A119313 -id:A119313 - cp's OEIS frontend

A119314 Complement of A119313.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 47, 49, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 76, 79, 80, 81, 83, 88, 89, 92, 97, 99, 100, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 136, 137

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) <= 1 or (A067029(m) > 1 and A020639(m)^2 <= A119288(m)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A119315 and A008578.
(Intersection with A119316) = A008578.
A000961 and A092259 are subsequences.
Cf. A001221, A020639, A067029, A119288, A119313.

Select[Range[140], !CompositeQ[#] || ((f = FactorInteger[#])[[1, 2]] > 1 && (Length[f] == 1 || f[[1, 1]]^2 < f[[2, 1]])) &] (* Amiram Eldar, Jul 02 2022 *)

A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 04 2022

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 a(n) gapless divisors of n = 1..24:
  1  2  1  4  1  6  1  8  1  2  1  12  1  2  1  16  1  18  1  4  1  2  1  24
     1     2     2     4     1     6      1     8      6      2     1     12
           1     1     2           4            4      2      1           8
                       1           2            2      1                  6
                                   1            1                         4
                                                                          2
                                                                          1
For example, the divisors of 12 are {1,2,3,4,6,12}, of which {1,2,4,6,12} belong to A055932, so a(12) = 5.

These divisors belong to A055932, a subset of A073491 (complement A073492).
The complement is A356225.
A001223 lists the prime gaps.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A000005, A001222, A028334, A029709, A055874, A056239, A070824, A112798, A119313, A137921, A287170, A289509, A356223.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
Table[Length[Select[Divisors[n],normQ[primeMS[#]]&]],{n,100}]

A356225 Number of divisors of n whose prime indices do not cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 1, 2, 3, 0, 1, 2, 1, 3, 3, 2, 1, 1, 2, 2, 3, 3, 1, 4, 1, 0, 3, 2, 3, 2, 1, 2, 3, 4, 1, 5, 1, 3, 5, 2, 1, 1, 2, 4, 3, 3, 1, 3, 3, 4, 3, 2, 1, 5, 1, 2, 5, 0, 3, 5, 1, 3, 3, 6, 1, 2, 1, 2, 5, 3, 3, 5, 1, 5, 4, 2, 1, 7, 3, 2, 3

Offset: 1

Gus Wiseman, Aug 13 2022

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 a(70) = 6 divisors: 5, 7, 10, 14, 35, 70.

These divisors belong to the complement of A055932, a subset of A073491.
These divisors belong to A080259, a superset of A073492.
The complement is counted by A356224.
A001223 lists the prime gaps.
A328338 has third-largest divisor prime, smallest A119313.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A000005, A001222, A028334, A055874, A056239, A070824, A112798, A137921, A287170, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
Table[Length[Select[Divisors[n],!normQ[primeMS[#]]&]],{n,100}]

a(n) = A000005(n) - A356224(n).

A328166 Heinz number of the run-lengths of the divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 4, 10, 4, 12, 8, 12, 4, 28, 4, 12, 16, 24, 4, 40, 4, 36, 16, 12, 4, 112, 8, 12, 16, 48, 4, 120, 4, 48, 16, 12, 16, 224, 4, 12, 16, 144, 4, 120, 4, 48, 64, 12, 4, 448, 8, 48, 16, 48, 4, 160, 16, 144, 16, 12, 4, 832, 4, 12, 64, 96, 16, 160, 4, 48, 16

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

			Splitting the divisors of 30 into runs gives {{1, 2, 3}, {5, 6}, {10}, {15}, {30}}, and the Heinz number of {1, 1, 1, 2, 3} is 120, so a(30) = 120.
More examples from _Antti Karttunen_, Dec 09 2021: (Start)
Splitting the divisors of 1 into runs gives {1}, and the Heinz number of that is 2.
Splitting the divisors of 2 into runs gives {1, 2}, and the Heinz number of that is 3. [one run of length 2, therefore a(2) = prime(2)^1].
Splitting the divisors of 3 into runs gives {1} and {3}, and the Heinz number of that is 4. [two runs of length 1, therefore a(3) = prime(1)^2].
Splitting the divisors of 4 into runs gives {1, 2} and {4}, and the Heinz number of that is 6. [one run of length 1, and other run of length 2, therefore a(4) = prime(1)*prime(2)].
Splitting the divisors of 5 into runs gives {1} and {5}, and the Heinz number of that is 4. [two runs of length 1, therefore a(5) = prime(1)^2].
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Wikipedia, Run (cards)
Index entries for sequences related to Heinz numbers

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the set of divisors of n is A275700(n).
Numbers whose divisors do not have weakly decreasing run-lengths are A328165.
Cf. A000005, A056239, A060680, A060775, A119313, A137921, A181063, A199970.

