A082020 -id:A082020 - cp's OEIS frontend

A157290 Decimal expansion of 105/Pi^4.

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

Offset: 1

R. J. Mathar, Feb 26 2009

The Product_{p = primes = A000040} (1+1/p^4), the quartic analog to A082020.

			1.077928136741855194... = (1+1/2^4)*(1+1/3^4)*(1+1/5^4)*(1+1/7^4)*...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 230.

Index entries for transcendental numbers

Cf. A005117, A013662, A013666, A030514, A082020, A113849, A347329.

Maple
```
evalf(105/Pi^4) ;
```

RealDigits[105/\[Pi]^4,10,150][[1]]  (* Harvey P. Dale, Mar 20 2011 *)

105/Pi^4 \\ Charles R Greathouse IV, Sep 30 2022

Equals A013662/A013666 = Product_{i>=1} (1+1/A030514(i)).
Equals Sum_{k>=1} 1/A005117(k)^4 = 1 + Sum_{k>=1} 1/A113849(k). - Amiram Eldar, May 22 2020
Equals 1/A347329. - Hugo Pfoertner, Jul 01 2024

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Antti Karttunen, May 28 2021

First negative term occurs as a(1440) = -18.

Antti Karttunen, Table of n, a(n) for n = 1..16560
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

f[p_, e_] := (p^(e+1) - 1)/(p-1); a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, n * (Times @@ (1 + 1/fct[[;; , 1]]) + 1) - Times @@ f @@@ fct]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023

A157291 Decimal expansion of Zeta(5)/Zeta(10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 26 2009

The product_{p = primes = A000040} (1+1/p^5), the fifth-power analog to A082020.

			1.035897477277500224... = (1+1/2^5)*(1+1/3^5)*(1+1/5^5)*(1+1/7^5)*...

Cf. A013663, A013668, A050997, A082020.

Cf. A005117, A113850.

Maple
```
evalf(Zeta(5)/Zeta(10)) ;
```

RealDigits[Zeta[5]/Zeta[10],10,120][[1]] (* Harvey P. Dale, Apr 06 2013 *)

Equals A013663/A013668 = Product_{i>=1} (1+1/A050997(i)).
Equals Sum_{k>=1} 1/A005117(k)^5 = 1 + Sum_{k>=1} 1/A113850(k). - Amiram Eldar, May 22 2020
Equals 93555 * zeta(5) / Pi^10. - Vaclav Kotesovec, May 22 2020

A289320 a(n) = A289310(n)^2 + A289311(n)^2.

Original entry on oeis.org

1, 5, 10, 25, 26, 50, 50, 125, 100, 130, 122, 250, 170, 250, 260, 625, 290, 500, 362, 650, 500, 610, 530, 1250, 676, 850, 1000, 1250, 842, 1300, 962, 3125, 1220, 1450, 1300, 2500, 1370, 1810, 1700, 3250, 1682, 2500, 1850, 3050, 2600, 2650, 2210, 6250, 2500

Offset: 1

Rémy Sigrist, Jul 02 2017

This sequence is totally multiplicative.
a(n) > n^2 for any n > 1.
If n is a square, then a(n) is a square.
If a(n) and a(m) are squares, then a(n*m) is a square.
a(n) is also a square for nonsquares n = 42, 168, 246, 287, 378, 672, 984, 1050, 1148, 1434, 1512, 1673, 2058, 2214, 2583, 2688, ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A066872, A082020, A289310, A289311.

f[p_, e_] := (p^2 + 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 13 2022 *)

a(n) = my (f=factor(n)); return (prod(i=1, #f~, (1 + f[i,1]^2) ^ f[i,2]))

from sympy import factorint
from operator import mul
from functools import reduce
def a(n): return 1 if n==1 else reduce(mul, [(1 + p**2)**k for p, k in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 03 2017

Totally multiplicative, with a(p^k) = (1 + p^2)^k for any prime p and k > 0.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 - 1/p^4)) = 0.4778963213... . - Amiram Eldar, Nov 13 2022
Sum_{n>=1} 1/a(n) = 15/Pi^2 (A082020). - Amiram Eldar, Dec 15 2022

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A082017 First row of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 2, 7, 6, 18, 15, 28, 29, 49, 45, 75, 66, 94, 91, 130, 120, 155, 153, 201, 190, 232, 231, 288, 276, 325, 326, 391, 378, 462, 435, 508, 496, 589, 561, 641, 630, 732, 703, 790, 780, 891, 861, 955, 946, 1066, 1035, 1136, 1128, 1257, 1225, 1333, 1326, 1464, 1431

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Cf. A074135, A082018, A082019, A082020, A082002, A082003, A082004, A082005.

Edited and more terms from Alois P. Heinz, Oct 26 2011

A082018 First column of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 4, 3, 13, 10, 21, 22, 40, 36, 64, 55, 81, 78, 115, 105, 138, 136, 182, 171, 211, 210, 265, 253, 300, 301, 364, 351, 433, 406, 477, 465, 556, 528, 606, 595, 695, 666, 751, 741, 850, 820, 912, 903, 1021, 990, 1089, 1081, 1208, 1176, 1282, 1275, 1411, 1378

Offset: 1

Amarnath Murthy, Apr 05 2003

nonn

This is the boustrophedon method of filling an array. Sums of antidiagonals of T are in A074132. Sums of antidiagonals of T divided by number of antidiagonals are in A074133. Diagonal of T is in A082019.

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Cf. A074135, A082017, A082019, A082020, A082002, A082003, A082004, A082005.

