A075860 -id:A075860 - cp's OEIS frontend

A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

0, 2, 5, 7, 5, 3, 5, 13, 5, 7, 19, 7, 5, 13, 7, 31, 7, 7, 19, 5, 7, 43, 13, 19, 7, 13, 2, 5, 7, 61, 7, 19, 3, 73, 7, 7, 7, 43, 13, 19, 7, 13, 5, 7, 2, 103, 109, 3, 5, 31, 61, 7, 13, 19, 13, 31, 7, 139, 19, 2, 73, 151, 7, 5, 3, 43, 13, 31, 19, 13, 181, 19, 13, 7, 193, 23, 199, 29, 103, 73

Offset: 1

Labos Elemer, Apr 16 2003

nonn

Cf. A075860, A008472, A082087, A082088, A082881, 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[Max[0, Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]], {n, 80}]

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

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)).

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.

Original entry on oeis.org

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}]

A376070 a(n) is the number of distinct terms reached by iterating the function x->2+A075860(x), starting from x=n, with n>0.

Original entry on oeis.org

3, 2, 4, 1, 3, 4, 3, 2, 3, 4, 4, 4, 3, 4, 2, 2, 6, 4, 5, 4, 4, 3, 5, 4, 4, 2, 4, 4, 6, 4, 5, 2, 4, 5, 4, 4, 3, 4, 2, 4, 4, 4, 3, 3, 2, 4, 5, 4, 4, 4, 4, 2, 3, 4, 2, 4, 3, 5, 6, 4, 5, 4, 4, 2, 4, 2, 3, 5, 2, 4, 4, 4, 3, 2, 2, 4, 4, 4, 5, 4, 4, 3, 4, 4, 3, 2, 2, 3, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4

Offset: 1

Rafik Khalfi, Sep 08 2024

nonn

The sequence has another definition: a(n)= The number of distinct elements in the set A(n)={f^{k}(n);k>=0}, where f^{k} is the k-th iteration of the function f defined by f(n)=2+A075860(n), f^{0}(n)=n and n>0.
For all n>0, the set A(n) contains either the fixed point 4 or a cyclic component {5,7,9}.
For all n>1 and h in A(n)\{n}, h-2 is a prime number.
a(n)=1 if and only if n=4.
If (p,p+2) is a twin prime pair with p>7, then a(p+2)=a(p)-1.

			For n=3, 3->5->7->9->5->7->9-> ... and {5,7,9} is a cyclic component, then a(n)=number of distinct terms = 4.
For n=66, 66->4->4->4-> ... and 4 is a fixed point, then a(n)= number of distinct terms = 2.
For n=25, 25->7->9->5->7->9->5->7->9->... and {5,7,9} is a cyclic component, then a(n)=number of distinct terms = 4.

Cf. A075860.

f := proc(n) option remember:
    if isprime(n) then
        n
    else
        procname(convert(numtheory:-factorset(n), `+`))
    end if
end proc:
f(1) := 0:
g := proc(n)
    2 + f(n)
end proc:
A376070 := proc(n)
    local k, result:
    k := 1:
    result := n:
    while not (result = 4 or result = 5 or result = 7 or result = 9) do
        result := g(result):
        k := k + 1:
    end do:
    if result = 5 or result = 7 or result = 9 then
        return k + 2;
    else
        return k:
    end if
end proc:
map(A376070, [$1..200]);

from sympy import  primefactors
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(primefactors(n)), n)
def A376070(n):
    k = 1
    result = n
    while result not in {4, 5, 7, 9}:
        result = 2 + a(result, None)
        k += 1
    if result in {5, 7, 9}:
        return k + 2
    else:
        return k
print([A376070(i) for i in range(1, 200)])

A369558 a(n) is the minimum value of k > 0 such that A075860(n) = A075860(n+k) with n > 1.

Original entry on oeis.org

2, 6, 4, 1, 6, 3, 7, 5, 10, 110, 6, 9, 13, 1, 10, 193, 6, 15, 1, 9, 22, 250, 1, 10, 6, 1, 5, 370, 8, 27, 7, 23, 34, 1, 6, 398, 2, 6, 9, 610, 4, 39, 13, 7, 2, 730, 6, 1, 1, 9, 3, 850, 11, 9, 6, 28, 58, 1586, 3, 57, 8, 13, 2, 7, 3, 818, 25, 5, 11, 1210, 5, 69, 1, 11

Offset: 2

Rafik Khalfi, Jan 25 2024

nonn

			For n=5, A075860(5) = A075860(5+1), so a(5)=1.
For n=15, A075860(15) = A075860(15+1), so a(15)=1.

Cf. A075860.

f := proc(n) option remember;
    if isprime(n) then
        n;
    else
        procname(convert(numtheory:-factorset(n), `+`));
    fi;
end proc:
g := proc(n)
    local k;
    for k from 1 do
        if f(n+k) = f(n) then
            return k;
        fi;
    end do;
end proc:
map(g, [$2..100]);

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
f(n) = if (n==1, 0, fp(n, 0));
a(n) = my(k=1, fn=f(n)); while(f(n+k) != fn, k++); k; \\ Michel Marcus, Feb 20 2024

A371058 Numbers k divisible by A075860(k).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 41, 43, 47, 49, 52, 53, 59, 61, 63, 64, 65, 66, 67, 71, 73, 74, 79, 81, 83, 86, 89, 91, 95, 97, 99, 101, 103, 104, 106, 107, 109, 110, 113, 121, 125, 127, 128, 131, 132, 134, 137, 138, 139

Offset: 1

Rafik Khalfi, Mar 09 2024

nonn

Cf. A075860.

f := proc (n) option remember; if isprime(n) then n else procname(convert(numtheory:-factorset(n), `+`)) end if end proc:
g := proc (n) if `mod`(n, f(n)) = 0 then n end if end proc:
map(g, [$2 .. 100]);

a[n_] := If[n == 1, 0, FixedPoint[Total[FactorInteger[#][[All, 1]]]&, n]];r=140;Select[Range[2,r], Divisible[#,Part[Array[a,r],#]]&] (* James C. McMahon, Mar 10 2024 *)

A374104 a(n) = A075860(A075860(n) + 2) + 2.

