A007916 -id:A007916 - cp's OEIS frontend

A052485 Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Henry Bottomley, Mar 16 2000

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001694 (complement), A112526. Not the same as A007916.

a052485 n = a052485_list !! (n-1)
a052485_list = filter ((== 0) . a112526) [1..]
-- Reinhard Zumkeller, Sep 16 2011

Select[Range[1000], Min[FactorInteger[#][[All, 2]]] <= 1 &] (* Geoffrey Critzer, Feb 11 2015 *)

is(n)=n>1 && vecmin(factor(n)[,2])==1 \\ Charles R Greathouse IV, Mar 19 2014

is(n)=!ispowerful(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import integer_nthroot, factorint
def A052485(n):
    def f(x): return int(n+sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

A112526(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2011
a(n) ~ n. - Charles R Greathouse IV, Jul 19 2012
a(n) = n + O(sqrt(n)). - Charles R Greathouse IV, Jul 08 2022
Sum_{n>=1} 1/a(n)^s = zeta(s) - zeta(2*s)*zeta(3*s)/zeta(6*s), for s > 1. - Amiram Eldar, May 13 2023

A377468 Least perfect-power >= n.

Original entry on oeis.org

1, 4, 4, 4, 8, 8, 8, 8, 9, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 27, 32, 32, 32, 32, 32, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81

Offset: 1

Gus Wiseman, Nov 05 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

The version for prime-powers is A000015.
The union is A001597 (perfect-powers), without powers of two A377702.
Positions of last appearances are also A001597.
The version for primes is A007918 or A151800.
The version for squarefree numbers is A067535.
Run-lengths are A076412.
The opposite version (greatest perfect-power <= n) is A081676.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A014210, A014234, A023055, A031218, A045542, A052410, A065514, A188951, A216765, A336416, A345531.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&!perpowQ[#]&],{n,100}]

from sympy import mobius, integer_nthroot
def A377468(n):
    if n == 1: return 1
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m = n-f(n-1)
    return bisection(lambda x:f(x)+m,n-1,n) # Chai Wah Wu, Nov 05 2024

Positions of first appearances for n > 2 are A216765(n-2) = A001597(n-1) + 1.

A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 31 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			Between prime(4) = 7 and prime(5) = 11 we have perfect-powers 8 and 9, so a(4) = 2.

For prime-powers instead of perfect-powers we have A080101.
Non-perfect-powers in the same range are counted by A377433.
Positions of 1 are A377434.
Positions of 0 are A377436.
Positions of terms > 1 are A377466.
For powers of 2 instead of primes we have A377467, for prime-powers A244508.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A053707, A064113, A065514, A065890, A080769, A377051, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],perpowQ]],{n,100}]

a(n) + A377433(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A325238 First positive integer with each omega-sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 96, 120, 128, 192, 210, 216, 240, 256, 360, 384, 420, 480, 512, 720, 768, 840, 900, 960, 1024, 1260, 1296, 1440, 1536, 1680, 1920, 2048, 2310, 2520, 2880, 3072, 3360, 3840, 4096, 4620, 5040, 5760, 6144, 6720

Offset: 1

Gus Wiseman, Apr 14 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = frequency depth of n, and the k-th part is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, given by red(n = p^i*...*q^j) = prime(i)*...*prime(j), i.e., the product of primes indexed by the prime exponents of n.

			The sequence of terms together with their omega-sequences begins:
    1:
    2: 1
    4: 2 1
    6: 2 2 1
    8: 3 1
   12: 3 2 2 1
   16: 4 1
   24: 4 2 2 1
   30: 3 3 1
   32: 5 1
   36: 4 2 1
   48: 5 2 2 1
   60: 4 3 2 2 1
   64: 6 1
   96: 6 2 2 1
  120: 5 3 2 2 1
  128: 7 1
  192: 7 2 2 1
  210: 4 4 1
  216: 6 2 1
  240: 6 3 2 2 1
  256: 8 1
  360: 6 3 3 1
  384: 8 2 2 1
  420: 5 4 2 2 1

Cf. A001221, A001222, A007916, A011784, A070175, A071625, A118914, A181819, A181821, A303555, A304465, A323014, A323023, A325238, A325239.

tomseq[n_]:=If[n<=1,{},Most[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]];
omseqs=Table[Total/@tomseq[n],{n,1000}];
Sort[Table[Position[omseqs,x][[1,1]],{x,Union[omseqs]}]]

A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90

Offset: 1

Gus Wiseman, Nov 02 2024

nonn

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

			Primes 8 and 9 are 19 and 23, and the interval (20,21,22) contains no prime-powers, so 8 is in the sequence.

For powers of 2 instead of primes see A377467, A013597, A014210, A014234, A244508.
For squarefree instead of perfect-power we have A068360, see A061398, A377430, A377431.
For just squares (instead of all perfect-powers) we have A221056, primes A224363.
For prime-powers (instead of perfect-powers) we have A377286.
These are the positions of 0 in A377432.
For one instead of none we have A377434, for prime-powers A377287.
For two instead of none we have A377466, for prime-powers A377288, primes A053706.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A046933 counts the interval from A008864(n) to A006093(n+1).
A065514 gives the nearest prime-power before prime(n)-1, difference A377289.
A080101 and A366833 count prime-powers between primes, see A377057, A053607, A304521.
A081676 gives the nearest perfect-power up to n.
A246655 lists the prime-powers not including 1, complement A361102.
A377468 gives the nearest perfect-power after n.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A276781, A345531, A376597, A377054, A377282.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Length[Select[Range[Prime[#]+1, Prime[#+1]-1],perpowQ]]==0&]

A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

Original entry on oeis.org

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075

Offset: 1

Henry Bottomley, Mar 16 2000

nonn

Number of terms < 10^n: 0, 1, 13, 60, 252, 916, 3158, 10553, 34561, 111891, 359340, 1148195, 3656246, 11616582, 36851965, ..., A118896(n) - A070428(n). - Robert G. Wilson v, Aug 11 2014

a(n) = (s(n))^2 * f(n), s(n) > 1, f(n) > 1, where s(n) is not a power of f(n), and f(n) is squarefree and gcd(s(n), f(n)) = f(n). - Daniel Forgues, Aug 11 2015

			a(3)=200 because 200=2^3*5^2, both 3 and 2 are greater than 1, and the highest common factor of 3 and 2 is 1.
Factorizations of a(1) to a(20):
    72 = 2^3  3^2,  108 = 2^2 3^3,  200 = 2^3 5^2,  288 = 2^5  3^2,
   392 = 2^3  7^2,  432 = 2^4 3^3,  500 = 2^2 5^3,  648 = 2^3  3^4,
   675 = 3^3  5^2,  800 = 2^5 5^2,  864 = 2^5 3^3,  968 = 2^3 11^2,
   972 = 2^2  3^5, 1125 = 3^2 5^3, 1152 = 2^7 3^2, 1323 = 3^3  7^2,
  1352 = 2^3 13^2, 1372 = 2^2 7^3, 1568 = 2^5 7^2, 1800 = 2^3  3^2 5^2.
Examples for a(n) = (s(n))^2 * f(n): (see above comment)
s(n) = 6,  6, 10, 12, 14, 12, 10, 18, 15, 20, 12, 22, 18, 15, 24, 21,
f(n) = 2,  3,  2,  2,  2,  3,  5,  2,  3,  2,  6,  2,  3,  5,  2,  3,

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 302: Strong Achilles Numbers.
Robert Israel, log-log plot of a(n).
Eric Weisstein's World of Mathematics, Achilles Number.
OEIS Wiki, Achilles numbers.

Cf. A001597, A001694, A007916, A072102, A082695.

filter:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:
select(filter,[$1..10000]); # Robert Israel, Aug 11 2014

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 5500, achillesQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmin(f)>1 && gcd(f)==1 \\ Charles R Greathouse IV, Sep 18 2015, replacing code by M. F. Hasler, Sep 23 2010

from math import gcd
from itertools import count, islice
from sympy import factorint
def A052486_gen(startvalue=1): # generator of terms >= startvalue
    return (n for n in count(max(startvalue,1)) if (lambda x: all(e > 1 for e in x) and gcd(*x) == 1)(factorint(n).values()))
A052486_list = list(islice(A052486_gen(),20)) # Chai Wah Wu, Feb 19 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A052486(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

a(n) = O(n^2). - Daniel Forgues, Aug 11 2015
a(n) = O(n^2 / log log n). - Daniel Forgues, Aug 12 2015
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 = A082695 - A072102 - 1 = 0.06913206841581433836...  - Amiram Eldar, Oct 14 2020

Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010

Name edited by M. F. Hasler, Jul 17 2019

A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, 1, 0, 1, -2, 4, -4, 0, 4, 2, -4, -2, 2, -2, 2, 4, -4, -2, -1, 2, 3, -4, 8, -8, 4, 0, -2, -2, 2, 2, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 0, 6, -3, -4, 5, 0, -4, 4, -2, -2

