A007416 -id:A007416 - cp's OEIS frontend

A291834 First positions of records in A252665.

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 432, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 25200, 30240, 40320, 50400, 60480, 75600, 80640, 90720, 100800, 120960, 151200, 181440

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.
The terms 2520, 7560, 25200, 221760, 665280, 8648640, ... are not in A033833 but are in A002182. The term 3600 is the smallest that is in neither A033833 nor A002182, but in A007416. The term 831600 is the smallest that is in none of the three aforementioned sequences.
Conjectures based on a(n) < 10^7:
1. Numbers in a(n) are products of the first several consecutive primes p.
2. Outside of a(1), the least prime factor of a(n) has multiplicity > 1. This implies no primes, primorials, or squarefree a(n) for n > 1.
3. The greatest prime factor of a(n) generally has multiplicity 1. Note, however, exceptions in a(n) for n = {1, 2, 3, 5, 7, 9, 12, 13, 15, 17, 19, 26, 29, 33, 73, ...}.
4. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.
See "Records and first positions of records in A252665" for more information. - Michael De Vlieger, Sep 03 2017

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A002182, A007416, A033833, A252665, A291833.

With[{s = Array[f[#, #, 5] &, 10^4]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]

A371181 Sorted list of positions of first appearances in the sequence A370820, which counts distinct divisors of prime indices.

Original entry on oeis.org

1, 2, 3, 7, 13, 37, 53, 89, 151, 223, 281, 311, 659, 827, 1069, 1163, 1511, 2045, 2423, 3241, 4211, 5443, 6473, 6997, 7561, 9037, 10271, 10627, 14323, 17611, 26203, 28181, 33613, 50543, 88099, 88483, 95603, 98965, 122119, 168281, 192709, 305107, 309073, 420167

Offset: 1

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.

			The terms together with their prime indices begin:
       1: {}
       2: {1}
       3: {2}
       7: {4}
      13: {6}
      37: {12}
      53: {16}
      89: {24}
     151: {36}
     223: {48}
     281: {60}
     311: {64}
     659: {120}
     827: {144}
    1069: {180}
    1163: {192}
    1511: {240}
    2045: {3,80}
    2423: {360}
    3241: {4,90}
    4211: {576}
    5443: {720}
    6473: {840}
    6997: {900}
    7561: {960}
    9037: {4,210}

Counting prime factors instead of divisors (see A303975) gives A062447(>0).
The unsorted version is A371131.
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. A000079, A000720, A000792, A002110, A005179, A007416, A355739, A370348, A370802, A370808, 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}]]//Sort

A036484 a(n) is the minimal number of binary order n which has maximal number of divisors in this interval.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 60, 120, 240, 360, 840, 1680, 2520, 7560, 15120, 27720, 55440, 110880, 221760, 498960, 720720, 1441440, 3603600, 7207200, 14414400, 32432400, 61261200, 122522400, 245044800, 367567200, 735134400, 2095133040

Offset: 0

Labos Elemer

Compare with A007416, where terms of this sequence are present.

			For n=9, with 256 < k <= 512, d(k) takes 17 distinct values, of which d(k)=24 is the greatest (see A036451 and A036470) and occurs first at k=360, so a(9)=360.

Cf. A000005, A029837, A005179, A007416, A036470, A036492, A036493.

Block[{nn = 22, s}, s = TakeList[Array[DivisorSigma[0, # + 1] &, 2^nn - 1], 2^Range[0, nn - 1]]; {1}~Join~Map[2^(#1 - 1) + #2 & @@ FirstPosition[s, #] &, Map[Max, s]]] (* Michael De Vlieger, Nov 04 2020 *)

a(22)-a(31) from Sean A. Irvine, Nov 04 2020

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 12, 30, 60, 120, 240, 504, 840, 1680, 3960, 7560, 15120, 32760, 65520, 131040, 262080, 498960, 997920, 1965600, 3603600, 7207200, 14414400, 32432400, 64864800, 122522400, 245044800, 514594080, 1029188160, 2095133040, 4227022800, 8454045600

Offset: 0

Labos Elemer

nonn

This sequence differs from A036451 only at n = 3, 5, 9, 12, and 15, which are the values of n for which there exists more than one k such that g(k) = n and d(k) has the maximum possible value.

a(n) is the largest term k in A067128 such that log_2(k) <= n. - Jon E. Schoenfield, May 13 2018

			For n = 9, k is in {257, 512}, max(d(k)) = 24 (see A036451); this holds for four different numbers (360, 420, 480, and 504); a(9) = 504 since it is the largest.

Cf. A000005, A029837, A005179, A007416, A036451, A036470, A036484.

{1}~Join~Table[Max@ MaximalBy[Range[2^(n - 1) + 1, 2^n], DivisorSigma[0, #] &], {n, 24}] (* Michael De Vlieger, Aug 01 2017 *)

a(22)-a(24) from Michael De Vlieger, Aug 01 2017

a(25)-a(33) from Jon E. Schoenfield, May 12 2018

A095921 Intersection of A002182 (highly composite numbers) and A003418 (LCMs of first n positive integers).

