A092759 -id:A092759 - cp's OEIS frontend

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.

Cf. A085964, A085965, A085969.

fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)

list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179702(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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 n+x-sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

with(numtheory):nn:=100:
for n from 1 to nn do:
x:=divisors(n):n0:=nops(x):it:=0:
for i from 1 to n0 do:
  for j from i+1 to n0 do:
   if gcd(x[i],x[j])>1
    then
    it:=it+1:
    else
   fi:
  od:
od:
  printf(`%d, `,it):
od:

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A232107 Number of groups of order prime(n)^7.

Original entry on oeis.org

2328, 9310, 34297, 113147, 750735, 1600573, 5546909, 9380741, 23316851, 71271069, 98488755, 233043067, 384847485, 485930975, 751588475, 1356370173, 2299880351, 2710679045, 4306310927, 5734323819, 6578172579, 9721485395, 12413061671, 17537591045, 26866372821

Offset: 1

Eric M. Schmidt, Nov 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
E. A. O'Brien and M. R. Vaughan-Lee, The groups of order p^7 for odd prime p, J. Algebra 292, 243-258, 2005.
Index entries for sequences related to groups

Cf. A000001, A000679, A090091, A090130, A090140, A232105, A232106.

Cf. A092759.

A232107 := Concatenation([2328, 9310, 34297], List(Filtered([7..10^5], IsPrime), p -> 3 * p^5 + 12 * p^4 + 44 * p^3 + 170 * p^2 + 707 * p + 2455 + (4 * p^2 + 44 * p + 291) * Gcd(p-1, 3) + (p^2 + 19 * p + 135) * Gcd(p-1, 4) + (3 * p + 31) * Gcd(p-1, 5) + 4 *  Gcd(p-1, 7) + 5 * Gcd(p-1, 8) +  Gcd(p-1, 9))); # Muniru A Asiru, Nov 16 2017

a:= n-> `if`(n<4, [2328, 9310, 34297][n], (c-> 3391 +(1242+
    (404 +(122 +(27 +3*c)*c)*c)*c)*c +(339 +(52 +4*c)*c)*igcd(c, 3)+
    (155 +(21 +c)*c)*igcd(c, 4) +(34 +3*c)*igcd(c, 5) +4*igcd(c, 7)+
     5*igcd(c, 8) +igcd(c, 9))(ithprime(n)-1)):
seq(a(n), n=1..25);  # Alois P. Heinz, Nov 17 2017

def A232107(n) : p = nth_prime(n); return 2328 if p==2 else 9310 if p==3 else 34297 if p==5 else 3*p^5 + 12*p^4 + 44*p^3 + 170*p^2 + 707*p + 2455 + (4*p^2 + 44*p + 291)*gcd(p - 1, 3) + (p^2 + 19*p + 135)*gcd(p - 1, 4) + (3*p + 31)*gcd(p - 1, 5) + 4*gcd(p - 1, 7) + 5*gcd(p - 1, 8) + gcd(p - 1, 9)

For a prime p > 5, the number of groups of order p^7 is 3p^5 + 12p^4 + 44p^3 + 170p^2 + 707p + 2455 + (4p^2 + 44p + 291)gcd(p - 1, 3) + (p^2 + 19p + 135)gcd(p - 1, 4) + (3p + 31)gcd(p - 1, 5) + 4 gcd(p - 1, 7) + 5 gcd(p - 1, 8) + gcd(p - 1, 9).

A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106

Offset: 1

Amiram Eldar, Dec 01 2023

First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.
First differs from A077377 at n = 63, and from A348506 at n = 68.
Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).
The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

Amiram Eldar, Table of n, a(n) for n = 1..12232 (terms below 20000)
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A268335 and A270428.
Cf. A367802, A367803, A367804.
Subsequences: A005117, A092759.
Cf. A092246.
Cf. A077377, A348506.

odQ[n_] := OddQ[n] && OddQ[DigitCount[n, 2, 1]]; Select[Range[150], AllTrue[FactorInteger[#][[;;, 2]], odQ] &]

is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2 && hammingweight(f[i, 2])%2), return (0))); 1;}

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A365179 a(1) = 2; for n >= 2, a(n) = p^6 if p == 2 (mod 3), p^7 if p = 3 or p == 1 (mod 3), where p = prime(n).

Original entry on oeis.org

2, 2187, 15625, 823543, 1771561, 62748517, 24137569, 893871739, 148035889, 594823321, 27512614111, 94931877133, 4750104241, 271818611107, 10779215329, 22164361129, 42180533641, 3142742836021, 6060711605323, 128100283921, 11047398519097, 19203908986159, 326940373369

Offset: 1

Jianing Song, Aug 25 2023

Conjecture 1: a(n) is the smallest nontrivial power of p such that there exists a finite nontrivial group whose automorphism group is of order a(n).

