A025475 -id:A025475 - cp's OEIS frontend

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset: 1

Daniel Forgues, May 27 2008

nonn

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
From Michael De Vlieger, Aug 11 2025: (Start)
This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)

Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.

With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* Michael De Vlieger, Aug 11 2025 *)

isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013

from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - Amiram Eldar, Aug 15 2025

A053607 Primes p such that a pure prime power occurs between p and the next prime.

Original entry on oeis.org

3, 7, 13, 23, 31, 47, 61, 79, 113, 127, 167, 241, 251, 283, 337, 359, 509, 523, 619, 727, 839, 953, 1021, 1327, 1367, 1669, 1847, 2039, 2179, 2207, 2399, 2803, 3121, 3469, 3719, 4093, 4483, 4909, 5039, 5323, 6229, 6553, 6857, 6883, 7919, 8191, 9403, 10193

Offset: 1

Labos Elemer, Feb 09 2000

nonn

			127 belongs here because 128 = 2^7 occurs between 127 and 131.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A051701, A025475.

lim = 10^4; pwrs = Sort[Flatten[Table[Prime[n]^i, {n, 1, PrimePi[Sqrt[lim]]}, {i, 2, Log[Prime[n], lim]}]]]; Union[NextPrime[pwrs, -1]] (* T. D. Noe, Mar 11 2013 *)

A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 4, 1, 0, 0, 5, 1, 0, 0, 0, 7, 2, 1, 1, 1, 1, 8, 1, -1, -2, -3, -4, -5, 9, 1, 0, 1, 3, 6, 10, 15, 11, 2, 1, 1, 0, -3, -9, -19, -34, 13, 2, 0, -1, -2, -2, 1, 10, 29, 63, 16, 3, 1, 1, 2, 4, 6, 5, -5, -34, -97, 17, 1, -2, -3, -4, -6, -10, -16, -21, -16, 18, 115

Offset: 0

Gus Wiseman, Oct 20 2024

Row k of the array is the k-th differences of A000961.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     4     5     7     8     9    11
  k=1:   1     1     1     1     2     1     1     2     2
  k=2:   0     0     0     1    -1     0     1     0     1
  k=3:   0     0     1    -2     1     1    -1     1    -3
  k=4:   0     1    -3     3     0    -2     2    -4     6
  k=5:   1    -4     6    -3    -2     4    -6    10    -8
  k=6:  -5    10    -9     1     6   -10    16   -18     5
  k=7:  15   -19    10     5   -16    26   -34    23     9
  k=8: -34    29    -5   -21    42   -60    57   -14   -42
  k=9:  63   -34   -16    63  -102   117   -71   -28   104
Triangle form:
    1
    2    1
    3    1    0
    4    1    0    0
    5    1    0    0    0
    7    2    1    1    1    1
    8    1   -1   -2   -3   -4   -5
    9    1    0    1    3    6   10   15
   11    2    1    1    0   -3   -9  -19  -34
   13    2    0   -1   -2   -2    1   10   29   63
   16    3    1    1    2    4    6    5   -5  -34  -97

Row k=0 is A000961, exclusive A246655.
Row k=1 is A057820.
Row k=2 is A376596.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, nonsquarefree A377046.
Triangle row-sums are A377052, absolute version A377053.
Column n = 1 is A377054, for primes A007442 or A030016.
First position of 0 in each row is A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A025475, A053707, A093555, A174965, A361102, A376340, A376598, A376653.

nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k)*binomial(j,k)*A000961(i+k).

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

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 01 2024

sign

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

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  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, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros are A376588, complement A376589.
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
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A064113, A069623, A093555, A174965, A182853, A336416, A336417, A361102.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ],2]

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A376562(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)
    r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)
    return next(i for i in count(1-k) if not perfect_power(r+i)) # Chai Wah Wu, Oct 02 2024

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375735, ones A375713(n) - 1.
Positions of zeros are A376600, complement A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.
A321346/A321378 count integer partitions without prime-powers, factorizations A322452.
For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]

from sympy import primepi, integer_nthroot
def A376599(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+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

A353834 Nonprime numbers whose prime indices have all equal run-sums.

Original entry on oeis.org

1, 4, 8, 9, 12, 16, 25, 27, 32, 40, 49, 63, 64, 81, 112, 121, 125, 128, 144, 169, 243, 256, 289, 325, 343, 351, 352, 361, 512, 529, 625, 675, 729, 832, 841, 931, 961, 1008, 1024, 1331, 1369, 1539, 1600, 1681, 1728, 1849, 2048, 2176, 2187, 2197, 2209, 2401

Offset: 1

Gus Wiseman, May 26 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    40: {1,1,1,3}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   112: {1,1,1,1,4}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
For example, 675 is in the sequence because its prime indices {2,2,2,3,3} have run-sums (6,6).

