A082880 -id:A082880 - cp's OEIS frontend

A082883 Primes p(x) satisfying the following conditions: [1]# A082882(x)=1; [2]# {p(x),p(x+1)} are not twin primes; [3]# values of A075860(j) for j composites between these two non-twin primes are identical. See also A075860, A082880-A082882.

103, 457, 1009, 1663, 2953, 3079, 6043, 12007, 17707, 20749, 21499, 25579, 28537, 30703, 41227, 54367, 55663, 59443, 66973, 70309, 81547, 83557, 90019, 97003, 101359, 102559, 105367, 108499, 116239, 120847, 126019, 129733, 133873, 138403

Offset: 1

Labos Elemer, Apr 16 2003

nonn

			p[2033]=17007 is here because next prime is 17013;
for the five j inter-prime composites
i.e. if j is from {17008,..,17012} the values
of A075860 are identical: {7,7,7,7,7}, so A082882(2033)=1;
Smallest such example is a(1)=103 with this sophisticated
property:for i={104,105,106} the fixed points of A008472(i)
i.e. values of A075860(i) are uniformly equal to 2.

Cf. A008472, A075860, A082880-A082882.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Do[s=Length[Union[tik=Table[FixedPoint[sopf, j], {j, 1+Prime[n], -1+Prime[n+1]}]]]; If[Equal[s, 1]&&!PrimeQ[2+Prime[n]], Print[Prime[n]]], {n, 1, 100000}]

A082881 Least value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the smallest fixed-point[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

0, 2, 5, 2, 5, 2, 5, 7, 2, 7, 2, 2, 5, 2, 2, 2, 7, 2, 2, 5, 2, 3, 2, 5, 3, 13, 2, 5, 3, 2, 2, 2, 3, 2, 7, 5, 3, 13, 2, 3, 7, 2, 5, 3, 2, 2, 2, 2, 5, 7, 2, 7, 2, 2, 2, 2, 7, 2, 3, 2, 2, 2, 2, 5, 2, 2, 5, 2, 19, 2, 2, 2, 5, 2, 2, 3, 2, 3, 2, 2, 17, 2, 5, 5, 2, 2, 2, 7, 23, 2, 2, 3, 3, 3, 5, 2, 2, 19, 2, 5, 2, 3, 2

Offset: 1

Labos Elemer, Apr 16 2003

nonn

			Between p(23)=83 and p(24)=89, the relevant fixed points are {5,13,2,2,13}, of which the smallest is 2=a(24).

Cf. A075860, A008472, A082087, A082088, A082880, A029908.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; sopf[x_] := Apply[Plus, ba[x]]; Table[Min[Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]], {n, 2, 103}]

a(n) = Min_{x=1+prime(n)..prime(n+1)-1} A075860(x).

A082882 Number of distinct values of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the counts of different fixed-points[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Apr 16 2003

nonn

This count is smaller than A001223[n]-1 and albeit not frequently but it can be one even if primes of borders are not twin primes. Such primes are collected into A082883.

			Between p(23)=83 and p(24)=89, the relevant fixed points are
{5,13,2,2,13}, i.e., four are distinct from the 5 values, a(24)=4;
between p(2033)=17707 and p(2034)=170713, the fixed-point set is {5,5,5,5,5}, so a(2033)=1, so a(88)=1.

Cf. A075860, A008472, A082087, A082088, A082880, A082081, A001223.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[Length[Union[Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]]], {n, 1, 1000}]

a(n) = Card(Union(A075860(x)); x=1+p(n), ..., -1+p(n+1)).

cp's OEIS Frontend

A082883 Primes p(x) satisfying the following conditions: [1]# A082882(x)=1; [2]# {p(x),p(x+1)} are not twin primes; [3]# values of A075860(j) for j composites between these two non-twin primes are identical. See also A075860, A082880-A082882.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A082881 Least value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the smallest fixed-point[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula