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 1-4 of 4 results.

A210452 Number of integers k

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 3, 1, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 5, 5, 4, 5, 5, 2, 4, 5, 5, 1, 5, 2, 6, 6, 5, 2, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 5, 2, 6, 3, 5, 7, 7, 7, 7, 3, 7, 7, 7, 3, 7, 4, 6, 8, 8, 8, 8, 3, 8, 8, 6, 3, 8, 8, 6, 8, 8, 3, 8, 8, 8, 7, 6, 8, 8, 3, 8, 8, 8
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 20 2013

Keywords

Comments

Conjecture: a(n)>0 for all n>4.
This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1.
Zhi-Wei Sun also made the following conjectures:
(1) For each integer n>197, there is a practical number k
(2) For every n=9,10,... there is a practical number k
(3) For any integer n>26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all practical.

Examples

			a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are both practical.

Links

Crossrefs

Cf. A005153, A210444, A210445, A071558, A208243, A208244, A208246, A208249, A219185, A209253, A209254, A219312, A219315, A219320.

Programs

  • Mathematica
    f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A275121 a(n) is the smallest multiple of n that is a practical number.

Original entry on oeis.org

1, 2, 6, 4, 20, 6, 28, 8, 18, 20, 66, 12, 78, 28, 30, 16, 204, 18, 228, 20, 42, 66, 276, 24, 100, 78, 54, 28, 348, 30, 496, 32, 66, 204, 140, 36, 666, 228, 78, 40, 820, 42, 860, 88, 90, 276, 1128, 48, 196, 100, 204, 104, 1272, 54, 220, 56, 228, 348, 1416, 60
Offset: 1

Views

Author

Lee A. Newberg, Jul 18 2016

Keywords

Comments

A rational in (0,1) as a fraction in lowest terms with denominator n, if expressed with denominator a(n) will have a practical-number denominator and can be written as an Egyptian fraction.
Note that a(n) exists for each n; a trivial upper bound is n * gpf(n)# = n * A034386(A006530(n)). - Charles R Greathouse IV, Jul 25 2016

Examples

			For example a(5)=20, indicating that a fraction with denominator 5 can be rewritten as a fraction with denominator 20, which is a practical number. Thus a fraction such as 4/5 can be written as 16/20. The new numerator 16 can be written as the sum of distinct divisors of 20 (16=10+5+1) because 20 is a practical number. The fractions 10/20, 5/20, and 1/20 are each a reciprocal: 1/2, 1/4, and 1/20. Thus 4/5 can be written as the sum of distinct reciprocals (Egyptian fraction expansion) as 4/5 = 1/2 + 1/4 + 1/20.

Links

Crossrefs

Cf. A005153 (practical numbers), A210445.
Cf. A006530, A034386.

Programs

  • PARI
    /* First declare the function is_a005153(n) as in A005153 */
a(n) = my(k=1); while(!is_a005153(k*n), k++); k*n \\ Felix Fröhlich, Jul 18 2016

Formula

a(n) = n * A210445(n).

Extensions

More terms from Felix Fröhlich, Jul 18 2016

A377311 Least positive integer k with k*n primitive practical.

Original entry on oeis.org

1, 1, 2, 5, 4, 1, 4, 11, 34, 2, 6, 17, 6, 2, 2, 17, 12, 17, 12, 1, 2, 3, 12, 31, 188, 3, 82, 1, 12, 1, 16, 37, 2, 6, 4, 41, 18, 6, 2, 47, 20, 1, 20, 2, 158, 6, 24, 67, 236, 94, 4, 2, 24, 41, 4, 59, 4, 6, 24, 79, 24, 8, 202, 67, 4, 1, 30, 3, 4, 2, 30, 97, 30, 9, 158, 3, 4, 1, 36, 97, 254, 10, 36, 101, 4, 10, 4, 1, 36, 79, 4, 3, 6, 12, 4, 127, 42, 118, 298, 47
Offset: 1

Views

Author

Frank M Jackson, Oct 24 2024

Keywords

Examples

			a(9) = 34. Consider the following sequence of 16 even multiples of 9 namely (18, 36, 54, . . . , 288, 306), all are practical numbers but only 9*34 = 306 is a primitive practical number. This is because 306 when divided by 3 is no longer practical whereas the other 15 even multiples remain practical when divided by 3.

Links

Crossrefs

Cf. A005153, A210445, A267124.

Programs

  • Mathematica
    PracticalQ[n_] := Module[{f,p,e,prod=1,ok=True},If[n<1||(n>1&&OddQ[n]),False,If[n==1,True,f=FactorInteger[n]; {p,e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1,prod],ok=False; Break[]]; prod=prod*p[[i]]^e[[i]],{i,Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n],#[[2]]>1&],m,ok=False},Do[If[!PracticalQ[n/plst[[m]]],ok=True,ok=False; Break[]],{m,1,Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
lst={}; Do[m=1; While[!PPracticalQ[n*m],m++]; AppendTo[lst,m],{n,1,100}]; lst

A362784 Least positive integer k with k primitive practical and k*n practical.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 6, 1, 2, 2, 6, 1, 6, 2, 2, 1, 20, 1, 20, 1, 2, 6, 20, 1, 6, 6, 2, 1, 20, 1, 20, 1, 2, 6, 6, 1, 20, 6, 2, 1, 20, 1, 20, 2, 2, 6, 28, 1, 6, 2, 6, 2, 28, 1, 6, 1, 6, 6, 30, 1, 30, 20, 2, 1, 6, 1, 30, 6, 6, 2, 30, 1, 30, 20, 2, 6, 6, 1, 42, 1, 2, 20, 42, 1, 6, 20, 6, 1, 42, 1, 6, 6, 6, 20, 6, 1, 42, 2, 2, 1
Offset: 1

Views

Author

Frank M Jackson, May 03 2023

Keywords

Comments

For all integers n>0 there exists k such that k*n is practical and k is primitive practical. For example, n*prime(f)# is practical where k = prime(f)# = A002110(f) is a primorial number and f is the prime index of the largest prime number in the factorization of n. All primorials are primitive practical numbers. The sequence above gives least k.

Examples

			a(5)=6 since 6*5=30 is practical and 6 is primitive practical. Also 4*5=20 is practical but 4 is not primitive practical.

Crossrefs

Cf. A005153, A210445, A267124.

Programs

  • Mathematica
    PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[!PracticalQ[n/plst[[m]]], ok = True, ok = False; Break[]], {m, 1, Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
lst = {}; Do[m=0; While[!PPracticalQ[m]||(!PracticalQ[m*n]&&m<10000), m++]; AppendTo[lst, m], {n, 1, 500}]; lst
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.