Offset: 1

Gus Wiseman, Oct 02 2024

sign

For the exclusive version, shift left once.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
Positions of zeros are A376597, complement A376598.
Sorted positions of first appearances are A376653, exclusive A376654.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A045542, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2]

from sympy import primepi, integer_nthroot
def A376596(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

The first term in this sequence but absent from A179983 is 180.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 78, 820, 8379, 84440, 846646, 8473868, 84763404, 847714834, 8477408261, ... . Apparently, the asymptotic density of this sequence exists and equals 0.8477... . - Amiram Eldar, Aug 04 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Cf. A001222, A007916, A055932, A056239, A069799, A112798, A124010, A133808.
Cf. A317087, A317088, A317089, A317091, A317092.
Subsequences: A129912\{1}, A179983\{1}.
Subsequence of A337533.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],normalQ[FactorInteger[#][[All,2]]]&]

is(k) = {my(e = Set(factor(k)[,2])); k > 1 && vecmax(e) == #e;} \\ Amiram Eldar, Aug 04 2024

A305563 Number of reducible integer partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 7, 15, 16, 27, 30, 56, 56, 100, 105, 157, 188, 287, 303, 470, 524, 724, 850, 1197, 1339, 1856, 2135, 2814, 3305, 4360, 4951, 6532, 7561, 9563, 11195, 14165, 16328, 20631, 23866, 29471, 34320, 42336, 48672, 59872, 69139, 83625, 96911, 117153

Offset: 1

Gus Wiseman, Jun 05 2018

nonn

A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1, ..., m_k) = 1 and the multiset {y_1, ..., y_k} is also reducible.

			The a(6) = 7 reducible integer partitions are (6), (51), (411), (321), (3111), (21111), (111111). Missing from this list are (42), (33), (222), (2211).

Cf. A007916, A071625, A181819, A182850, A182857, A275870, A304465, A304660, A304687, A304818, A305564, A305565, A305566.

ptnredQ[y_]:=Or[Length[y]==1,And[GCD@@y==1,ptnredQ[Sort[Length/@Split[y],Greater]]]];
Table[Length[Select[IntegerPartitions[n],ptnredQ]],{n,20}]

A336416 Number of perfect-power divisors of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 7, 7, 11, 18, 36, 36, 47, 47, 84, 122, 166, 166, 221, 221, 346, 416, 717, 717, 1001, 1360, 2513, 2942, 4652, 4652, 5675, 5675, 6507, 6980, 13892, 17212, 20408, 20408, 39869, 45329, 51018, 51018, 68758, 68758, 105573, 138617, 284718, 284718, 338126, 421126

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

			The a(1) = 0 through a(9) = 18 divisors:
       1: 1
       2: 1
       6: 1
      24: 1,4,8
     120: 1,4,8
     720: 1,4,8,9,16,36,144
    5040: 1,4,8,9,16,36,144
   40320: 1,4,8,9,16,32,36,64,128,144,576
  362880: 1,4,8,9,16,27,32,36,64,81,128,144,216,324,576,1296,1728,5184

David A. Corneth, Table of n, a(n) for n = 0..9999

The maximum among these divisors is A090630, with quotient A251753.
The version for distinct prime exponents is A336414.
The uniform version is A336415.
Replacing factorials with Chernoff numbers (A006939) gives A336417.
Prime powers are A000961.
Perfect powers are A001597, with complement A007916.
Prime power divisors are counted by A022559.
Cf. A000005, A001221, A001222, A011371, A032741, A118914, A124010, A130091, A327527.
Factorial numbers: A000142, A007489, A027423, A048656, A071626, A181796, A325272, A325273, A325617.

perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[n!],perpouQ]],{n,0,15}]

a(n) = sumdiv(n!, d, (d==1) || ispower(d)); \\ Michel Marcus, Aug 19 2020

addhelp(val, "exponent of prime p in n!")
val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = {if(n<=3, return(1)); my(pr = primes(primepi(n\2)), v = vector(#pr, i, val(n, pr[i])), res = 1, cv); for(i = 2, v[1], if(issquarefree(i), cv = v\i; res-=(prod(i = 1, #cv, cv[i]+1)-1)*(-1)^omega(i) ) ); res } \\ David A. Corneth, Aug 19 2020

a(p) = a(p-1) for prime p. - David A. Corneth, Aug 19 2020

a(26)-a(34) from Jinyuan Wang, Aug 19 2020

a(35)-a(49) from David A. Corneth, Aug 19 2020

cp's OEIS Frontend

A052485 Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.

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A377468 Least perfect-power >= n.

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A377432 Number of perfect-powers x in the range prime(n) < x < prime(n+1).

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A325238 First positive integer with each omega-sequence.

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A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

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A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

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A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

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A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

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