Rafik Khalfi - cp's OEIS frontend

A380098 Numbers whose sum of cubes of distinct prime factors is prime.

165, 210, 390, 399, 420, 462, 495, 561, 570, 595, 615, 630, 651, 780, 798, 825, 840, 924, 957, 1050, 1085, 1140, 1170, 1173, 1197, 1218, 1235, 1245, 1260, 1302, 1386, 1435, 1470, 1482, 1485, 1495, 1554, 1560, 1596, 1615, 1680, 1683, 1705, 1710, 1767, 1771, 1815, 1845, 1848, 1885, 1890, 1938, 1950, 1953

Offset: 1

Rafik Khalfi, Jan 12 2025

nonn

			165=3*5*11 and 3^3 + 5^3 + 11^3 = 1483, which is prime. Therefore, 165 is included.

Cf. A005064, A114522, A114989.

a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^3, j=1..nops(DPF)))=true then n else fi end: seq(a(n), n=1..2000);

Select[Range[2000], PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^3]]&]

from sympy import isprime, factorint
def ok(n): return isprime(sum(p**3 for p in factorint(n)))
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Jan 12 2025

A378423 a(n) is the number of distinct terms reached by iterating the function f(x) = 2 + A008472(x), starting from x=n.

Original entry on oeis.org

3, 2, 4, 1, 3, 4, 3, 2, 3, 4, 7, 4, 6, 8, 5, 2, 7, 4, 6, 4, 5, 6, 5, 4, 4, 8, 4, 8, 5, 5, 4, 2, 3, 6, 9, 4, 6, 6, 5, 4, 7, 9, 6, 6, 5, 5, 5, 4, 4, 4, 7, 8, 6, 4, 5, 8, 5, 4, 7, 5, 6, 10, 5, 2, 5, 5, 10, 6, 9, 3, 7, 4, 6, 8, 5, 6, 5, 5, 5, 4, 4, 6, 6, 9, 5, 6, 7, 6, 8, 5, 7, 5, 5, 8, 9, 4, 4, 8, 3, 4

Offset: 1

Rafik Khalfi, Nov 25 2024

nonn

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 f.

The set A(n) contains either the fixed point 4 or a cyclic component {5,7,9}.

			For n=33, 33->16->4->4-> ... and 4 is a fixed point, then a(n)= number of distinct terms = 3.
For n=66, 66->18->7->9->5->7 ... and {5,7,9} is a cyclic component, then a(n)= number of distinct terms = 5.

Cf. A008472.

f:= proc(n)
add( d, d= numtheory[factorset](n)):
end proc: f(1) := 0:
g:= proc(n)
   2 + f(n)
end proc:
 a:= 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(a, [$1..100]);

a[n_] := -1 + Length@ NestWhileList[2 + If[# == 1, 0, Total[FactorInteger[#][[;; , 1]]]] &, n, UnsameQ, All]; Array[a, 100] (* Amiram Eldar, Nov 26 2024 *)

from sympy import factorint
def a(n):
    reach = set()
    while n not in reach:
        reach.add(n)
        n = 2 + sum(factorint(n))
    return len(reach)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 26 2024

A377643 a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, with sopfr = A001414.

Original entry on oeis.org

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

Offset: 1

Rafik Khalfi, Nov 03 2024

nonn

			For n=1, the trajectory from n down to 8 comprises a(1) = 7 terms: 1 -> 2 -> 4 -> 6 -> 7 -> 9 -> 8.

Cf. A001414 (sopfr).

f := proc(n)
  add(op(1, i) * op(2, i), i = ifactors(n)[2]):
end proc:
g := proc(n)
  2 + f(n):
end proc:
A377643 := proc(n)
local k, result:
 k := 1:
result := n:
while result <> 8 do
result := g(result):
k := k + 1:
end do:
k:
end proc:
A377643(8) := 1:
map(A377643, [$1..100]);

s[n_] := 2 + Plus @@ Times @@@ FactorInteger[n]; s[1] = 2; a[n_] := Length@ NestWhileList[s, n, # != 8 &]; Array[a, 100] (* Amiram Eldar, Nov 07 2024 *)

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)])

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)])

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

A374163 a(1) = 1; for n>1 a(n) is the minimum value of k > 0 such that sigma^{k}(n)-1 is prime, if such a k exists; otherwise -1, where sigma^{k} is the k-th iteration of sigma=A000203.

Original entry on oeis.org

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

Offset: 1

Rafik Khalfi, Jun 29 2024

nonn

			For n=12, sigma^{4}(12)-1 = 360-1 = 359 is prime, and there is no positive k<4 such that sigma^{k}(12)-1 is prime, so a(12)=4.

