A065483 -id:A065483 - cp's OEIS frontend

A126272 a(1)=27; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+2}^{e_i+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

Offset: 1

Jonathan Vos Post, Mar 09 2007

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

Cf. A000040, A045967, A065483, A353502.

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

f[p_, e_] := NextPrime[p, 2]^(e + 2); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Sum_{n>=1} 1/a(n) = (72/95)*A065483 - 26/27. - Amiram Eldar, Aug 11 2022

More terms from R. J. Mathar, Apr 20 2007

A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/(zeta(2*n))^2 = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.81(0781476121562952243125904486251808972503617945007235890014471780028943
5600578871201157742402315484804630969609261939218523878437047756874095
5137481910274963820549927641099855282199710564399421128798842257597684
51519536903039073806).

			1.8107814761215629522431259...

Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

Mathematica
```
RealDigits[N[5005/2764,105]][[1]]
```

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Equals 5005/2764 = 5*7*11*13/(2^2*691).
Equals Product_{n>=1} 1+A000040(n)^2/A084920(n)^2.
Equals (13/9)*A340066.
From Vaclav Kotesovec, Dec 29 2020: (Start)
Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).
Equals 7*zeta(6)^2 / (4*zeta(12)).
Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)
Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024

A381824 Odd cubefull numbers: odd numbers that are divisible by the cube of any of their prime factors.

Original entry on oeis.org

1, 27, 81, 125, 243, 343, 625, 729, 1331, 2187, 2197, 2401, 3125, 3375, 4913, 6561, 6859, 9261, 10125, 12167, 14641, 15625, 16807, 16875, 19683, 24389, 27783, 28561, 29791, 30375, 35937, 42875, 50625, 50653, 59049, 59319, 64827, 68921, 78125, 79507, 83349, 83521, 84375, 91125

Offset: 1

Amiram Eldar, Mar 08 2025

Numbers whose prime factorization has primes and exponents that are larger than 2 (except for 1 whose prime factorization is empty).

Numbers k such that A020639(k) >= 3 and A051904(k) >= 3.

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

Intersection of A005408 and A036966.
Subsequences: A016755 (odd cubes), A381825 (odd cubefull exponentially odd numbers).
Cf. A020639, A051904, A065483.

Join[{1}, Select[Range[3, 10000, 2], Min[FactorInteger[#][[;; , 2]]] > 2 &]]

isok(k) = k == 1 || (k % 2 && vecmin(factor(k)[, 2]) > 2);

Sum_{n>=1} 1/a(n) = Product_{p prime >= 3} (1 + 1/(p^2*(p-1))) = (4/5) * A065483 = 1.07182732285947779727... .

A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Dec 28 2020

This is a rational number.
This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),
which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).
This number has decimal period length 230:
1.25(3617945007235890014471780028943560057887120115774240231548480463096960
9261939218523878437047756874095513748191027496382054992764109985528219
9710564399421128798842257597684515195369030390738060781476121562952243
12590448625180897250).

			1.25361794500723589001447178...

Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340065.

Mathematica
```
RealDigits[N[3465/2764, 105]][[1]]
```

default(realprecision, 105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

Equals 3465/2764 = 3^2*5*7*11/(2^2*691).
Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.
Equals (9/13)*A340065.

A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

Original entry on oeis.org

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200

Offset: 1

Amiram Eldar, Feb 19 2025

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

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

Subsequence of A001694, A377816 and A377818.
Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.
Cf. A036966, A065483.

With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]

isok(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x==2), e) == 1);

Sum_{n>=1} 1/a(n) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

cp's OEIS Frontend

A126272 a(1)=27; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+2}^{e_i+2}.

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A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A381824 Odd cubefull numbers: odd numbers that are divisible by the cube of any of their prime factors.

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A340066 Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

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A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

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