cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 22 results. Next

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2022

Keywords

Comments

First differs from A053149 and A356193 at n=16.

Links

Crossrefs

Cf. A001694, A013664, A268335, A335988.
Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

Programs

  • Mathematica
    f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
  • PARI
    a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Formula

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A335988.
a(n) = A356191(n) iff n is a powerful number (A001694).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022

A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.
Also, the largest exponentially odd unitary divisor of the powerful part on n.
Also, the powerful part of the largest exponentially odd unitary divisor of n.

Links

Crossrefs

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.
Cf. A056191, A366126, A367513.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1 || EvenQ[e], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, f[i, 1]^f[i, 2]));}

Formula

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.
a(n) = A350389(A057521(n)).
a(n) = A057521(A350389(n)).
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.

A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A359411 and A367516 at n = 64.
Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

Links

Crossrefs

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.
Cf. A359411, A367516.

Programs

  • Mathematica
    f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, 2));}

Formula

a(n) = A034444(A368167(n)).
Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

A365298 a(n) is the smallest number k such that k*n is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 2, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 18, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364
Offset: 1

Views

Author

Amiram Eldar, Aug 31 2023

Keywords

Links

Crossrefs

Cf. A157292, A335988, A356192.

Programs

  • Mathematica
    f[p_, e_] := p^If[OddQ[e], Max[e, 3] - e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i, 2]%2, max(f[i, 2], 3) - f[i,2], 1))};

Formula

Multiplicative with a(p) = p^2, a(p^e) = p if e is even, and a(p^e) = 1 is e is odd and > 1.
a(n) = A356192(n)/n.
a(n) = 1 if and only if n is in A335988.
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(3*s-2) - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (2*Pi^4/315) * Product_{p prime} (1 - p^2 - p^3 + p^4 + p^8 + p^9)/(p^8*(p+1)) = 0.207915752545... .

A371415 Dedekind psi function applied to the cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 12, 36, 48, 150, 192, 432, 324, 392, 768, 1728, 1800, 1452, 3888, 3072, 2916, 2366, 4704, 3750, 5400, 6912, 7200, 5202, 7220, 15552, 12288, 14112, 17424, 18816, 12696, 27648, 28800, 19208, 34992, 28392, 26244, 25230, 45000, 64800, 30752, 48600, 62208, 49152
Offset: 1

Views

Author

Amiram Eldar, Mar 22 2024

Keywords

Links

Crossrefs

Cf. A001615, A065464, A098198, A335988.
Similar sequences: A323332, A371413, A371414.

Programs

  • Mathematica
    psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[40000], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]
  • PARI
    dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f, ans); print1(1, ", "); for(k = 2, max, f = factor(k); ans = 1; for (i = 1, #f~, if (f[i, 2] == 1 || !(f[i, 2] % 2), ans = 0; break)); if(ans, print1(dedpsi(f), ", ")));}

Formula

a(n) = A001615(A335988(n)).
Sum_{n>=1} 1/a(n) = (Pi^4/36) * Product_{p prime} (1 - (2*p-1)/p^3) = A098198 * A065464 = 1.158760974549073218921828... .

A371414 Euler phi function applied to the cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 4, 18, 16, 100, 64, 72, 162, 294, 256, 288, 400, 1210, 648, 1024, 1458, 2028, 1176, 2500, 1800, 1152, 1600, 4624, 6498, 2592, 4096, 5292, 4840, 4704, 11638, 4608, 6400, 14406, 5832, 8112, 13122, 23548, 10000, 7200, 28830, 16200, 10368, 16384, 21780, 18496, 19360
Offset: 1

Views

Author

Amiram Eldar, Mar 22 2024

Keywords

Links

Crossrefs

Cf. A000010, A098198, A335988.
Similar sequences: A323333, A371414, A371415.

Programs

  • Mathematica
    Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]
  • PARI
    lista(max) = {my(f, ans); print1(1, ", "); for(k = 2, max, f = factor(k); ans = 1; for (i = 1, #f~, if (f[i, 2] == 1 || !(f[i, 2] % 2), ans = 0; break)); if(ans, print1(eulerphi(f), ", ")));}

Formula

a(n) = A000010(A335988(n)).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*(p+1))) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 2/p^4) = 1.43921640806700099050... .

A368171 a(n) is the smallest divisor d of n such that n/d is a cubefull exponentially odd number (A335988).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 6, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2023

Keywords

Comments

First differs from A050985 at n = 32, and from A367699 at n = 64.

Links

Crossrefs

Cf. A004709, A335988, A368170.
Cf. A050985, A367699.

Programs

  • Mathematica
    f[p_, e_] := If[e <= 2, p^e, If[EvenQ[e], p, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2] <= 2, f[i,1]^f[i,2], if(f[i,2]%2, 1, f[i,1])))};

Formula

Multiplicative with a(p^e) = p^e if e <= 2, a(p^e) = 1 if e is odd and e > 1, and p otherwise.
a(n) = n/A368170(n).
a(n) >= 1, with equality if and only if n is in A335988.
a(n) <= n, with equality if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^5 + 1/p^6) = 0.42246686366220037592... .

A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

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, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Sep 07 2019

Keywords

Comments

First differs from A050361 at n = 64.
From Amiram Eldar, Sep 08 2023: (Start)
The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024

Links

Crossrefs

Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.
Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.

Programs

  • Mathematica
    fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); Array[a, 100]
  • PARI
    a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023

Formula

Multiplicative with a(p^e) = floor((e+1)/2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022
a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023
From Amiram Eldar, Sep 08 2023: (Start)
a(n) = A046951(A003557(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)

Extensions

Name corrected by Amiram Eldar, Sep 08 2023

A374459 Nonsquarefree exponentially odd numbers.

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, 216, 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
Offset: 1

Views

Author

Amiram Eldar, Jul 09 2024

Keywords

Comments

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.
Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.
All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.
The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

Examples

			8 = 2^3 is a term since 3 is odd and larger than 1.

Links

Crossrefs

Intersection of A013929 (or A046099) and A268335.
Subsequence of A301517.
Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.
Cf. A005117, A374460.

Programs

  • Mathematica
    q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]
  • PARI
    is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

Formula

a(n) = A268335(A374460(n)).
Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

A355038 a(n) = n^2 times the squarefree kernel of n.

Original entry on oeis.org

1, 8, 27, 32, 125, 216, 343, 128, 243, 1000, 1331, 864, 2197, 2744, 3375, 512, 4913, 1944, 6859, 4000, 9261, 10648, 12167, 3456, 3125, 17576, 2187, 10976, 24389, 27000, 29791, 2048, 35937, 39304, 42875, 7776, 50653, 54872, 59319, 16000, 68921, 74088, 79507, 42592, 30375
Offset: 1

Views

Author

Peter Munn, Jun 16 2022

Keywords

Links

Crossrefs

The range of values is A335988.
Cf. A007947, A064549, A065463, A102631, A356191.

Programs

  • Mathematica
    a[n_] := n^2 * Times @@ FactorInteger[n][[;; , 1]]; Array[a, 50] (* Amiram Eldar, Jun 18 2022 *)
  • PARI
    a(n) = n^2 * factorback(factor(n)[,1]);

Formula

Multiplicative with a(p^e) = p^(2e+1).
a(n) = n^2 * A007947(n).
a(n) = A064549(n^2). - Amiram Eldar, Jun 20 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - 1/(p*(p+1))) = A065463 / 4 = 0.1761105502... . - Amiram Eldar, Nov 13 2022
a(n) = A356191(n^2). - Amiram Eldar, Nov 30 2023
Showing 1-10 of 22 results. Next

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