Table[Times@@Prime/@Length/@Split[Divisors[n],#2==#1+1&],{n,30}]

A328166(n) = { my(rl=0,pd=0,v=vector(numdiv(n)),m=1); fordiv(n, d, if(d>(1+pd), v[rl]++; rl=0); pd=d; rl++); v[rl]++; for(i=1,#v, m *= prime(i)^v[i]); (m); }; \\ Antti Karttunen, Dec 09 2021

A001222(a(n)) = A137921(n).

A056239(a(n)) = A000005(n).

A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 28 2022

nonn

First differs from A000005 at 10, 14, 20, 21, 22, ... = A307516.

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 a(n) counted divisors of n = 1, 2, 4, 6, 12, 16, 24, 30, 36, 48, 72, 90:
  1   2   4   6  12  16  24  30  36  48  72  90
      1   2   3   6   8  12  15  18  24  36  45
          1   2   4   4   8   6  12  16  24  30
              1   3   2   6   5   9  12  18  18
                  2   1   4   3   6   8  12  15
                  1       3   2   4   6   9   9
                          2   1   3   4   8   6
                          1       2   3   6   5
                                  1   2   4   3
                                      1   3   2
                                          2   1
                                          1

These divisors belong to A073491, a superset of A055932, complement A073492.
The initial case is A356224.
The complement in the initial case is counted by A356225.
A000005 counts divisors.
A001223 lists the prime gaps.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A328338 has third-largest divisor prime.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A028334, A029709, A055874, A070824, A119313, A137921, A287170, A307516, A356223, A356233-A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Divisors[n],nogapQ[primeMS[#]]&]],{n,100}]

A087134 Smallest number having exactly n divisors that are not greater than the number's greatest prime factor.

Original entry on oeis.org

1, 2, 6, 20, 42, 84, 156, 312, 684, 1020, 1380, 1860, 3480, 3720, 4920, 7320, 10980, 14640, 16920, 21960, 26280, 34920, 45720, 59640, 69840, 89880, 106680, 125160, 145320, 177240, 213360, 244440, 269640, 354480, 320040, 375480, 435960, 456120, 531720, 647640

Offset: 1

Reinhard Zumkeller, Aug 17 2003

nonn

A087133(a(n))=n.
Also smallest number such that the n-th divisor is prime. - Reinhard Zumkeller, May 15 2006
From David A. Corneth, Jan 22 2019: (Start)
For the first 10000 terms except 1, a(n) is of the form A025487(k) * p where p is the smallest prime larger than the n-th divisor and, if the (n+1)-th divisor exists, less than that divisor.
This sequence isn't a sequence of indices of records to A087133 as it's not monotonically increasing; 354480 = a(34) > a(35) = 320040. (End)

			a(3) = A119313(1) = 6.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A025487, A087133, A119311, A119312.

See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{s = Array[Function[{d, p}, LengthWhile[d, # < p &]] @@ {#, SelectFirst[Reverse@ #, PrimeQ]} &@ Divisors@ # &, 10^6]}, Array[FirstPosition[s, #][[1]] &, Max@ s + 1, 0]] (* Michael De Vlieger, Jan 23 2019 *)

a087133(n) = if (n==1, 1, my(f = factor(n), gpf = f[#f~,1]); sumdiv(n, d, d <= gpf));
a(n) = my(k = 1); while (a087133(k) != n, k++); k; \\ Michel Marcus, Sep 21 2014

More terms from Reinhard Zumkeller, May 15 2006

More terms from Michel Marcus, Sep 21 2014

A328165 Numbers whose divisors do not have weakly decreasing run-lengths.

