A305630 -id:A305630 - 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

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of first appearances (A376268):
  1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...

These are the sorted positions of first appearances in A053289 (union A023055).
The complement is A376519.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443

Offset: 1

Gus Wiseman, Sep 28 2024

nonn

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]

A305635 1 and odd numbers that are not perfect powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 127, 129, 131, 133, 135, 137

Offset: 1

Gus Wiseman, Jun 07 2018

nonn

Cf. A001597, A005117, A007916, A039956, A056911.

Cf. A305630, A305631, A305632, A305633, A305634.

[1] cat  [n : n in [3..200 by 2] | not IsPower(n) ]; // Vincenzo Librandi, Jul 06 2018

radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
Select[Range[200],OddQ[#]&&radQ[#]&]

isok(n) = (n==1) || ((n % 2) && !ispower(n)); \\ Michel Marcus, Jun 08 2018

A323053 Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 25, 30, 38, 47, 58, 71, 87, 106, 131, 156, 190, 228, 275, 328, 394, 468, 556, 661, 784, 923, 1089, 1283, 1507, 1766, 2068, 2416, 2821, 3284, 3822, 4438, 5148, 5961, 6898, 7968, 9195, 10593, 12198, 14019, 16102, 18472

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(2) = 1 through a(11) = 12 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)      (B)
            (22)  (32)  (33)   (43)   (44)    (54)    (55)     (65)
                        (222)  (52)   (53)    (63)    (64)     (74)
                               (322)  (62)    (72)    (73)     (83)
                                      (332)   (333)   (433)    (92)
                                      (2222)  (522)   (532)    (443)
                                              (3222)  (622)    (533)
                                                      (3322)   (632)
                                                      (22222)  (722)
                                                               (3332)
                                                               (5222)
                                                               (32222)

Cf. A001597, A002865, A007916, A052410, A101417, A102430, A108917, A305148, A305630, A305631, A321346, A323093.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 4, 6, 5, 4, 7, 10, 8, 7, 11, 15, 13, 12, 17, 22, 19, 18, 25, 30, 28, 26, 35, 42, 39, 38, 49, 59, 56, 54, 69, 81, 77, 76, 94, 110, 105, 105, 127, 147, 141, 142, 171, 195, 189, 190, 227, 257, 250, 254, 299, 335, 328, 334, 390, 432

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

			O.g.f.: 1/((1 - x)(1 + x^2)(1 - x^3)(1 - x^5)(1 + x^6)(1 - x^7)(1 + x^10)...).

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1+(-x)^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

Original entry on oeis.org

0, 0, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -2, 1, 1, 3, -1, 1, 2, 1, -2, 3, 1, 1, -3, 1, 1, 1, -2, 1, 1, 1, -1, 3, 1, 3, -3, 1, 1, 3, -3, 1, 1, 1, -2, 4, 1, 1, -4, 1, 2, 3, -2, 1, 3, 3, -3, 3, 1, 1, -4, 1, 1, 4, -1, 3, 1, 1, -2, 3, 1, 1, -3, 1, 1, 4, -2, 3, 1, 1, -4

Offset: 0

Gus Wiseman, Jun 07 2018

sign

Cf. A000607, A001597, A007916, A048165, A081362, A091050, A303707, A304779, A304817, A305614, A305630-A305635.

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Sum[x^p/(1+x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

A305634 Even numbers that are not perfect powers.

Original entry on oeis.org

2, 6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138

Offset: 1

Gus Wiseman, Jun 07 2018

nonn

Perfect powers are of the form m^k where m > 0 and k > 1 (A001597).

			10 is in the sequence since it is even and is not a power of an integer.  17 is not in the sequence since it is odd, and 36 is not in the sequence since it is a power of an integer (36 = 6^2).

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

Cf. A001597, A005117, A007916, A039956.
Cf. A305630, A305631, A305632, A305633, A305635.
Cf. A005843, A001597.

N:= 1000:
S:={seq(i,i=2..N,2)} minus {seq(seq(e^m,m=2..floor(log[e](N))),e=2..floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Jan 24 2019

radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
Select[Range[200],EvenQ[#]&&radQ[#]&]

isok(n) = !(n % 2) && !ispower(n); \\ Michel Marcus, Jun 08 2018

A005843 \ A001597. - Eric Chen, Jun 14 2018

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).

cp's OEIS Frontend

A375706 First differences of non-perfect-powers.

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A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

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Maple

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A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

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Mathematica

A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

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Mathematica

A305635 1 and odd numbers that are not perfect powers.

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Magma

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A323053 Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.

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Mathematica

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

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Mathematica

A305633 Expansion of Sum_{r not a perfect power} x^r/(1 + x^r).

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A305634 Even numbers that are not perfect powers.

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