Cf. A000203, A066421.

sigma_iterate := proc (n, k)
    local sigma_result, i:
    sigma_result := n:
    for i to k do
        sigma_result := sigma(sigma_result)
    end do:
    return sigma_result
end proc:
find_min_k := proc (n)
    local k, sigma_k_n, prime_candidate:
    k := 0:
    do
        k := k+1:
        sigma_k_n := sigma_iterate(n, k):
        prime_candidate := sigma_k_n - 1:
        if isprime(prime_candidate) then
            return k
        end if
    end do
end proc:
map(find_min_k, [$ 2 .. 100]);

A374163[n_] := If[n==1, 1, Length[NestWhileList[DivisorSigma[1, #]&, n, !PrimeQ[# - 1]&, {2, 1}]] - 1]; Array[A374163, 100] (* Paolo Xausa, Jul 24 2024 *)

a(n) = my(k=1, s=sigma(n)); while(!isprime(s-1), k++; s = sigma(s)); k; \\ Michel Marcus, Jun 29 2024

Offset corrected by N. J. A. Sloane, Jul 25 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

A374066 a(n) is the number of terms in the trajectory when the map x -> A067240(x) is iterated, starting from x = n until x = 0.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 7, 6, 6, 7, 7, 8, 6, 6, 8, 9, 6, 7, 7, 8, 6, 7, 7, 8, 7, 6, 8, 7, 6, 7, 9, 8, 6, 7, 7, 8, 6, 7, 10, 11, 7, 8, 7, 8, 8, 9, 9, 8, 7, 7, 8, 9, 6, 7, 9, 6, 8, 7, 7, 8, 8, 7, 8, 9, 7, 8, 8, 9, 7, 7, 7, 8, 6, 10, 8, 9, 7, 7, 9, 8, 8, 9

Offset: 1

Rafik Khalfi, Jun 27 2024

nonn

			For n=11, the trajectory from n down to 0 comprises a(11) = 7 terms: 11 -> 10 -> 5 -> 4 -> 2 -> 1 -> 0.

Cf. A000010, A067240.

f := proc(n)
    local e, j:
    e := ifactors(n)[2]:
    add((e[j][1] - 1) * e[j][1]^(e[j][2] - 1), j = 1 .. nops(e))
end proc:
 A374066:= proc(n)
    local count, current:
    count := 1:
    current := n:
    while current <> 0 do
        current := f(current):
        count := count + 1
    end do:
    return count
end proc:
map(A374066, [$1..200]);

f[p_, e_] := (p - 1)*p^(e - 1); s[n_] := s[n] = Plus @@ f @@@ FactorInteger[n]; a[n_] := Length[NestWhileList[s, n, # > 0 &]]; Array[a, 100] (* Amiram Eldar, Jun 27 2024 *)

A373437 Integers k such that sigma(sigma(2k))=2sigma(sigma(k)); sigma=A000203.

Original entry on oeis.org

2, 6, 14, 18, 38, 42, 50, 54, 62, 74, 86, 114, 122, 126, 134, 146, 150, 158, 162, 186, 206, 218, 222, 254, 258, 266, 302, 314, 326, 342, 350, 366, 378, 386, 398, 402, 422, 434, 438, 450, 458, 474, 482, 518, 542, 554, 558, 566, 578, 602, 618, 626, 654, 662, 666, 674, 686, 734, 746, 758, 762, 774, 794

Offset: 1

Rafik Khalfi, Jun 04 2024

nonn

Graeme L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Exp. Math., 5 (1996), 91-100.

Cf. A000203, A051027.

with(numtheory):
P := proc (q)
    local n, result:
    result := []:
    for n to q do
        if sigma(sigma(2*n)) = 2*sigma(sigma(n)) then
            result := [op(result), n]:
        end if
    end do:
    print(result):
end proc:
P(10^3);

Select[Range[800],DivisorSigma[1,DivisorSigma[1,2#]]==2DivisorSigma[1,DivisorSigma[1,#]]&] (* Stefano Spezia, Jun 05 2024 *)

from sympy import divisor_sigma as sigma
def P(q):
    result = []
    for n in range(1, q + 1):
        if sigma(sigma(2 * n)) == 2 * sigma(sigma(n)):
            result.append(n)
    print(result)
P(10**3)

cp's OEIS Frontend

User: Rafik Khalfi

A380098 Numbers whose sum of cubes of distinct prime factors is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

A378423 a(n) is the number of distinct terms reached by iterating the function f(x) = 2 + A008472(x), starting from x=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Python

A377643 a(n) is the number of terms in the trajectory when the map x -> 2+sopfr(x) is iterated, starting from x = n until x = 8, with sopfr = A001414.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

A374163 a(1) = 1; for n>1 a(n) is the minimum value of k > 0 such that sigma^{k}(n)-1 is prime, if such a k exists; otherwise -1, where sigma^{k} is the k-th iteration of sigma=A000203.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

A373437 Integers k such that sigma(sigma(2k))=2sigma(sigma(k)); sigma=A000203.