A113851 -id:A113851 - cp's OEIS frontend

A062503 Squarefree numbers squared.

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329, 5476

Offset: 1

Jason Earls, Jul 09 2001

nonn

Also, except for the initial term, numbers whose prime factors are squared. - Cino Hilliard, Jan 25 2006
Also cubefree numbers that are squares. - Gionata Neri, May 08 2016
All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (term of this sequence), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020
Powerful numbers (A001694) all of whose nonunitary divisors are not powerful (A052485). - Amiram Eldar, May 13 2023

Harry J. Smith, Table of n, a(n) for n = 1..1000

Characteristic function is A227291.
Other powers of squarefree numbers: A005117(1), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7), A072774(all).
Cf. A000290, A001694, A052485, A062320.
Cf. A001248 (a subsequence).
A329332 column 2 in ascending order.

a062503 = a000290 . a005117  -- Reinhard Zumkeller, Jul 07 2013

Mathematica
```
Select[Range[100], SquareFreeQ]^2
```

je=[]; for(n=1,200, if(issquarefree(n),je=concat(je,n^2),)); je

n=0; for (m=1, 10^5, if(issquarefree(m), write("b062503.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 08 2009

is(n)=issquare(n,&n) && issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A062503(n):
    def f(x): return n-1+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    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**2 # Chai Wah Wu, Aug 19 2024

Numbers k such that Sum_{d|k} mu(d)*mu(k/d) = 1. - Benoit Cloitre, Mar 03 2004
a(n) = A000290(A005117(n)); A227291(a(n)) = 1. - Reinhard Zumkeller, Jul 07 2013
A000290 \ A062320. - R. J. Mathar, Jul 27 2013
a(n) ~ (Pi^4/36) * n^2. - Charles R Greathouse IV, Nov 24 2015
a(n) = A046692(a(n))^2. - Torlach Rush, Jan 05 2019
For all k in the sequence, Omega(k) = 2*omega(k). - Wesley Ivan Hurt, Apr 30 2020
Sum_{n>=1} 1/a(n) = zeta(2)/zeta(4) = 15/Pi^2 (A082020). - Amiram Eldar, May 22 2020

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14

Offset: 0

Peter Munn, Nov 10 2019

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.

Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

			Square array A(n,k) begins:
n\k |  0   1     2      3        4          5           6             7
----+------------------------------------------------------------------
   0|  1   1     1      1        1          1           1             1
   1|  1   2     4      8       16         32          64           128
   2|  1   3     9     27       81        243         729          2187
   3|  1   6    36    216     1296       7776       46656        279936
   4|  1   5    25    125      625       3125       15625         78125
   5|  1  10   100   1000    10000     100000     1000000      10000000
   6|  1  15   225   3375    50625     759375    11390625     170859375
   7|  1  30   900  27000   810000   24300000   729000000   21870000000
   8|  1   7    49    343     2401      16807      117649        823543
   9|  1  14   196   2744    38416     537824     7529536     105413504
  10|  1  21   441   9261   194481    4084101    85766121    1801088541
  11|  1  42  1764  74088  3111696  130691232  5489031744  230539333248
  12|  1  35  1225  42875  1500625   52521875  1838265625   64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
(End)

The range of values is A072774.
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
Other subtables: A182944, A319075, A329050.
Re-ordered subtable of A297845, A306697, A329329.
A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
Cf. A285322.

A(n,k) = A019565(n)^k.
A(k,n) = A225546(A(n,k)).
A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
A(2n,k) = A003961(A(n,k)).
A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
A(2^n,k) = A319075(k,n+1).
A(2^n, 2^k) = A329050(n,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022

A113849 Numbers whose prime factors are raised to the fourth power.

Original entry on oeis.org

16, 81, 625, 1296, 2401, 10000, 14641, 28561, 38416, 50625, 83521, 130321, 194481, 234256, 279841, 456976, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2825761, 3111696, 3418801, 4477456, 4879681, 6765201, 7890481

Offset: 1

Cino Hilliard, Jan 25 2006

This is essentially A005117 (the squarefree numbers) raised to the fourth power. - T. D. Noe, Mar 13 2013

All positive integers have a unique factorization into powers of squarefree numbers with distinct exponents that are powers of two. So every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (term of this sequence), at most one 8th power of a squarefree number, and so on. - Peter Munn, Mar 12 2020

			1296 = 16*81 = 2^4*3^4 so the prime factors of 1296, 2 and 3, are raised to the fourth power.

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

Proper subset of A000583.
Other powers of squarefree numbers: A005117(1), A062503(2), A062838(3),  A113850(5), A113851(6), A113852(7), A072774(all).
Cf. A000351, A225546.

