A331048 -id:A331048 - cp's OEIS frontend

A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 11, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 11, 1, 1, 1, 7, 1, 11, 1, 4, 1, 1, 1, 19, 1, 4, 4, 9, 1, 1, 1, 7, 1

Offset: 1

Gus Wiseman, Jan 08 2020

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 1's are A005117.
Positions of 2's appear to be A001248.
The denominators are A331024.
The rounded quotients are A331048.
The same for integer partitions is A330994.
Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]],{n,100}]//Numerator

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A331023(n) = numerator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

a(2^n) = A330994(n).

More terms from Antti Karttunen, May 27 2021

A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 08 2020

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.
The numerators are A331023.
The rounded quotients are A331048.
The same for integer partitions is A330995.
Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]],{n,100}]//Denominator

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A331024(n) = denominator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

a(2^n) = A330995(n).

More terms from Antti Karttunen, May 27 2021

A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Paolo Xausa, Feb 19 2025

The corresponding largest elements are given by A081812.
The positions of terms > 1 are given by A001694.
Records of a(n) = 2, 3, 4, 5,.. appear at n=4=2^2, 27=3^3, 625=5^4, 3125=5^5, 117649=7^6, 823543=7^7 ,... (subsequence A051647).- R. J. Mathar, Mar 05 2025

			a(36) = 2 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and its smallest element is 2.
a(31500) = 1 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its smallest element is 1.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A001694, A081812, A288636, A331048, A381201, A381202, A381203, A381204, A381205, A381214.

A381212 := proc(n)
    local a,pe;
    a := n ;
    for pe in ifactors(n)[2] do
        a := min(a,op(1,pe),op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A381212(n),n=2..100) ; # R. J. Mathar, Mar 05 2025

A381212[n_] := Min[Flatten[FactorInteger[n]]];
Array[A381212, 100, 2]

a(n) = my(f=factor(n)); vecmin(setunion(Set(f[,1]), Set(f[,2]))); \\ Michel Marcus, Feb 20 2025

A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 44, 46, 48, 51, 53, 56, 59, 61, 64, 68, 71, 74, 78, 81, 85, 89, 93, 97, 101, 106, 111, 115, 120, 126

Offset: 0

Gus Wiseman, Jan 08 2020

nonn

Conjecture: This sequence is nondecreasing. More generally, the rational sequence A000041(n)/A000009(n) is nondecreasing for n > 5.

The numerators are A330994.
The denominators are A330995.
The same for factorizations is A331048.
Cf. A000009, A000041, A001055, A003238, A035359, A045778, A046063, A331022.

Table[Round[PartitionsP[n]/PartitionsQ[n]],{n,0,100}]

A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

Original entry on oeis.org

128, 2187, 10368, 34992, 78125, 80000, 307328, 823543, 1250000, 1366875, 1874048, 3655808, 5250987, 6328125, 10690688, 13176688, 16681088, 19487171, 32019867, 35819648, 62462907, 62748517, 66706983, 90531968, 118210688, 182660427, 187578125, 239892608, 285012027

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

Contains p^7 for all primes p.

			The 15 factorizations and 5 strict factorizations of 2187:
  (2187)           (2187)
  (27*81)          (27*81)
  (3*729)          (3*729)
  (9*243)          (9*243)
  (3*9*81)         (3*9*81)
  (9*9*27)
  (3*27*27)
  (3*3*243)
  (3*9*9*9)
  (3*3*3*81)
  (3*3*9*27)
  (3*3*3*9*9)
  (3*3*3*3*27)
  (3*3*3*3*3*9)
  (3*3*3*3*3*3*3)

Factorizations are A001055.
Strict factorizations are A045778.
Taking "twice" instead of "three times" gives A001248.
Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.

facsm[n_]:=facsm[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsm[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100000],3==Length[facsm[#]]/Length[Select[facsm[#],UnsameQ@@#&]]&]

a(7)-(10) from Alois P. Heinz, Jan 17 2020

a(11)-a(29) from Giovanni Resta, Jan 20 2020

cp's OEIS Frontend

A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

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A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

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A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

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A330996 Nearest integer to P(n)/Q(n) = A000041(n)/A000009(n).

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A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

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