A048133 -id:A048133 - cp's OEIS frontend

A233341 Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.

1, 4, 8, 12, 25, 30, 32, 48, 63, 70, 75, 80, 165, 176, 189, 192, 289, 306, 315, 320, 385, 392, 507, 520, 575, 598, 621, 644, 841, 858, 957, 968, 1015, 1044, 1071, 1088, 1105, 1122, 1425, 1444, 1463, 1470, 1771, 1782, 1935, 1978, 2145, 2156, 2303, 2350, 2397

Offset: 1

Clark Kimberling, Dec 07 2013

r(m) = 1 if and only if m = 1 or m is a prime. Conjecture: Every positive integer divides infinitely many terms of this sequence.
Sequence is empirically observed to be strictly increasing for n <= 1000, in contrast to similar map in A050710. - Christian N. K. Anderson, May 05 2023
Observe that for ~4/7 of the first thousand terms, r(a(n)) = a(n-1); e.g., a(12)=80, r(80)=75=a(11) -> 70=a(10) -> 63=a(9). However, the other ~3/7 take a different route to zero; e.g., a(9)=63 decreases by 7 at all 9 steps. Contrast A048133, where every term's iteration ends when r(k)=5. - Christian N. K. Anderson, May 05 2023

			r(8) = 8 - 2 = 6; r(6) = 6 - 3 = 3; r(3) = 3 - 3 = 0. Thus 3 applications of r map 8 to 0, whereas 1 or 2 applications suffice for n < 8. Therefore, a(3) = 8.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000 (first 200 terms from Clark Kimberling)

Cf. A233342.

z = 10000; h[n_] := h[n] = n - FactorInteger[n][[-1, 1]]; t[n_] := Drop[FixedPointList[h, n], -2]; Table[t[n], {n, 1, z}]; a = Table[Length[t[n]], {n, 1, z}]; f[n_] := First[Flatten[Position[a, n]]]; Table[f[n], {n, 1, 80}]

For 228 <= n <= 1000, a(n) ~ 0.8526*n^2.023 to within 4% (empirical observation). - Christian N. K. Anderson, May 05 2023

A084931 Consider trajectory of n under repeated applications of the function f(x) = 'Sum of the prime factors of x (with multiplicity)' (see A029908). Sequence gives composite numbers n that end at a prime m that divides n and m is greater than any m's seen already.

Original entry on oeis.org

15, 21, 182, 494, 1219, 2852, 3182, 9782, 19339, 19982, 22454, 72836, 76814, 102134, 156782, 192182, 423182, 750979, 758894, 1364534, 1465454, 1548782, 2376182, 3379982, 4066934, 4204982

Offset: 1

Sven Simon, Jun 12 2003

With a prime triple (p,p+4,p+6), the number a(n) = 2*p*(p+6) is always in the sequence, f( f( 2*p*(p+6) )) = f( 2*(p+4) ) = p+6. Such prime triples can be found in sequence A022005.
As long as two successive triples (p1,p1 + 4,p1 + 6) and (p2,p2 + 4,p2 + 6) of A022005 have p2 < 1.2*p1, no other numbers occur in the sequence between a(n1) and a(n2), this holds at least for larger p1 > 500. Other types of prime sets occurring in the sequence: (p,p+4,3p-4) with F( F( (p+4)*(3p-4))) = F( 4p ) = p + 4 (p,p+6,p+8) with F( F( 4*p*(p+8) )) = F( 2*(p+6) ) = p + 8.
Large examples of (p,p+4,++6)-triples: (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, + 13 (4135 digits, David Broadhurst) (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, +13 (4134 digits, David Broadhurst) Record examples of prime triples can be found on Tony Forbes's web site. There are triples of type (p,p+4,p+6) too.

			a(10) = 19982: f(f(19982)) = f(f(2*97*103)) = f(2+97+103) = f(202) = f(2*101) = 2+101 = 103.

Cf. A022005, A048133, A084932 (primes reached).

A121360 Smallest k such that A002217(k)=n.

