A323090 -id:A323090 - cp's OEIS frontend

A375706 First differences of non-perfect-powers.

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

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The 5th non-perfect-power is 7, and the 6th is 10, so a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

For prime-powers (A000961) we have A057820.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A007916.
For nonsquarefree numbers (A013929) we have A078147.
For non-prime-powers (A024619) we have A375708.
Positions of 1s are A375740, complement A375714.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
Cf. A000040, A046933, A052410, A303707, A305630, A305631, A323054, A323088, A323090, A375736.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ]]

up_to = 112;
A375706list(up_to) = { my(v=vector(up_to), pk=2, k=2, i=0); while(i<#v, k++; if(!ispower(k), i++; v[i] = k-pk; pk = k)); (v); };
v375706 = A375706list(up_to);
A375706(n) = v375706[n]; \\ Antti Karttunen, Jan 19 2025

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A375706(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if not perfect_power(i))-m # Chai Wah Wu, Sep 09 2024

a(n) = A007916(n+1) - A007916(n).

More terms from Antti Karttunen, Jan 19 2025

A323087 Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 2, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(60) = 9 factorizations:
  (2*3*10), (2*5*6), (3*4*5),
  (2*30), (3*20), (4*15), (5*12), (6*10),
  (60).

Cf. A001597, A007916, A025147, A045778, A052410, A087897, A275972, A305150, A323054, A323086, A323090, A323091.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[strfacs[n],stableQ[#,IntegerQ[Log[#1,#2]]&]&]],{n,100}]

A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 7, 9, 11, 11, 15, 16, 18, 22, 24, 27, 32, 34, 41, 45, 51, 59, 64, 75, 82, 94, 105, 116, 132, 146, 163, 183, 202, 225, 251, 277, 309, 341, 378, 417, 463, 510, 564, 622, 685, 754, 830, 914, 1001, 1103, 1207, 1325

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			A list of all strict integer partitions using numbers that are not perfect powers begins:
   2: (2)        11: (6,3,2)    15: (13,2)       17: (12,5)
   3: (3)        12: (12)       15: (12,3)       17: (12,3,2)
   5: (5)        12: (10,2)     15: (10,5)       17: (11,6)
   5: (3,2)      12: (7,5)      15: (10,3,2)     17: (10,7)
   6: (6)        12: (7,3,2)    15: (7,6,2)      17: (10,5,2)
   7: (7)        13: (13)       15: (7,5,3)      17: (7,5,3,2)
   7: (5,2)      13: (11,2)     16: (14,2)       18: (18)
   8: (6,2)      13: (10,3)     16: (13,3)       18: (15,3)
   8: (5,3)      13: (7,6)      16: (11,5)       18: (13,5)
   9: (7,2)      13: (6,5,2)    16: (11,3,2)     18: (13,3,2)
   9: (6,3)      14: (14)       16: (10,6)       18: (12,6)
  10: (10)       14: (12,2)     16: (7,6,3)      18: (11,7)
  10: (7,3)      14: (11,3)     16: (6,5,3,2)    18: (11,5,2)
  10: (5,3,2)    14: (7,5,2)    17: (17)         18: (10,6,2)
  11: (11)       14: (6,5,3)    17: (15,2)       18: (10,5,3)
  11: (6,5)      15: (15)       17: (14,3)       18: (7,6,5)

Cf. A001597, A007916, A025147, A052410, A087897, A305631, A321346, A323054, A323089, A323090.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&FreeQ[#,1]&&And@@Not/@perpowQ/@#&]],{n,20}]

O.g.f.: Product_{n in A007916} (1 + x^n).

A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 08 2024

nonn

			The a(3456) = 28 factorizations are:
  (4*8*9*12)  (4*9*96)    (36*96)   (3456)
              (8*9*48)    (4*864)
              (4*12*72)   (48*72)
              (4*16*54)   (54*64)
              (4*18*48)   (8*432)
              (4*24*36)   (9*384)
              (4*27*32)   (12*288)
              (4*8*108)   (16*216)
              (8*12*36)   (18*192)
              (8*16*27)   (24*144)
              (8*18*24)   (27*128)
              (9*12*32)   (32*108)
              (9*16*24)
              (12*16*18)

Dominic McCarty, Table of n, a(n) for n = 1..10000

Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050326, non-strict A050320.
For prime-powers we have A050361, non-strict A000688.
For nonprime numbers we have A050372, non-strict A050370.
The version for partitions is A256012, non-strict A114374.
For perfect-powers we have A323090, non-strict A294068.
The non-strict version is A376657.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Cf. A008480, A053797, A053806, A061398, A089259, A120992, A303707, A322452, A375707, A376312.

function nextNonSquareFree(val){val+=1;for(let i=2;i*i<=val;i+=1){if(val%i==0&&val%(i*i)==0){return val}}return nextNonSquareFree(val)}function strictFactorCount(val,maxFactor){if(val==1){return 1}let sum=0;while(maxFactorDominic McCarty, Oct 19 2024

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}] (* corrected by Gus Wiseman, Jun 27 2025 *)

A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 51, 59, 66, 75, 86, 96, 110, 123, 139, 157, 176, 199, 221, 248, 278, 309, 346, 385, 427, 476, 528, 586, 650, 719, 795, 880, 973, 1074, 1186, 1307, 1439, 1584, 1744, 1915, 2104

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			A list of all strict integer partitions using 1 and numbers that are not perfect powers begins:
  1: (1)         8: (5,2,1)      12: (12)         14: (14)
  2: (2)         9: (7,2)        12: (11,1)       14: (13,1)
  3: (3)         9: (6,3)        12: (10,2)       14: (12,2)
  3: (2,1)       9: (6,2,1)      12: (7,5)        14: (11,3)
  4: (3,1)       9: (5,3,1)      12: (7,3,2)      14: (11,2,1)
  5: (5)        10: (10)         12: (6,5,1)      14: (10,3,1)
  5: (3,2)      10: (7,3)        12: (6,3,2,1)    14: (7,6,1)
  6: (6)        10: (7,2,1)      13: (13)         14: (7,5,2)
  6: (5,1)      10: (6,3,1)      13: (12,1)       14: (6,5,3)
  6: (3,2,1)    10: (5,3,2)      13: (11,2)       14: (6,5,2,1)
  7: (7)        11: (11)         13: (10,3)       15: (15)
  7: (6,1)      11: (10,1)       13: (10,2,1)     15: (14,1)
  7: (5,2)      11: (7,3,1)      13: (7,6)        15: (13,2)
  8: (7,1)      11: (6,5)        13: (7,5,1)      15: (12,3)
  8: (6,2)      11: (6,3,2)      13: (7,3,2,1)    15: (12,2,1)
  8: (5,3)      11: (5,3,2,1)    13: (6,5,2)      15: (11,3,1)

Cf. A001597, A007916, A052410, A087897, A276431, A303707, A305630, A323088, A323090.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Not/@perpowQ/@#&]],{n,65}]

O.g.f.: (1 + x) * Product_{n in A007916} (1 + x^n).

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A375706 First differences of non-perfect-powers.

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A323087 Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.

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A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

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A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

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A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.

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