Original entry on oeis.org

4, 4, 7, 4, 9, 9, 5, 4, 7, 5, 15, 9, 4, 7, 4, 4, 21, 9, 9, 5, 5, 4, 7, 9, 9, 4, 7, 7, 33, 5, 5, 4, 7, 9, 9, 9, 4, 5, 4, 5, 45, 9, 4, 4, 4, 9, 9, 9, 5, 5, 5, 4, 4, 9, 4, 7, 4, 5, 63, 5, 9, 7, 5, 4, 9, 4, 4, 9, 4, 7, 75, 9, 4, 4, 4, 5, 9, 9, 5, 5, 7, 4, 15, 9, 4, 4, 4, 4, 9, 5, 5, 9

Offset: 1

Rafik Khalfi, Jun 28 2024

nonn

(p,p+2) is a twin prime pair if and only if a(p)=p+4.

For all positive integers n, there exists a positive integer m such that a(n)

Cf. A075860, A008472.

g:= proc(n) option remember;
    if isprime(n) then n
   else procname(convert(numtheory:-factorset(n), `+`))
    fi
 end proc:
 g(1):= 0:
seq(2+g(2+g(i)),i=1..140);

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
f(n) = if (n==1, 0, fp(n, 0)); \\ A075860
a(n) = f(f(n)+2)+2; \\ Michel Marcus, Jun 28 2024

A374330 a(n) is the number of numbers k <= prime(n)^2 such that A075860(k) = prime(n).

Original entry on oeis.org

2, 2, 6, 8, 2, 10, 3, 14, 6, 8, 22, 7, 8, 21, 9, 14, 12, 45, 14, 17, 45, 17, 21, 20, 18, 17, 64, 21, 54, 28, 25, 22, 22, 72, 37, 82, 26, 28, 31, 43, 36, 93, 44, 95, 38, 95, 41, 38, 33, 106, 36, 49, 111, 65, 53, 53, 49, 113, 55, 68, 138, 80, 49, 50, 152, 61, 55, 43, 73, 120

Offset: 1

Rafik Khalfi, Jul 04 2024

nonn

For all n>=1, a(n)>=2.

			For n=3, prime(3)=5. The only integers k <= 5^2 such that A075860(k)=5 are 5,6,12,18,24 and 25. Therefore a(3)=6.

Cf. A001248, A075860.

f := proc (n)
    option remember;
    if isprime(n) then
        return n
    else
        return procname(convert(numtheory:-factorset(n), `+`))
    end if
end proc:
g := proc (n)
    local count, k;
    count := 0;
    for k from ithprime(n) to ithprime(n)^2 do
        if f(k) = ithprime(n) then
            count := count + 1
        end if
    end do;
    return count
end proc:
map(g, [$1 .. 80]);

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
f(n) = if (n==1, 0, fp(n, 0)); \\ A075860
a(n) = sum(k=1, prime(n)^2, f(k) == prime(n)); \\ Michel Marcus, Jul 04 2024

A375535 a(n) = n - A075860(n).

Original entry on oeis.org

1, 0, 0, 2, 0, 1, 0, 6, 6, 3, 0, 7, 0, 11, 13, 14, 0, 13, 0, 13, 14, 9, 0, 19, 20, 24, 24, 25, 0, 23, 0, 30, 30, 15, 30, 31, 0, 31, 37, 33, 0, 37, 0, 31, 43, 41, 0, 43, 42, 43, 44, 50, 0, 49, 53, 53, 44, 27, 0, 53, 0, 59, 56, 62, 60, 64, 0, 49, 67, 67, 0, 67, 0, 72, 73, 69, 72, 73

Offset: 1

Rafik Khalfi, Aug 18 2024

nonn

If p is a prime number, then a(p)=0.

			For n=15, a(15) = 15-2 = 13.

Cf. A075860.

f := proc(n)
    option remember:
    if isprime(n) then
        n
    else
        procname(convert(numtheory:-factorset(n), `+`))
    end if
end proc:
f(1) := 0:
seq(n - f(n), n = 1..100);

from sympy import primefactors
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(primefactors(n)), n)
print([i-a(i, None) for i in range(1, 100)])

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cp's OEIS Frontend

A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

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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.

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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.

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Mathematica

A376070 a(n) is the number of distinct terms reached by iterating the function x->2+A075860(x), starting from x=n, with n>0.

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A369558 a(n) is the minimum value of k > 0 such that A075860(n) = A075860(n+k) with n > 1.

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PARI

A371058 Numbers k divisible by A075860(k).

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A374104 a(n) = A075860(A075860(n) + 2) + 2.

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A374330 a(n) is the number of numbers k <= prime(n)^2 such that A075860(k) = prime(n).

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