry on oeis.org

2, 6, 10, 22, 34, 42, 62, 82, 118, 134, 166, 218, 230, 254, 314, 358, 382, 390, 422, 482, 554, 566, 662, 706, 734, 798, 802, 862, 922, 1018, 1094, 1126, 1174, 1198, 1234, 1418, 1478, 1546, 1594, 1718, 1754, 1838, 1914, 1934, 1982, 2062, 2126, 2134, 2174, 2306

Offset: 1

Gus Wiseman, Mar 21 2024

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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.

			The terms together with their prime indices begin:
     2: {1}
     6: {1,2}
    10: {1,3}
    22: {1,5}
    34: {1,7}
    42: {1,2,4}
    62: {1,11}
    82: {1,13}
   118: {1,17}
   134: {1,19}
   166: {1,23}
   218: {1,29}
   230: {1,3,9}
   254: {1,31}
   314: {1,37}
   358: {1,41}
   382: {1,43}
   390: {1,2,3,6}

Joseph Likar, Table of n, a(n) for n = 1..10000

Partitions of this type are counted by A054973.
The unsorted version is A275700.
These numbers have products A371286, unsorted version A371285.
Squarefree case of A371288, counted by A371284.
A000005 counts divisors.
A001221 counts distinct prime factors.
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, A005179, A007416, A034729, A370820, A371131, A371177.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SameQ[prix[#],Divisors[Last[prix[#]]]]&]

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 42, 8, 36, 20, 22, 24, 390, 84, 60, 16, 34, 72, 798, 40, 252, 44, 230, 48, 100, 780, 216, 168, 1914, 120, 62, 32, 132, 68, 420, 144, 101010, 1596, 2340, 80, 82, 504, 4386, 88, 360, 460, 5170, 96, 1764, 200, 204, 1560, 42294, 432, 220, 336

Offset: 1

Gus Wiseman, Mar 21 2024

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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.

			The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
          1: {}
          2: {1}
          6: {1,2}
          4: {1,1}
         10: {1,3}
         12: {1,1,2}
         42: {1,2,4}
          8: {1,1,1}
         36: {1,1,2,2}
         20: {1,1,3}
         22: {1,5}
         24: {1,1,1,2}
        390: {1,2,3,6}
         84: {1,1,2,4}
         60: {1,1,2,3}
         16: {1,1,1,1}
         34: {1,7}
         72: {1,1,1,2,2}

Product of A275700 applied to each prime index.
The squarefree case is also A275700.
The sorted version is A371286.
A000005 counts divisors.
A001221 counts distinct prime factors.
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, A003963, A005179, A007416, A034729, A054973, A056239, A321898, A370820, A371165, A371181, A371284, A371288.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]

If n = prime(x_1)*...*prime(x_k) then a(n) = A275700(x_1)*...*A275700(x_k).

A371286 Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 60, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 118, 120, 124, 128, 132, 134, 136, 144, 160, 164, 166, 168, 176, 192, 200, 204, 216, 218, 220, 230, 236, 240, 248, 252, 254, 256, 264, 268, 272, 288

Offset: 1

Gus Wiseman, Mar 22 2024

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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.

			The terms together with their prime factorizations and unique factorizations into terms of A275700 begin:
   1 =             = ()
   2 = 2           = (2)
   4 = 2*2         = (2*2)
   6 = 2*3         = (6)
   8 = 2*2*2       = (2*2*2)
  10 = 2*5         = (10)
  12 = 2*2*3       = (2*6)
  16 = 2*2*2*2     = (2*2*2*2)
  20 = 2*2*5       = (2*10)
  22 = 2*11        = (22)
  24 = 2*2*2*3     = (2*2*6)
  32 = 2*2*2*2*2   = (2*2*2*2*2)
  34 = 2*17        = (34)
  36 = 2*2*3*3     = (6*6)
  40 = 2*2*2*5     = (2*2*10)
  42 = 2*3*7       = (42)
  44 = 2*2*11      = (2*22)
  48 = 2*2*2*2*3   = (2*2*2*6)
  60 = 2*2*3*5     = (6*10)
  62 = 2*31        = (62)
  64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
  68 = 2*2*17      = (2*34)
  72 = 2*2*2*3*3   = (2*6*6)
  80 = 2*2*2*2*5   = (2*2*2*10)
  82 = 2*41        = (82)
  84 = 2*2*3*7     = (2*42)
  88 = 2*2*2*11    = (2*2*22)
  96 = 2*2*2*2*2*3 = (2*2*2*2*6)

Products of elements of A275700.
The squarefree case is A371283.
The unsorted version is A371285.
A000005 counts divisors.
A001221 counts distinct prime factors.
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, A003963, A005179, A007416, A034729, A054973, A056239, A370820, A371284, A371288.

nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1, {{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
s=Table[Times@@Prime/@Divisors[n],{n,nn}];
m=Max@@Table[Select[Range[2,k],prix[#] == Divisors[Last[prix[#]]]&],{k,nn}];
Join@@Position[Table[Length[Select[facs[n], SubsetQ[s,Union[#]]&]],{n,m}],1]

A036451 Maximal value of d(x) (the number of divisors of x, A000005) if the binary order (see A029837) of x, the value g(x) = n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 120, 144, 168, 200, 240, 288, 360, 432, 504, 600, 720, 864, 1008, 1152, 1344, 1600, 1920, 2304, 2688, 3072, 3584, 4096, 4800, 5760, 6720, 7680, 8640, 10080, 11520, 13824, 16128, 18432, 20736, 23040

Offset: 0

Labos Elemer

nonn

g(x) <= n can be replaced by g(x) = n.

			In the range of g(x) <= 5, the values of d(x) can be: 1, 2, 3, 4, 5, 6, 8 of which 8 is the maximal, so a(n) = a(g(x)) = 8.

Giovanni Resta, Table of n, a(n) for n = 0..250 (based on A002182 b-file)

Cf. A000005, A002182, A029837, A005179, A007416, A036470-A036472.

Max /@ Table[DivisorSigma[0, Floor[2^(n - 1) + k]], {n, 0, 22}, {k, Ceiling[2^(n - 1)]}] (* Michael De Vlieger, May 10 2017 *)

a(22)-a(32) from Alex Ratushnyak, Jun 06 2013

a(33)-a(49) from Giovanni Resta, Jun 06 2013

cp's OEIS Frontend

A140635 Smallest positive integer having the same number of divisors as n.

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A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

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A275700 a(n) = Product_{d|n} prime(d).

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A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

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A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

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A099316 Greatest 3-smooth number dividing the n-th minimal number.

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A371283 Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.

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Mathematica

A371285 Heinz number of the multiset union of the divisor sets of each prime index of n.

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