Displaying 1-10 of 14 results found.
Continued fraction for constant defined in A065483.
+20
1
1, 2, 1, 16, 1, 1, 3, 1, 5, 17, 1, 5, 2, 10, 3, 2, 14, 4, 19, 2, 3, 1, 20, 1, 1, 1, 3, 4, 1, 1, 5, 1, 1, 7, 1, 5, 3, 2, 1, 1, 1, 2, 16, 1, 2, 31, 3, 1, 2, 1, 2, 27, 11, 1, 27, 6, 6, 1, 1, 10, 1, 3, 14, 3, 1, 1, 1, 3, 10, 1, 1, 2, 8, 3, 1, 1, 2, 1, 2, 5, 4, 3, 1, 2, 1, 1, 2, 1, 4, 1, 2, 1
PROG
(PARI) contfrac(prodeulerrat(1 + 1/(p^2*(p-1)))) \\ Amiram Eldar, Jul 08 2024
3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.
+10
88
1, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 432, 512, 625, 648, 729, 864, 1000, 1024, 1296, 1331, 1728, 1944, 2000, 2048, 2187, 2197, 2401, 2592, 2744, 3125, 3375, 3456, 3888, 4000, 4096, 4913, 5000, 5184, 5488, 5832, 6561, 6859, 6912, 7776, 8000
COMMENTS
Also called powerful_3 numbers.
REFERENCES
M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) ( A065483). - Amiram Eldar, Jun 23 2020 Numbers of the form x^5*y^4*z^3. There is a unique representation with x,y squarefree and coprime. - Charles R Greathouse IV, Jan 12 2022
MAPLE
isA036966 := proc(n)
local p ;
for p in ifactors(n)[2] do
if op(2, p) < 3 then
return false;
end if;
end do:
return true ;
end proc:
option remember;
if n = 1 then
1 ;
else
for a from procname(n-1)+1 do
if isA036966(a) then
return a;
end if;
end do:
end if;
MATHEMATICA
Select[ Range[2, 8191], Min[ Table[ # [[2]], {1}] & /@ FactorInteger[ # ]] > 2 &]
Join[{1}, Select[Range[8000], Min[Transpose[FactorInteger[#]][[2]]]>2&]] (* Harvey P. Dale, Jul 17 2013 *)
PROG
(Haskell)
import Data.Set (singleton, deleteFindMin, fromList, union)
a036966 n = a036966_list !! (n-1)
a036966_list = 1 : f (singleton z) [1, z] zs where
f s q3s p3s'@(p3:p3s)
| m < p3 = m : f (union (fromList $ map (* m) ps) s') q3s p3s'
| otherwise = f (union (fromList $ map (* p3) q3s) s) (p3:q3s) p3s
where ps = a027748_row m
(m, s') = deleteFindMin s
(z:zs) = a030078_list
(PARI) list(lim)=my(v=List(), t); for(a=1, sqrtnint(lim\1, 5), for(b=1, sqrtnint(lim\a^5, 4), t=a^5*b^4; for(c=1, sqrtnint(lim\t, 3), listput(v, t*c^3)))); Set(v) \\ Charles R Greathouse IV, Nov 20 2015 (PARI) list(lim)=my(v=List(), t); forsquarefree(a=1, sqrtnint(lim\1, 5), my(a5=a[1]^5); forsquarefree(b=1, sqrtnint(lim\a5, 4), if(gcd(a[1], b[1])>1, next); t=a5*b[1]^4; for(c=1, sqrtnint(lim\t, 3), listput(v, t*c^3)))); Set(v) \\ Charles R Greathouse IV, Jan 12 2022 (Python)
from math import gcd
from sympy import integer_nthroot, factorint
def f(x):
c = n+x
for w in range(1, integer_nthroot(x, 5)[0]+1):
if all(d<=1 for d in factorint(w).values()):
for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
c -= integer_nthroot(z//y**4, 3)[0]
return c
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
Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).