Original entry on oeis.org

1, 2, 6, 12, 60, 840, 2520, 27720, 720720, 80313433200

Offset: 1

Matthew Vandermast, Jul 11 2004

Also the intersection of A003418 and A007416.

J. Roberts, Lure Of The Integers. Math. Assoc. Of America, 1992, p. 86.

M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

Cf. A055492.

A099317 Minimal numbers having in canonical prime factorization at least one factor p^e such that e+1 is not prime, p prime and e>0.

Original entry on oeis.org

24, 120, 360, 840, 2520, 7560, 10080, 15120, 27720, 50400, 60480, 83160, 110880, 166320, 332640, 352800, 554400, 665280, 967680, 1081080, 1441440, 2162160, 2822400, 3870720, 3880800, 4324320, 7207200, 8648640, 10644480, 10810800, 17297280, 21621600, 31046400

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

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

A000005(a(n)) = A072066(m) for some m.

			A007416(38) = A005179(64) = 7560 = 2^3*3^3*5*7;
A007416(39) = A005179(72) = 10080 = 2^5*3^2*5*7.

David A. Corneth and Amiram Eldar, Table of n, a(n) for n = 1..9000 (terms <= 10^45)

Cf. A000005, A005179, A007416, A072066.

More terms from Amiram Eldar, Jul 09 2019

A099318 Smallest k>1 such that k*(n-th minimal number) is also a minimal number.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 3, 2, 4, 2, 3, 3, 2, 5, 4, 2, 4, 3, 3, 2, 4, 3, 2, 5, 3, 4, 3, 2, 5, 4, 4, 3, 2, 4, 2, 3, 7, 3, 3, 3, 2, 5, 2, 4, 5, 4, 4, 4, 5, 2, 2, 3, 3, 3, 4, 2, 7, 3, 2, 7, 2, 2, 5, 5, 3, 3, 4, 4, 5, 3, 2, 3, 5, 2, 2, 4, 3, 4, 2, 4, 3, 4, 2

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

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

Is a(n) in A000961? - David A. Corneth, Aug 31 2019

			A007416(38) = A005179(64) = 7560 = 2^3*3^3*5*7;

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 844 terms from Amiram Eldar)

Cf. A000961, A005179, A007416.

a(n)*A007416(n) = A007416(m) for some m.

More terms from Amiram Eldar, Aug 31 2019

A099385 Number of partitions of 2*n into minimal numbers.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 24, 33, 46, 62, 82, 106, 138, 174, 220, 274, 339, 414, 507, 611, 737, 881, 1049, 1239, 1466, 1717, 2012, 2344, 2724, 3148, 3641, 4182, 4806, 5499, 6282, 7148, 8137, 9216, 10441, 11789, 13297, 14951, 16814, 18837, 21106, 23584, 26331

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

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

			a(4) = A099383(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. A099386, A000041, A007416, A005179.

A099386 Number of partitions of 2*n into distinct minimal numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 4, 3, 4, 5, 3, 4, 5, 3, 5, 6, 4, 6, 7, 5, 8, 9, 6, 9, 9, 6, 10, 10, 8, 12, 11, 9, 12, 11, 9, 12, 11, 9, 13, 12, 10, 14, 12, 10, 14, 12, 11, 15, 14, 13, 17, 16, 14, 18, 15, 13, 18, 15

Offset: 0

Reinhard Zumkeller, Oct 14 2004

nonn

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

a(n) = A099384(2*n) = A099384(2*n+1).

			a(9) = A099384(18) = #{16+2, 12+6, 12+4+2} = 3.

Cf. A099384, A099385, A000009, A007416, A005179.

A131903 Integers x such that d(k)=d(x) for some 0A000005 is the number of divisors.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Complement of A007416. - T. D. Noe, Jul 26 2007

			This sequence contains 8 because 8 has |{1,2,4,8}|=4 divisors and 6<8 has |{1,2,3,6}|=4 divisors.

Cf. A069822, A131902-A131908.

Clear[tmp]; Function[n, If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], n] & [tmp[DivisorSigma[0, n]]]] /@ Range[64]

isok(n) = {my(nd = numdiv(n)); for (k=1, n-1, if (numdiv(k) == nd, return (1)););}

a(n) = n-th element of the set {x>0 : there exists a k with 0A000005 is the number of divisors.

a(54)-a(67) from Michel Marcus, Apr 03 2015

Edited by Danny Rorabaugh, Apr 03 2015

cp's OEIS Frontend

A291834 First positions of records in A252665.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A371181 Sorted list of positions of first appearances in the sequence A370820, which counts distinct divisors of prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A036484 a(n) is the minimal number of binary order n which has maximal number of divisors in this interval.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A095921 Intersection of A002182 (highly composite numbers) and A003418 (LCMs of first n positive integers).

Views

Author

Keywords

Comments

References

Links

Crossrefs

A099317 Minimal numbers having in canonical prime factorization at least one factor p^e such that e+1 is not prime, p prime and e>0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A099318 Smallest k>1 such that k*(n-th minimal number) is also a minimal number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A099385 Number of partitions of 2*n into minimal numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

A099386 Number of partitions of 2*n into distinct minimal numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs