A034684 -id:A034684 - cp's OEIS frontend

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)

Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Factorization

A027748, A124010 are used in a formula defining this sequence.
Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
Cf. A060175, A141810, A213925.

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012

A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 128, 6, 256, 9, 10, 512, 1024, 17, 2048, 12, 18, 33, 4096, 34, 8192, 65, 16384, 20, 32768, 7, 65536, 131072, 66, 129, 24, 36, 262144, 257, 130, 40, 524288, 11, 1048576, 68, 72, 513, 2097152, 258, 4194304, 1025, 514, 132

Offset: 1

Rémy Sigrist, Jun 30 2017

For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0):
- let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 },
- a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }.
In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples.
a(A002110(n)) = 2^n-1 for any n >= 0.
a(A000961(n+1)) = 2^(n-1) for any n > 0.
A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 representation of a(n)).
A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1).
This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence.
The numbers 4, 36, 40 and 532 equal their image; are there other such numbers?
This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective).

			For n = 204 = 2^2 * 3 * 17:
- S_204 = A000961 \ { 2^3, 2^4, ..., 3^2, ... }
        = { 1, 2, 3, 4, 5, 7, 11, 13, 17, ... },
- a(204) = 2^#{ 2, 3 } + 2^#{ 2 } + 2^#{ 2, 3, 4, 5, 7, 11, 13 }
         = 2^2 + 2^1 + 2^7
         = 134.
See also the illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, PARI program for A289271
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse), A322988, A322990, A322991, A322992, A322995.

Cf. also A156552, A052331 for similar constructions.

PARI
```
See Links section.
```

A289271(n) = { my(f = factor(n), pps = vecsort(vector(#f~, i, f[i, 1]^f[i, 2])), s=0, x=1, pp=1, k=-1); for(i=1,#f~, while(pp < pps[i], pp++; while(!isprimepower(pp)||(gcd(pp,x)>1), pp++); k++); s += 2^k; x *= pp); (s); }; \\ Antti Karttunen, Jan 01 2019

A289272 Inverse to A289271.

Original entry on oeis.org

1, 2, 3, 6, 4, 10, 12, 30, 5, 14, 15, 42, 20, 70, 60, 210, 7, 18, 21, 66, 28, 90, 84, 330, 35, 126, 105, 462, 140, 630, 420, 2310, 8, 22, 24, 78, 36, 110, 132, 390, 40, 154, 120, 546, 180, 770, 660, 2730, 56, 198, 168, 858, 252, 990, 924, 4290, 280, 1386, 840

Offset: 0

Rémy Sigrist, Jun 30 2017

a(2^n-1) = A002110(n) for any n >= 0.
a(2^(n-1)) = A000961(n+1) for any n > 0.
A001221(a(n)) = A000120(n) for any n >= 0.
From Antti Karttunen, Jan 01 2019: (Start)
A034684(a(n)) = A000961(1+A001511(n)) for any n >= 1. (See also Rémy Sigrist's comment in A289271).
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A322991(k), and each right hand child contains A322992(k), when their parent contains k:
                                     1
                                     |
                  ...................2...................
                 3                                       6
       4......../ \........10                 12......../ \........30
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   5       14         15       42         20       70         60       210
  7 18   21  66     28  90   84  330    35  126 105  462   140  630  420 2310
etc.
The leftmost edge is A000961, the next lefmost is A278568 (after 2: 6, 10, 14, 18, ...), the righmost edge is A002110, the next rightmost A088860 but with 3 instead of 4.
Compare also to trees like A005940 (A163511) and A052330.
(End)

			A289271(1) = 0, hence a(0) = 1.
A289271(2) = 1, hence a(1) = 2.
A289271(3) = 2, hence a(2) = 3.
A289271(4) = 4, hence a(4) = 4.
A289271(5) = 8, hence a(8) = 5.
A289271(6) = 3, hence a(3) = 6.
A289271(7) = 16, hence a(16) = 7.
A289271(8) = 32, hence a(32) = 8.
A289271(9) = 64, hence a(64) = 9.
A289271(10) = 5, hence a(5) = 10.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A289272
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A000961, A001221, A002110, A034684, A088860, A278568, A289271 (inverse), A322989, A322990, A322991, A322992.

Cf. also A005940, A163511, A052330.

PARI
```
See Links section.
```

A289272(n) = { my(m=1, pp=1); while(n>0, pp++; while(!isprimepower(pp)||(gcd(pp,m)>1), pp++); if(n%2, m *= pp); n >>=1); (m); }; \\ Antti Karttunen, Jan 01 2019

A067695 Smallest prime factor with minimum exponent in canonical prime factorization of n, a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 3, 13, 2, 3, 2, 17, 2, 19, 5, 3, 2, 23, 3, 5, 2, 3, 7, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 5, 41, 2, 43, 11, 5, 2, 47, 3, 7, 2, 3, 13, 53, 2, 5, 7, 3, 2, 59, 3, 61, 2, 7, 2, 5, 2, 67, 17, 3, 2, 71, 3, 73, 2, 3, 19, 7, 2, 79, 5

Offset: 1

Reinhard Zumkeller, Feb 23 2002

nonn

			a(12) = a(2^2 * 3^1) = 3, but A020639(12) = 2;
a(36) = a(2^2 * 3^2) = 2 = A020639(36).

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A020639, A034684, A028233, A051904, A067029.

a:= n-> `if`(n=1, 1, (l-> (m-> min(map(i-> i[1], select(y->
      y[2]=m, l))))(min(map(x-> x[2], l))))(ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 25 2023

a[n_] := Module[{f = FactorInteger[n], p, e}, Min[Select[f, Last[#] == Min[f[[;;, 2]]] &][[;;, 1]]]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n) = if (n==1, 1, my(f=factor(n), i=vecmin(f[,2])); f[vecmin(select(x->(x==i), f[,2], 1)), 1]); \\ Michel Marcus, Jul 17 2023

from sympy import factorint
def A067695(n):
    if n == 1: return 1
    f, g = map(tuple,zip(*sorted(factorint(n).items())))
    return f[g.index(min(g))] # Chai Wah Wu, Feb 07 2023

A117180 Lowest prime-power dividing the n-th nonsquarefree positive integer.

Original entry on oeis.org

4, 8, 9, 3, 16, 2, 4, 3, 25, 27, 4, 32, 4, 5, 4, 5, 3, 49, 2, 4, 2, 7, 3, 7, 64, 4, 8, 3, 4, 5, 81, 3, 8, 2, 4, 3, 2, 9, 4, 8, 4, 7, 4, 9, 3, 121, 4, 125, 2, 128, 3, 5, 8, 4, 9, 3, 4, 2, 8, 9, 3, 5, 2, 4, 3, 169, 9, 4, 7, 11, 4, 8, 4, 7, 3, 4, 2, 8, 3, 9, 13, 4, 8, 4, 7, 9, 3, 8, 2, 4, 3, 2, 243, 4, 5, 8

Offset: 1

Leroy Quet, Mar 01 2006

nonn

			12, the 4th nonsquarefree positive integer, is 2^2 * 3. 3 is the smallest prime power dividing 12. So a(4) = 3.

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

Cf. A034684, A013929, A117181, A117182.

A013929 := proc(nmax) local a,n ; a := [] ; n :=1 ; while nops(a) < nmax do if not numtheory[issqrfree](n) then a := [op(a),n] ; fi ; n := n+1 ; od ; a ; end : A034684 := proc(n) local ifs; if n = 1 then 1 ; else ifs := ifactors(n)[2] ; seq(op(1,op(i,ifs))^op(2,op(i,ifs)), i=1..nops(ifs)) ; min(%) ; fi ; end: a013929 := A013929(200) : for n from 1 to nops(a013929) do printf("%d, ",A034684(op(n,a013929))) ; od ; # R. J. Mathar, May 10 2007

s[n_] := Min @@ Power @@@ FactorInteger[n]; s /@ Select[Range[200], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 11 2021 *)

a(n) = A034684(A013929(n)).

More terms from R. J. Mathar, May 10 2007

A117182 a(n) = A117181(n) - A117180(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 7, 1, 5, 0, 0, 3, 0, 5, 3, 7, 4, 13, 0, 23, 9, 25, 1, 2, 2, 0, 13, 1, 22, 15, 11, 0, 4, 3, 7, 19, 29, 47, 2, 21, 5, 23, 9, 25, 4, 5, 0, 27, 0, 7, 0, 8, 22, 9, 3, 7, 46, 33, 23, 11, 8, 10, 27, 79, 37, 5, 0, 10, 39, 18, 5, 5, 15, 43, 20, 61, 45, 9, 17, 14, 14, 3, 49, 19, 7, 25, 16, 16

Offset: 1

Leroy Quet, Mar 01 2006

nonn

			12, the 4th nonsquarefree positive integer, is 2^2 * 3. 2^2 = 4 is the largest prime power dividing 12. 3 is the smallest prime power dividing 12. So a(4) = 4 - 3 = 1.

