A325240 -id:A325240 - cp's OEIS frontend

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Positions of 1's in A062977. Supersequence of A054753, A096156.
Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
Cf. A059956, A088453.

Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021

A051903(a(n)) - A051904(a(n)) = 1.

A244515 Number of partitions of n where the minimal multiplicity of any part is 2.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 4, 2, 6, 4, 9, 6, 16, 9, 23, 18, 34, 27, 51, 40, 75, 63, 103, 90, 152, 130, 208, 191, 286, 267, 402, 368, 546, 518, 730, 709, 998, 954, 1322, 1305, 1751, 1740, 2330, 2299, 3056, 3074, 3968, 4031, 5202, 5249, 6721, 6877, 8642, 8888, 11147, 11432, 14248, 14747, 18097, 18838, 23093, 23938, 29186, 30489

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

			From _Gus Wiseman_, Jul 03 2019: (Start)
The a(2) = 1 through a(12) = 9 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325240.
  11  22  33    22111  44      33111    55        33311      66
          2211         3311    2211111  3322      44111      4422
                       22211            4411      3311111    5511
                       221111           222211    221111111  33222
                                        331111               332211
                                        22111111             441111
                                                             2222211
                                                             33111111
                                                             2211111111
(End)

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column k = 2 of A243978.

Cf. A000041, A007690, A008284, A116608, A325240, A325242.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
a:= n-> b(n$2, 2) -b(n$2, 3):
seq(a(n), n=1..80);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1, k], {j, Max[1, k], n/i}]]];
a[n_] := b[n, n, 2] - b[n, n, 3];
Array[a, 80] (* Jean-François Alcover, May 01 2018, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[#]==2&]],{n,0,30}] (* Gus Wiseman, Jul 03 2019 *)

A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 75, 81, 105, 135, 225, 243, 315, 375, 405, 525, 675, 729, 735, 945, 1125, 1155, 1215, 1575, 1875, 2025, 2187, 2205, 2625, 2835, 3375, 3465, 3645, 3675, 4725, 5145, 5625, 5775, 6075, 6561, 6615, 7875, 8085, 8505, 9375, 10125, 10395, 10935

Offset: 1

Gus Wiseman, Apr 20 2021

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 sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   15: {2,3}
   27: {2,2,2}
   45: {2,2,3}
   75: {2,3,3}
   81: {2,2,2,2}
  105: {2,3,4}
  135: {2,2,2,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  315: {2,2,3,4}
  375: {2,3,3,3}
  405: {2,2,2,2,3}
  525: {2,3,3,4}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  735: {2,3,4,4}
  945: {2,2,2,3,4}

The version starting at 1 is A055932.
The partitions with these Heinz numbers are counted by A264396.
Positions of 1's in A339662.
A000009 counts partitions covering an initial interval.
A000070 counts partitions with a selected part.
A016945 lists numbers with smallest prime index 2.
A034296 counts gap-free (or flat) partitions.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A107428 counts gap-free compositions (initial: A107429).
A286469 and A286470 give greatest difference for Heinz numbers.
A325240 lists numbers with smallest prime multiplicity 2.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A006128, A007052, A124010, A257989, A257993, A264401, A317090, A317589, A339737.

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

A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 36, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 141, 142, 143, 145, 146, 147, 148, 153, 155, 158, 159, 161

Offset: 1

Antti Karttunen, Dec 18 2017

nonn

Except for a(1) = 1, these appear to be cubefree numbers with two distinct prime factors, or Heinz numbers of integer partitions with two distinct parts, none appearing more than twice. The enumeration of these partitions by sum is given by A307370. Equivalently, except for a(1) = 1, this sequence is the intersection of A004709 and A007774. - Gus Wiseman, Jul 03 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000196, A295666, A296204.
Cf. A006881, A054753, A085986 (seem to be subsequences).
Cf. A004709, A007774, A056239, A112798, A118914, A307370, A325240.

filter:= proc(k) local d,r,v;
   r:= 1;
   for d in numtheory:-divisors(k^2) do
     v:= igcd(d,k^2/d);
     if isprime(v) then r:= r*v fi
   od;
   r = k^2
end proc:
select(filter, [$1..200]); # Robert Israel, Feb 20 2024

a(n) = A000196(A296204(n)).

A376172 Numbers whose prime factorization has an even minimum exponent.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 72, 81, 100, 108, 121, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1372, 1444

Offset: 1

Amiram Eldar, Sep 13 2024

Numbers k such that A051904(k) is even.

The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Subsequence of A001694.
Complement of A376173 within A001694.
Subsequences: A001248, A062503, A325240.
Cf. A051904.

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # == 1 || EvenQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[2000]

is(k) = {my(f = factor(k), e = f[,2]); !(#e) || (ispowerful(f) && !(vecmin(e) % 2));}

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (-1)^(k+1) * s(k) = 1.70559662202357112914..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).

cp's OEIS Frontend

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

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A244515 Number of partitions of n where the minimal multiplicity of any part is 2.

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A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

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A296205 Numbers k such that Product_{d|k^2, gcd(d,k^2/d) is prime} gcd(d,k^2/d) = k^2.

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A376172 Numbers whose prime factorization has an even minimum exponent.

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