Original entry on oeis.org

2, 1, 8, 14, 26, 62, 134, 393, 1257, 4659, 9314, 27933, 167073, 334142, 4677862, 14033577, 56134292, 2133102298, 11788191218, 52541651594, 536226160966, 2681130804805, 8512643845707, 129535245187539

Offset: 1

Klaus Brockhaus, Jul 27 2006

Least number which becomes prime or 0 or 4 after n-1 iterations of f(x) = sum of prime factors of x (cf. A001414). Agrees with A048133 except for the second term.
a(21) <= 536226160966. a(22) <= 2681130804805. a(23) <= 8512643845707. - Donovan Johnson, Oct 16 2010
a(25) <= 1679460336114602. - Danila Potapov, Dec 01 2024

Cf. A002217, A001414, A048133.

a(19)-a(20) from Donovan Johnson, Oct 16 2010

a(21)-a(24) from Danila Potapov, Dec 01 2024

A362684 a(n) is the index at which n first occurs in A362681.

Original entry on oeis.org

1, 3, 5, 7, 26, 49, 632, 1682

Offset: 0

Kevin L. Schwartz and Christian N. K. Anderson, May 01 2023

Among terms n=1..7, each number iterates through the previous terms. For example, the first iteration takes 1682 to 632=a(6), the second takes 632 to 49=a(5) and so forth.
1682 is the only number < 10^6 that requires 7 iterations to reach completion.
a(8), if it exists, is larger than 10^6.
a(8) > 10^9. - Michel Marcus, Oct 11 2023

			a(7) = 1682 because the first number to require 7 iterations to reach 2 or less is 1682.
Each iteration step is n <- 2n - {sum of proper factors of 2n} = A157449(2n); for n = 1682, this gives 2n = 3364, 3364 - (2 + 4 + 29 + 58 + 116 + 841 + 1682) = 632, which is a(6).
Note that the proper factors of x are all divisors of x other than 1 and x.

Cf. A362681, A157449.

Cf. A048133.

a(0)=1 prepended by Michel Marcus, Oct 11 2023

A318387 Starting with a(1) = 6, a(n) is the smallest number whose sum of prime divisors (taken with multiplicity) is a(n-1). In other words, a(n) = A056240(a(n-1)).

Original entry on oeis.org

6, 8, 15, 26, 69, 134, 393, 1556, 4659, 9314, 27933, 921327, 85680249, 171360494, 2227686253, 17821489976, 124750429783, 19336316610785, 4544034403522255, 3567067006764843005, 203322819385596050031, 25008706784428314148401, 825287323886134366896771, 91606892951360914725537141, 1923744751978579209236279751

Offset: 1

David James Sycamore, Aug 25 2018

nonn

Any nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a nontrivial sequence (because A056240(r)=r for r=4 or a prime). In this sequence a(1) is set to 6 because it is the smallest composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non-stationary sequence of this kind.

Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).

			a(2) = 8, the smallest number whose sopfr is 6: A056240(8) = 6;
a(3) = 15, the smallest number whose sopfr is 8: A056240(8) = 15; etc.

Cf. A001414, A048133, A002217, A056240.

With[{s = Array[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &,10^6]}, Nest[Append[#, First@ FirstPosition[s, #[[-1]] ]] &, {6}, 11]] (* Michael De Vlieger, Aug 25 2018 *)

a(n) = A056240(a(n-1)); A002217(a(n)) = 1 + A002217(a(n - 1))

a(13)-a(17) from Giovanni Resta, Aug 28 2018

Terms a(18) onward from Max Alekseyev, Sep 20 2024

cp's OEIS Frontend

A233341 Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.

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A084931 Consider trajectory of n under repeated applications of the function f(x) = 'Sum of the prime factors of x (with multiplicity)' (see A029908). Sequence gives composite numbers n that end at a prime m that divides n and m is greater than any m's seen already.

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A121360 Smallest k such that A002217(k)=n.

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A362684 a(n) is the index at which n first occurs in A362681.

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A318387 Starting with a(1) = 6, a(n) is the smallest number whose sum of prime divisors (taken with multiplicity) is a(n-1). In other words, a(n) = A056240(a(n-1)).

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