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

Cf. A117180, A117181.

A013929 := proc(nmax) local a,n ; a := [] ; n :=1 ; while nops(a) < nmax do if not numtheory[issqrfree](n) then a := [op(a),n] ; fi ; n := n+1 ; od ; a ; end :
A034699 := proc(n) local ifs,res; if n = 1 then 1 ; else ifs := ifactors(n)[2] ; seq(op(1,op(i,ifs))^op(2,op(i,ifs)), i=1..nops(ifs)) ; max(%) ; fi ; end:
A034684 := proc(n) local ifs,res; if n = 1 then 1 ; else ifs := ifactors(n)[2] ; seq(op(1,op(i,ifs))^op(2,op(i,ifs)), i=1..nops(ifs)) ; min(%) ; fi ; end:
a013929 := A013929(200) : for n from 1 to nops(a013929) do printf("%d, ",A034699(op(n,a013929))-A034684(op(n,a013929))) ; od ; # R. J. Mathar, May 10 2007

s[n_] := Differences[MinMax[Power @@@ FactorInteger[n]]][[1]]; s /@ Select[Range[250], !SquareFreeQ[#] &] (* Amiram Eldar, Feb 11 2021 *)

More terms from R. J. Mathar, May 10 2007

A126855 Numbers k such that, if k = product{p|k} p^c(k,p), each c(k,p) is a positive integer and each p is a distinct prime, then the smallest prime-power p^c(k, p) is not a power of the smallest prime dividing k.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 60, 63, 80, 84, 96, 112, 120, 132, 135, 144, 156, 160, 168, 175, 176, 189, 192, 204, 208, 224, 228, 240, 264, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 351, 352, 360, 372, 384, 405, 408, 416, 420, 425, 440, 444, 448

Offset: 1

Leroy Quet, Mar 23 2007

nonn

The numbers of terms not exceeding 10^k, for k=1,2,...., are 0, 11, 128, 1245, 12474, 124052, 1240434, 12398594, 123976845, 1239840735, ... Apparently this sequence has an asymptotic density 0.1239... - Amiram Eldar, Mar 20 2021

			3600 is included because 3600 = 2^4 * 3^2 * 5^2 and the smallest prime-power (which is largest prime-power of its prime to divide 3600), 3^2 = 9, is not a power of the smallest prime to divide 3600, which is 2.

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

Cf. A020639, A034684, A102749.

fQ[n_] := Block[{p = Power @@@ FactorInteger[n]},First[p] != Min[p]];Select[Range[460], fQ] (* Ray Chandler, Mar 25 2007 *)

Extended by Ray Chandler, Mar 25 2007

A139084 a(n) = (smallest prime-power among the largest powers dividing n of each prime dividing n) * (smallest prime-power among the largest powers dividing (n+1) of each prime dividing (n+1)).

Original entry on oeis.org

2, 6, 12, 20, 10, 14, 56, 72, 18, 22, 33, 39, 26, 6, 48, 272, 34, 38, 76, 12, 6, 46, 69, 75, 50, 54, 108, 116, 58, 62, 992, 96, 6, 10, 20, 148, 74, 6, 15, 205, 82, 86, 172, 20, 10, 94, 141, 147, 98, 6, 12, 212, 106, 10, 35, 21, 6, 118, 177, 183, 122, 14, 448, 320, 10, 134

Offset: 1

Leroy Quet, Apr 07 2008

nonn

The largest powers dividing 44 of each prime dividing 44 are 2^2 and 11^1. The least of these is 2^2 =4. The largest powers dividing 45 of each prime dividing 45 are 3^2 and 5^1. The least of these is 5^1 = 5. So a(44) = 4 * 5 = 20.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A139082, A139083.

f[{a_,b_}]:=a^b;a[n_]:=Min[f/@FactorInteger[n]]*Min[f/@FactorInteger[n+1]];Array[a,66] (* James C. McMahon, Jun 28 2025 *)

minpp(n)=local(m,r,pp);if(n==1,1,m=factor(n);r=m[1,1]^m[1,2];for(i=2,matsize(m)[1],pp=m[i,1]^m[i,2];if(pp

a(n) = A034684(n) * A034684(n+1). [From Franklin T. Adams-Watters, Apr 09 2009]

More terms from Franklin T. Adams-Watters, Apr 09 2009

A100574 If n = product{p|n, p=prime} p^b(p,n), where each b(p,n) is a positive integer and the product is over distinct prime divisors of n, a(n) = difference between the maximum p^b(p,n) and minimum p^b(p,n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 5, 2, 0, 0, 7, 0, 1, 4, 9, 0, 5, 0, 11, 0, 3, 0, 3, 0, 0, 8, 15, 2, 5, 0, 17, 10, 3, 0, 5, 0, 7, 4, 21, 0, 13, 0, 23, 14, 9, 0, 25, 6, 1, 16, 27, 0, 2, 0, 29, 2, 0, 8, 9, 0, 13, 20, 5, 0, 1, 0, 35, 22, 15, 4, 11, 0, 11, 0, 39, 0, 4, 12, 41, 26, 3, 0, 7, 6, 19, 28

Offset: 1

Leroy Quet, Jan 02 2005

nonn

a(n)=0 iff n a prime or a power of a prime. - Robert G. Wilson v, Jan 10 2005

			For 24 = 2^3 *3, 2^3 and 3 are separated by 5, so a(30) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034684, A034699.

pf[n_] := Block[{pb = Flatten[ Table[ #[[1]]^#[[2]], {1}] & /@ FactorInteger[n]]}, Max[pb] - Min[pb]]; Table[ pf[n], {n, 2, 100}] (* Robert G. Wilson v, Jan 10 2005 *)

A100574(n) = if(1==n,0,my(f=factor(n), v = vector(#f[, 1], i, f[i, 1]^f[i, 2])); vecmax(v)-vecmin(v)); \\ Antti Karttunen, Aug 06 2018

a(n) = A034699(n) - A034684(n). - Antti Karttunen, Aug 06 2018

More terms from Robert G. Wilson v, Jan 10 2005

A139083 a(n) = (smallest prime-power among the largest powers of each prime dividing n) + (smallest prime-power among the largest powers of each prime dividing (n+1)).

Original entry on oeis.org

3, 5, 7, 9, 7, 9, 15, 17, 11, 13, 14, 16, 15, 5, 19, 33, 19, 21, 23, 7, 5, 25, 26, 28, 27, 29, 31, 33, 31, 33, 63, 35, 5, 7, 9, 41, 39, 5, 8, 46, 43, 45, 47, 9, 7, 49, 50, 52, 51, 5, 7, 57, 55, 7, 12, 10, 5, 61, 62, 64, 63, 9, 71, 69, 7, 69, 71, 7, 5, 73, 79, 81, 75, 5, 7, 11, 9, 81

Offset: 1

Leroy Quet, Apr 07 2008

nonn

The largest powers of each prime dividing 44 are 2^2 and 11^1. The least of these is 2^2 =4. The largest powers of each prime dividing 45 are 3^2 and 5^1. The least of these is 5^1 = 5. So a(44) = 4 + 5 = 9.

Cf. A139081, A139084.

f[{a_,b_}]:=a^b;a[n_]:=Min[f/@FactorInteger[n]]+Min[f/@FactorInteger[n+1]];Array[a,78] (* James C. McMahon, Jun 28 2025 *)

minpp(n)=local(m,r,pp);if(n==1,1,m=factor(n);r=m[1,1]^m[1,2];for(i=2,matsize(m)[1],pp=m[i,1]^m[i,2];if(pp

a(n) = A034684(n) + A034684(n+1). [Franklin T. Adams-Watters, Apr 09 2009]

More terms from Franklin T. Adams-Watters, Apr 09 2009

cp's OEIS Frontend

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

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A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition).

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A289272 Inverse to A289271.

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A067695 Smallest prime factor with minimum exponent in canonical prime factorization of n, a(1)=1.

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A117180 Lowest prime-power dividing the n-th nonsquarefree positive integer.

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A117182 a(n) = A117181(n) - A117180(n).

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A126855 Numbers k such that, if k = product{p|k} p^c(k,p), each c(k,p) is a positive integer and each p is a distinct prime, then the smallest prime-power p^c(k, p) is not a power of the smallest prime dividing k.

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