Conjecture 2: for n >= 2, if |Aut(G)| = a(n), then |G| = a(n)/p, where p = prime(n). Moreover, G is unique up to isomorphism if p == 2 (mod 3).

			By the Peter Hegarty and Desmond MacHale link we have |Aut(G)| = 3^r => |Aut(G)| = 2187 = 3^7. It seems that if |Aut(G)| = 2187, then G = SmallGroup(729,m) for m = 90, 92 or 414.
It seems that |Aut(G)| = 5^r => |Aut(G)| >= 15625 = 3^6, and |Aut(G)| = 15625 => G = SmallGroup(3125,38).
It seems that |Aut(G)| = 7^r => |Aut(G)| >= 823543 = 7^7, and |Aut(G)| = 823543 => G = SmallGroup(117649,m) for m = 199, 824, 831 through 836.
It seems that |Aut(G)| = 11^r => |Aut(G)| >= 1771561 = 11^6, and |Aut(G)| = 1771561 => G = SmallGroup(161051,40).

Jianing Song, Table of n, a(n) for n = 1..10000
Peter Hegarty and Desmond MacHale, Minimal odd order automorphism groups, arXiv:0905.0993 [math.GR], 2009.

Cf. A030516 (sixth powers of primes), A092759 (seventh powers of primes).

a(n) = if(n==1, 2, my(p=prime(n)); if(p%3==2, p^6, p^7))

from sympy import prime
def A365179(n): return 2 if n == 1 else (p:=prime(n))**(6 if p%3 == 2 else 7) # Chai Wah Wu, Aug 26 2023

A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 500, 512, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000

Offset: 1

Amiram Eldar, Sep 13 2024

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.
Powerful numbers k such that A051903(k) is odd.
Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.
The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Complement of A376170 within A001694.
Intersection of A001694 and A376142.
Subsequence of A102834.
Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.
Cf. A051903, A082695, A376092.

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]

is(k) = {my(f = factor(k), e = f[,2]); #e && ispowerful(f) && vecmax(e) % 2;}

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).

A236209 Sum of the seventh powers of the first n primes.

Original entry on oeis.org

128, 2315, 80440, 903983, 20391154, 83139671, 493478344, 1387350083, 4792175530, 22042051839, 49554665950, 144486543083, 339240816964, 611059428071, 1117682548534, 2292393688371, 4781045173190, 7923788009211, 13984499614534, 23079619772925, 34127018292022

Offset: 1

Robert Price, Jan 20 2014

Robert Price, Table of n, a(n) for n = 1..1000
OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.
Partial sums of A092759.

Table[Sum[Prime[k]^7, {k, n}], {n, 1000}]

a(n) = Sum_{k=1..n} prime(k)^7.

A257659 Numbers that are not seventh powers, but can be written as the sum of the seventh powers of two or more of their prime factors.

Original entry on oeis.org

275223438741, 4561072096211306682, 9306119954843409393442022085025276

Offset: 1

Felix Fröhlich, Jul 26 2015

From Robert Israel, Nov 02 2016: (Start)
Each term is the sum of the seventh powers of three or more of its prime factors (since the sum of seventh powers of two distinct primes would not be divisible by those primes).
It is possible that the three terms shown are just the smallest examples presently known - there may be smaller ones.
Other terms include the following (and these too may not be the next terms):
48174957112005843444270083236899591347874 = 2^7 + 1259^7 + 648383^7.
343628633008268493930426179988576850614546787655 = 5^7 + 97^7 + 6178313^7.
1556588247952374145751498792380776025975963817566087335 = 5^7 + 941^7 + 55174589^7.
6777869034345885139001456808449377853222864558972446987604 = 2^7 + 337^7 + 182635307^7.
8652931112104420195217156139788964690213217995925746635175635 = 5^7 + 29^7 + 507351601^7.
33684756195335243623428442147352712728560450053586233129585039130540009686445977 = 3^7 + 2731^7 + 229647602339^7.
4218418507660286246537768294375414778864666339784229288571328866079146694717894140 = 5^7 + 7^7 + 2677^7 + 457863123059^7.
(End)

			275223438741 is not a seventh power, i.e., not a term of A001015, but is equal to the product of prime numbers 3 * 23 * 43 * 92761523, and 3^7 + 23^7 + 43^7 = 275223438741, so 275223438741 is a term of the sequence.

J. M. De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 362, ISBN 978-0-8218-4807-4.

Jean-Marie De Koninck & Florian Luca, Partial Sums of Powers of Prime Factors, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.6 (see p. 7).

Cf. A001015, A092759.

Edited by Robert Israel, Nov 02 2016

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