Select[ Range@50^4, Union[Last /@ FactorInteger@# ] == {4} &] (* Robert G. Wilson v, Jan 26 2006 *)
nn = 50; t = Select[Range[2, nn], Union[Transpose[FactorInteger[#]][[2]]] == {1} &]; t^4 (* T. D. Noe, Mar 13 2013 *)
Rest[Select[Range[100], SquareFreeQ]^4] (* Vaclav Kotesovec, May 22 2020 *)

allpwrfact(n,p) = { local(x,j,ln,y,flag); for(x=4,n, y=Vec(factor(x)); ln = length(y[1]); flag=0; for(j=1,ln, if(y[2][j]==p,flag++); ); if(flag==ln,print1(x",")); ) } \\ All prime factors are raised to the power p

from math import isqrt
from sympy import mobius
def A113849(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    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**4 # Chai Wah Wu, Aug 19 2024

From Peter Munn, Oct 31 2019: (Start)
a(n) = A005117(n+1)^4.
{a(n)} = {A225546(A000351(n)) : n >= 0} \ {1}, where {a(n)} denotes the set of integers in the sequence.
(End)
Sum_{k>=1} 1/a(k) = zeta(4)/zeta(8) - 1 = 105/Pi^4 - 1. - Amiram Eldar, May 22 2020

More terms from Robert G. Wilson v, Jan 26 2006

A153160 a(n) = A007916(n)^6.

Original entry on oeis.org

64, 729, 15625, 46656, 117649, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 308915776, 481890304, 594823321, 729000000, 887503681

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

Cf. A007916, A153147, A153157, A153158, A153159, A113851.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^6

from sympy import mobius, integer_nthroot
def A153160(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 m**6 # Chai Wah Wu, Nov 21 2024

Edited and extended by Ray Chandler, Dec 22 2008

A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

Original entry on oeis.org

1, 0, 1, 7, 0, 9, 2, 7, 6, 9, 1, 3, 0, 4, 9, 9, 2, 7, 6, 6, 4, 3, 2, 7, 2, 1, 3, 3, 0, 9, 7, 9, 0, 9, 9, 2, 0, 4, 9, 2, 2, 1, 9, 0, 7, 9, 4, 9, 4, 1, 0, 1, 1, 3, 4, 6, 6, 4, 6, 5, 1, 7, 9, 3, 8, 1, 8, 9, 3, 5, 3, 3, 5, 8, 3, 4, 2, 2, 7, 9, 4, 3, 1, 8, 1, 5, 1, 5, 9, 6, 4, 7, 8, 5, 0, 6, 6, 8, 9, 7, 8, 4, 5, 4, 6, 5, 1, 0, 6, 4, 0, 2, 6, 1, 3, 3, 6, 9, 3, 0

Offset: 1

Ilya Gutkovskiy, Feb 25 2016

More generally, Product_{k >= 1} (1 + 1/prime(k)^m) = zeta(m)/zeta(2*m), where zeta(m) is the Riemann zeta function.

			1.0170927691304992766432721330979099204922190794941...

Eric Weisstein's World of Mathematics, Prime Products.

Cf. A005117, A082020, A113851, A157289, A157290, A157291.

RealDigits[Zeta[6]/Zeta[12], 10, 120][[1]]
RealDigits[675675/(691 Pi^6), 10, 120][[1]]

zeta(6)/zeta(12) \\ Amiram Eldar, Jun 11 2023

Equals zeta(6)/zeta(12).
Equals 675675/(691*Pi^6).
Equals Sum_{k>=1} 1/A005117(k)^6 = 1 + Sum_{k>=1} 1/A113851(k). - Amiram Eldar, Jun 27 2020

A336596 Numbers whose number of divisors is divisible by 7.

Original entry on oeis.org

64, 192, 320, 448, 576, 704, 729, 832, 960, 1088, 1216, 1344, 1458, 1472, 1600, 1728, 1856, 1984, 2112, 2240, 2368, 2496, 2624, 2752, 2880, 2916, 3008, 3136, 3264, 3392, 3520, 3645, 3648, 3776, 3904, 4032, 4160, 4288, 4416, 4544, 4672, 4800, 4928, 5056, 5103

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is 1 - zeta(7)/zeta(6) = 0.0088404638... (Sathe, 1945).

			64 is a term since A000005(64) = 7 is divisible by 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A008589, A059269, A336595.

Cf. A030516, A113851 and A138031 are subsequences.

q:= n-> is(irem(numtheory[tau](n), 7)=0):
select(q, [$1..5500])[];  # Alois P. Heinz, Jul 26 2020

Select[Range[5000], Divisible[DivisorSigma[0, #], 7] &]

A030516 UNION A030632 UNION A137484 UNION A137491 UNION A175745 UNION A175750 UNION ... - R. J. Mathar, May 05 2023

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A062503 Squarefree numbers squared.

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A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

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A113849 Numbers whose prime factors are raised to the fourth power.

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A153160 a(n) = A007916(n)^6.

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A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

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A336596 Numbers whose number of divisors is divisible by 7.

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