Edited and more terms from Alois P. Heinz, Oct 26 2011

A295657 Multiplicative with a(p^e) = p^floor((e-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 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

Offset: 1

Antti Karttunen, Nov 28 2017

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000188, A003557, A004526, A020639, A028234, A067029, A295658, A295659.
Cf. A004709 (positions of ones), A082020, A212793.
Cf. also A008834, A053150, A061704.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> p^Floor[(e - 1)/2]] &, 105] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^floor((f[i,2]-1)/2));} \\ Amiram Eldar, Nov 30 2022

a(1) = 1; for n > 1, a(n) = A020639(n)^A004526(A067029(n)-1) * a(A028234(n)).
a(n) = A000188(A003557(n)).
a(n) = 1 iff A212793(n) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/Pi^2 = 1.519817... (A082020). - Amiram Eldar, Nov 30 2022

A361810 a(n) is the sum of divisors of n that are both infinitary and exponential.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 25, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 36, 37, 38, 39, 50, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 68, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Mar 25 2023

The number of these divisors is A359411(n).

The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.

			a(8) = 10 since 8 has 2 divisors that are both infinitary and exponential, 2 and 8, and 2 + 8 = 10.

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

Cf. A002110, A049417, A051377, A082020, A115964, A138302, A359411.

f[p_, e_] := DivisorSum[e, p^# &, BitOr[#, e] == e &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(p,e) = sumdiv(e, d, p^d*(bitor(d, e) == e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); }

Multiplicative with a(p^e) = Sum_{d|e, bitor(d, e) == e} p^d.
a(n) >= n, with equality if and only if n is in A138302.
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, bitor(d, e) == e} p^(d-2*e))) = 0.51015879911178031024... .

A378769 Intersection of A375055 and A376936.

Original entry on oeis.org

5400, 9000, 10584, 10800, 13500, 16200, 18000, 21168, 21600, 24696, 26136, 27000, 31752, 32400, 36000, 36504, 37044, 40500, 42336, 43200, 45000, 48600, 49000, 49392, 52272, 54000, 62424, 63504, 64800, 67500, 68600, 72000, 73008, 74088, 77976, 78408, 81000, 84672

Offset: 1

Michael De Vlieger, Dec 13 2024

Let omega = A001221, bigomega = A001222, rad = A007947.
Powerful numbers k with bigomega(k) > omega(k) > 2 that are divisible by two distinct prime cubes p^3 and q^3.
Numbers k such that there exists (d, k/d), d | k, such that d neither divides nor is coprime to k/d and vice versa in the following 3 ways:
Type A: rad(d) does not divide d/k and rad(d/k) does not divide d
Type B: rad(d) divides d/k but rad(d/k) does not divide d
Type C: rad(d) | d/k and rad(d/k) | d, hence rad(d) = rad(d/k) = rad(k), a kind of coreful divisor pair.
Since (d, d/k) are noncoprime and do not divide one another, both must be composite, thus k is also composite.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
A378767 = { k : omega(k) > 1, p^3 | k for some prime p }, and
A376936 = { k : rad(k)^2 | k, p^3 | k and q^3 | k for distinct primes p, q }.
Therefore, we need only take intersection of A375055 and A376936.

			Table of the first 12 terms of this sequence, showing examples of types A, B, and C described in Comments.
   n     a(n)  Factors of a(n)    Type A      Type B      Type C
  ----------------------------------------------------------------
   1    5400   2^3 * 3^3 * 5^2    24 * 225    4 * 1350    60 * 90
   2    9000   2^3 * 3^2 * 5^3    18 * 500    4 * 2250    60 * 150
   3   10584   2^3 * 3^3 * 7^2    24 * 441    4 * 2646    84 * 126
   4   10800   2^4 * 3^3 * 5^2    48 * 225    8 * 1350    90 * 120
   5   13500   2^2 * 3^3 * 5^3    12 * 1125   9 * 1500    90 * 150
   6   16200   2^3 * 3^4 * 5^2    24 * 675    4 * 4050    60 * 270
   7   18000   2^4 * 3^2 * 5^3    18 * 1000   8 * 2250   120 * 150
   8   21168   2^4 * 3^3 * 7^2    48 * 441    8 * 2646   126 * 168
   9   21600   2^5 * 3^3 * 5^2    50 * 432    8 * 2700    90 * 240
  10   24696   2^3 * 3^2 * 7^3    18 * 1372   4 * 6174    84 * 294
  11   26136   2^3 * 3^3 * 11^2   24 * 1089   4 * 6534   132 * 198
  12   27000   2^3 * 3^3 * 5^3    24 * 1125   4 * 6750    60 * 450

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..16, showing divisor pairs of type A in light gray, type B in blue and gold, and type C in black.

Cf. A001694, A024619, A126706, A375055, A376936, A378767, A378900.

Cf. A082020, A082695.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^16],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[s, PrimeOmega[#] > PrimeNu[#] > 2 &]

Intersection of A375055, A376936, and A378767.
This sequence is { k : rad(k)^2 | k, bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.0025524144364532126894... . - Amiram Eldar, Dec 21 2024

cp's OEIS Frontend

A157290 Decimal expansion of 105/Pi^4.

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A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

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A157291 Decimal expansion of Zeta(5)/Zeta(10).

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A289320 a(n) = A289310(n)^2 + A289311(n)^2.

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A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

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A082017 First row of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

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A082018 First column of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

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A295657 Multiplicative with a(p^e) = p^floor((e-1)/2).

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