Original entry on oeis.org

56, 72, 110, 112, 132, 144, 156, 182, 210, 216, 224, 240, 264, 272, 288, 306, 312, 342, 364, 380, 392, 396, 420, 432, 440, 448, 462, 468, 480, 506, 528, 544, 550, 552, 576, 600, 612, 616, 624, 648, 650, 684, 702, 720, 728, 756, 760, 770, 780, 784, 792, 812

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, with runs {{1, 2}, {4}, {7, 8}, {14}, {28}, {56}}, with lengths (2, 1, 2, 1, 1, 1), which are not weakly decreasing, so 56 is in the sequence.

Wikipedia, Run (cards)

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A000005, A027750, A033676, A060680, A060775, A119313, A181063, A199970.

Select[Range[1000],!GreaterEqual@@Length/@Split[Divisors[#],#2==#1+1&]&]

A328338 Numbers whose third-largest divisor is prime.

Original entry on oeis.org

6, 8, 10, 14, 15, 20, 21, 22, 26, 27, 28, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 68, 69, 74, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134, 141, 142, 143, 145, 146, 148

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

			The sequence of terms together with their divisors begins:
   6: {1,2,3,6}
   8: {1,2,4,8}
  10: {1,2,5,10}
  14: {1,2,7,14}
  15: {1,3,5,15}
  20: {1,2,4,5,10,20}
  21: {1,3,7,21}
  22: {1,2,11,22}
  26: {1,2,13,26}
  27: {1,3,9,27}
  28: {1,2,4,7,14,28}
  33: {1,3,11,33}
  34: {1,2,17,34}
  35: {1,5,7,35}
  38: {1,2,19,38}
  39: {1,3,13,39}
  44: {1,2,4,11,22,44}
  46: {1,2,23,46}
  51: {1,3,17,51}
  52: {1,2,4,13,26,52}

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

A subset of A002808 and superset of A006881.
Numbers whose third-smallest divisor is prime are A119313.
Third-smallest divisor is A292269.
Cf. A000005, A020639, A027750, A033676, A060775, A067513, A088725.

q:= n-> (l-> nops(l)>2 and isprime(l[-3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[-3]]]&]

isA328338(n) = { my(u=numdiv(n)); ((u>2)&&isprime(divisors(n)[u-2])); }; \\ Antti Karttunen, Oct 17 2019

A356223 Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

Original entry on oeis.org

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

Offset: 1

Gus Wiseman, Aug 04 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

			We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

The first appearances are at A038664, seconds A356221.
Diagonal of A356222.
A001223 lists the prime gaps.
A073491 lists numbers with gapless prime indices.
A356224 counts divisors with gapless prime indices, complement A356225.
A356226 = gapless interval lengths of prime indices, run-lengths A287170.
Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

nn=1000;
gaps=Differences[Array[Prime,nn]];
Table[Position[gaps,2*n][[n,1]],{n,Select[Range[nn],Length[Position[gaps,2*#]]>=#&]}]

A119316 Complement of A119315.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A067029(m) = 1 or (A001221(m) > 1 and A119288(m) < A020639(m)^2).

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 77, 779, 7806, 78105, 780693, 7806565, 78062581, 780603128, 7806020219, ... . Apparently, the asymptotic density of this sequence exists and equals 0.780... . - Amiram Eldar, Jul 02 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A119313 and A008578.
(Intersection with A119314) = A008578.
Cf. A001221, A020639, A067029, A119288, A119315.

Select[Range[100], (f = FactorInteger[#])[[1, 2]] == 1 || (Length[f] > 1 && f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

cp's OEIS Frontend

A119314 Complement of A119313.

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A356224 Number of divisors of n whose prime indices cover an initial interval of positive integers.

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A356225 Number of divisors of n whose prime indices do not cover an initial interval of positive integers.

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A328166 Heinz number of the run-lengths of the divisors of n.

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A356069 Number of divisors of n whose prime indices cover an interval of positive integers (A073491).

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A087134 Smallest number having exactly n divisors that are not greater than the number's greatest prime factor.

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A328165 Numbers whose divisors do not have weakly decreasing run-lengths.

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Mathematica

A328338 Numbers whose third-largest divisor is prime.

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