+10
15
2, 2, 0, 3, 8, 5, 6, 5, 9, 6, 4, 3, 7, 8, 5, 9, 7, 8, 7, 8, 7, 2, 8, 2, 8, 3, 1, 6, 4, 8, 0, 0, 8, 9, 6, 6, 2, 5, 6, 7, 1, 7, 3, 1, 9, 3, 7, 8, 7, 8, 5, 8, 6, 3, 4, 1, 7, 0, 4, 9, 5, 5, 4, 4, 8, 7, 1, 6, 6, 8, 8, 6, 8, 1, 1, 8, 5, 2, 6, 9, 5, 4, 9, 7, 5, 7, 2, 6, 6, 0, 4, 1, 9, 0, 1, 3, 9, 5, 6
LINKS
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
FORMULA
Also defined as: Sum_{m>=1} 1/(m* A000010(m)). See Weisstein link.
EXAMPLE
2.203856596437859787872828316480...
MATHEMATICA
$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n];
LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]]; (Pi^2/6)*Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10] ] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)
PROG
(PARI) prodeulerrat(1 + p/((p-1)^2*(p+1))) \\ Hugo Pfoertner, Jun 02 2020
Numbers whose minimum prime exponent is 2.
+10
6
4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800
COMMENTS
Or barely powerful numbers, a subset of powerful numbers A001694. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
FORMULA
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023
EXAMPLE
The sequence of terms together with their prime indices begins:
4: {1,1}
9: {2,2}
25: {3,3}
36: {1,1,2,2}
49: {4,4}
72: {1,1,1,2,2}
100: {1,1,3,3}
108: {1,1,2,2,2}
121: {5,5}
144: {1,1,1,1,2,2}
169: {6,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
225: {2,2,3,3}
288: {1,1,1,1,1,2,2}
289: {7,7}
324: {1,1,2,2,2,2}
361: {8,8}
392: {1,1,1,4,4}
400: {1,1,1,1,3,3}
MATHEMATICA
Select[Range[1000], Min@@FactorInteger[#][[All, 2]]==2&]
PROG
(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023 (Python)
from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
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, 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
for w in range(1, integer_nthroot(x, 5)[0]+1):
if all(d<=1 for d in factorint(w).values()):
for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
c += integer_nthroot(z//y**4, 3)[0]
return c
return bisection(f, n, n**2) # Chai Wah Wu, Oct 02 2024
CROSSREFS
Maximum instead of minimum gives A067259. Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.
+10
5
8, 27, 125, 81, 343, 3375, 1331, 243, 625, 9261, 2197, 10125, 4913, 35937, 42875, 729, 6859, 16875, 12167, 27783, 166375, 59319, 24389, 30375, 2401, 132651, 3125, 107811, 29791, 1157625, 50653, 2187, 274625, 185193, 456533, 50625, 68921, 328509, 614125
MATHEMATICA
f[p_, e_] := NextPrime[p]^(e + 2); a[1] = 8; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)
a(n) = p^2*(p-1), where p = prime(n).
+10
5
4, 18, 100, 294, 1210, 2028, 4624, 6498, 11638, 23548, 28830, 49284, 67240, 77658, 101614, 146068, 201898, 223260, 296274, 352870, 383688, 486798, 564898, 697048, 903264, 1020100, 1082118, 1213594, 1283148, 1430128, 2032254, 2230930
FORMULA
Product_{n>=1} (1 + 1/a(n)) = A065483. Product_{n>=1} (1 - 1/a(n)) = A065414. (End)
EXAMPLE
a(4) = 294 because the 4th prime number is 7, 7^2 = 49, 7-1 = 6 and 49 * 6 = 294.
MATHEMATICA
Table[p^3-p^2, {p, Prime[Range[40]]}] (* Harvey P. Dale, Jan 15 2015 *)
Numbers with all prime indices and exponents > 2.
+10
5
1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911
COMMENTS
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.
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p-1))) = (72/95)* A065483 = 1.0154153584... . - Amiram Eldar, May 28 2022
EXAMPLE
The initial terms together with their prime indices:
1: {}
125: {3,3,3}
343: {4,4,4}
625: {3,3,3,3}
1331: {5,5,5}
2197: {6,6,6}
2401: {4,4,4,4}
3125: {3,3,3,3,3}
4913: {7,7,7}
6859: {8,8,8}
12167: {9,9,9}
14641: {5,5,5,5}
15625: {3,3,3,3,3,3}
16807: {4,4,4,4,4}
24389: {10,10,10}
28561: {6,6,6,6}
29791: {11,11,11}
42875: {3,3,3,4,4,4}
MATHEMATICA
Select[Range[10000], #==1||!MemberQ[FactorInteger[#], {_?(#<5&), _}|{_, _?(#<3&)}]&]
CROSSREFS
The version for <= 2 instead of > 2 is A018256, # of compositions A137200. The version for indices and exponents prime (instead of > 2) is:
The partitions with these Heinz numbers are counted by A353501. A000726 counts partitions with multiplicities <= 2, compositions A128695.