For equal run-lengths we have A072774\A000040, counted by A047966(n)-1.
These partitions are counted by A304442(n) - 1.
These are the nonprime positions of prime powers in A353832.
Including the primes gives A353833.
For distinct run-sums we have A353838\A000040, counted by A353837(n)-1.
For compositions we have A353848\A000079, counted by A353851(n)-1.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 pertain to partition run-sum trajectory.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000005, A000961, A007947, A025475, A071625, A073093, A323014, A353839.

Select[Range[100],!PrimeQ[#]&&SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

from itertools import count, islice
from sympy import factorint, primepi
def A353848_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or (sum((f:=factorint(n)).values()) > 1 and len(set(primepi(p)*e for p, e in f.items())) <= 1), count(max(startvalue,1)))
A353848_list = list(islice(A353848_gen(),30)) # Chai Wah Wu, May 27 2022

A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 119, 121, 125, 128, 161, 169, 243, 256, 289, 343, 351, 361, 378, 455, 512, 527, 529, 595, 625, 721, 729, 841, 845, 918, 959, 961, 1024, 1045, 1081, 1241, 1265, 1323, 1331, 1369, 1375, 1547, 1615, 1681, 1792, 1849, 1855

Offset: 1

Hieronymus Fischer, Nov 11 2007

nonn

All perfect prime powers (A025475) with power > 0 are included.

Originally, the definition started with "Nonprime numbers ..." and the first term was equal to 1. This is misleading, since 1 has no prime factors. - Hieronymus Fischer, Apr 20 2013

			a(5) = 25, since 25=5*5 and sqrt((5^2+5^2)/2)=5;
a(23) = 378, since 378=2*3*3*3*7 and sqrt((2^2+3*3^2+7^2)/5)=sqrt(16)=4.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A001597, A025475, A134333, A134344, A134376.

Cf. A134601, A134605, A134608, A134611, A134617, A134619, A134621.

f[{a_,b_}]:=Table[a,b];Select[Range[2,1855],!PrimeQ[#]&&IntegerQ[ RootMeanSquare[f/@FactorInteger[#]//Flatten] ]&] (* James C. McMahon, Apr 08 2025 *)

Definition clarified and edited by Hieronymus Fischer, Apr 20 2013

A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are zero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

			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, ...
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, ...
with zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...

Gus Wiseman, Inflection and undulation points in the prime-powers inclusive.

The first differences were A057820, see also A053707, A376340.
These are the zeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376598.
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), A376596 (second differences), 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, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]

A056798 Prime powers with even nonnegative exponents.

Original entry on oeis.org

1, 4, 9, 16, 25, 49, 64, 81, 121, 169, 256, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 16384

Offset: 1

Labos Elemer, Aug 28 2000

nonn

Also numbers whose geometric mean of divisors is an integer. - Ctibor O. Zizka, Sep 29 2008

This is just a special case. In fact, the numbers whose geometric mean of divisors is an integer are all the squares of integers (A000290). - Daniel Lignon, Nov 29 2014

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000290, A000961, A025475, A154945.

Take[Union[Flatten[Table[Prime[n]^k, {n, 31}, {k, 0, 14, 2}]]], 45] (* Alonso del Arte, Jul 05 2011 *)

is(n)=my(e=isprimepower(n)); if(e, e%2==0, n==1) \\ Charles R Greathouse IV, Sep 18 2015

from sympy import primepi, integer_nthroot
def A056798(n):
    if n==1: return 1
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0])for k in range(2,x.bit_length(),2)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 13 2024

a(n) = A025473(n)^(2*A025474(n)) = A000961(n)^2;
A001222(a(n)) mod 2 = 0;
A003415(a(n)) = A192083(n); A068346(a(n)) = A192084(n). - Reinhard Zumkeller, Jun 26 2011
Sum_{n>=2} 1/a(n) = A154945. - Amiram Eldar, Sep 21 2020

cp's OEIS Frontend

A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

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A053607 Primes p such that a pure prime power occurs between p and the next prime.

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A377051 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the powers of primes.

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

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A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

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A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

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A353834 Nonprime numbers whose prime indices have all equal run-sums.

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A134600 Composite numbers such that the square mean of their prime factors is an integer (where the prime factors are taken with multiplicity and the square mean of c and d is sqrt((c^2+d^2)/2)).

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