A295341 counts partitions with some multiplicity > 2, compositions A335464.
Count of the 3-full divisors of n.
+10
4
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1
COMMENTS
a(n) is the number of divisors d of n with d an element of A036966. This is the 3-full analog of the 2-full case A005361.
FORMULA
Multiplicative with a(p^e) = max(1,e-1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p^2*(p-1))) ( A065483). (Ivić, 1978). - Amiram Eldar, Jul 23 2022 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)). - Amiram Eldar, Sep 21 2023
EXAMPLE
a(16)=3 because the divisors of 16 are {1,2,4,8,16}, and three of these are 3-full: 1, 8=2^3 and 16=2^4.
MAPLE
f:= n -> convert(map(t -> max(1, t[2]-1), ifactors(n)[2]), `*`):
MATHEMATICA
Table[Product[Max[{1, i - 1}], {i, FactorInteger[n][[All, 2]]}], {n, 1, 200}] (* Geoffrey Critzer, Feb 12 2015 *) Table[1 + DivisorSum[n, 1 &, AllTrue[FactorInteger[#][[All, -1]], # > 2 &] &], {n, 120}] (* Michael De Vlieger, Jul 19 2017 *)
PROG
(PARI) A190867(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= max(1, f[k, 2]-1); ); m; } \\ Antti Karttunen, Jul 19 2017 (Python)
from sympy import factorint
from operator import mul
def a(n): return 1 if n==1 else reduce(mul, [max(1, e - 1) for e in factorint(n).values()])
Multiplicative with a(p^e) = p^(e + 2), e > 0.
+10
4
1, 8, 27, 16, 125, 216, 343, 32, 81, 1000, 1331, 432, 2197, 2744, 3375, 64, 4913, 648, 6859, 2000, 9261, 10648, 12167, 864, 625, 17576, 243, 5488, 24389, 27000, 29791, 128, 35937, 39304, 42875, 1296, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 21296, 10125
FORMULA
Dirichlet g.f.: Product_{primes p} (1 + p^3/(p^s - p)).
Dirichlet g.f.: zeta(s-3) * zeta(s-1) * Product_{primes p} (1 + p^(4-2*s) - p^(6-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * zeta(3) * n^4 / 4, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.53589615382833799980850263131854595064822237... Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) = A065483. (End)
MATHEMATICA
g[p_, e_] := p^(e+2); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + p^3*X/(1 - p*X))[n], ", "))
a(1)=27; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+2}^{e_i+2}.
+10
2
27, 125, 343, 625, 1331, 42875, 2197, 3125, 2401, 166375, 4913, 214375, 6859, 274625, 456533, 15625, 12167, 300125, 24389, 831875, 753571, 614125, 29791, 1071875, 14641, 857375, 16807, 1373125, 50653, 57066625, 68921, 78125, 1685159
COMMENTS
Analog of A045967 a(1)=4; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^{e_i+1}. In a sense, n is the zeroth sequence in a family of sequences, A045967 is the first sequence in a family of sequences and a(n) is the second sequence in a family of sequences. If we had a(1) = 1 (instead of 4), then this would be multiplicative and a permutation of A353502. - Amiram Eldar, Aug 11 2022
MAPLE
A126272 := proc(n) local pf, i, p, e, resul ; if n = 1 then 27 ; else pf := ifactors(n)[2] ; resul := 1 ; for i from 1 to nops(pf) do p := op(1, op(i, pf)) ; e := op(2, op(i, pf)) ; resul := resul * nextprime(nextprime(p))^(e+2) ; od ; resul ; fi ; end: for n from 1 to 40 do printf("%d, ", A126272(n)) ; od ; # R. J. Mathar, Apr 20 2007
MATHEMATICA
f[p_, e_] := NextPrime[p, 2]^(e + 2); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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