A121559 -id:A121559 - cp's OEIS frontend

A121560 Lengths of blocks of zeros in sequence A121559.

2, 1, 1, 3, 1, 3, 1, 3, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 2, 1, 2, 3, 1, 3, 1, 3, 2, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 4, 2, 1, 1, 4, 3, 1, 3, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 1, 1

Offset: 1

Kerry Mitchell, Aug 07 2006

			a(1) = 2 because A121559 begins 1, 0, 0, 1, with 2 zeros between 1's.

Kerry Mitchell, Table of n, a(n) for n = 1..2782

Cf. A121559.

A200947 Sequence A007924 expressed in decimal.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 20, 32, 33, 64, 65, 66, 68, 128, 129, 256, 257, 258, 260, 512, 513, 514, 516, 517, 520, 1024, 1025, 2048, 2049, 2050, 2052, 2053, 2056, 4096, 4097, 4098, 4100, 8192, 8193, 16384, 16385, 16386, 16388, 32768, 32769, 32770

Offset: 0

Frank M Jackson, Nov 24 2011

nonn

			8=7+1, hence A007924(8)=10001, so a(8)=17.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A003714, A007895, A007924, A027941, A066352, A121559, A345297.

a:= proc(n) option remember; local m, p, r; m:=n; r:=0;
      while m>0 do
        if m=1 then r:=r+1; break fi;
        p:= prevprime(m+1); m:= m-p;
        r:= r+2^numtheory[pi](p)
      od; r
    end:
seq(a(n), n=0..52);  # Alois P. Heinz, Jun 12 2023

cprime[n_Integer] := If[n==0, 1, Prime[n]]; gentable[n_Integer] := (m=n; ptable={}; While[m != 0, (i = 0; While[cprime[i] <= m, i++]; j=0; While[j

a(n) = decimal(A007924(n)).

a(n) mod 2 = A121559(n) for n>=1. - Alois P. Heinz, Jun 12 2023

Edited by N. J. A. Sloane, May 20 2023

A121561 The number of iterations of "subtract the largest prime less than or equal to the current value" to go from n to the limiting value 0 or 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2

Offset: 1

Kerry Mitchell, Aug 07 2006

Number of steps to go from n to A121559(n).

The sequence has the form of blocks of numbers; see A121562 for the lengths of those blocks.

			a(9) = 2 because there are 2 steps in going from 9 to 0 in A121559: 9 mod 7 = 2 and 2 mod 2 = 0.

Robert G. Wilson v, Table of n, a(n) for n = 1..100000

Cf. A121559, A064722, a(n)=1: A093515, a(n)=2: A093513, a(n)=3: A138026, a(n)=4: A138027.

LrgstPrm[n_] := Block[{k = n}, While[ !PrimeQ@ k, k-- ]; k]; f[n_] := Block[{c = 0, d = n}, While[d > 1, d = d - LrgstPrm@d; c++ ]; c]; Array[f, 105] (* Robert G. Wilson v, Feb 29 2008 *)

from sympy import prevprime
def a(n): return 0 if n == 0 or n == 1 else 1 + a(n - prevprime(n+1))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jul 26 2022

A175077 Final number reached by iterating r -> A049711(r) starting at r = n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

See A175071 for starting n that reach 1, and A175072 for starting n that reach 2.

			Iteration procedure for n = 6: 6 mod 5 = 1 = a(6).
Iteration procedure for n = 7: 7 mod 5 = 2 = a(7).

Cf. A049711, A121559, A151799, A175071, A175072.

A151799 := proc(n) prevprime(n) ; end proc:
A049711 := proc(n) if n <=2 then n; else n-A151799(n) ; end if; end proc:
A175077 := proc(n) local r ; r := n ; while r > 2 do r := A049711(r) ; end do: r ; end proc:
seq(A175077(n),n=1..100) ; # R. J. Mathar, Feb 19 2010

f[n_] := Switch[n, 1, 1, 2, 2, _, n - NextPrime[n, -1]];
a[n_] := FixedPoint[f, n];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 13 2023 *)

a(A175071(k)) = 1; a(A175072(k)) = 2, any k. - R. J. Mathar, Feb 19 2010

a(n) = A121559(n-1) + 1 for n >= 2. - Pontus von Brömssen, Jul 31 2022

More terms from R. J. Mathar, Feb 19 2010

A175090 Composites c with result 0 under iterations of {r mod (max prime p <= r)} starting at r = c.

Original entry on oeis.org

9, 10, 15, 16, 21, 22, 25, 26, 28, 33, 34, 36, 39, 40, 45, 46, 49, 50, 52, 55, 56, 58, 63, 64, 66, 69, 70, 75, 76, 78, 81, 82, 85, 86, 88, 91, 92, 94, 96, 99, 100, 105, 106, 111, 112, 115, 116, 118, 120, 122, 123, 124, 126, 129, 130, 133, 134, 136, 141, 142

Offset: 1

Jaroslav Krizek, Jan 28 2010

nonn

Intersection of A002808 and A175089.

Composites c such that A121559(c) = 0. - Michel Marcus, Aug 22 2014

			Iteration procedure for a(3) = 15: 15 mod 13 = 2, 2 mod 2 = 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007917 and A064722 (both for the iterations).

g:= proc(x) if isprime(x) then 0 else x mod prevprime(x) fi end proc:
f:= proc(x) local y; y:= x; while y > 1 do y:= g(y) od; y = 0 end proc:
select(not(isprime) and f, [$4..200]); # Robert Israel, Feb 09 2015

Composites := Select[Range[2, 200], ! PrimeQ[#] &]; Select[Composites, PrimeQ[# - NextPrime[#, -1]] &] (* Carlos Eduardo Olivieri, Feb 09 2015 *)

Missing term 55 inserted, more terms added, Michel Marcus, Aug 22 2014

A175078 Number of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

a(123) = 3 (first occurrence of value 3), a(1357324) = 4 (first occurrence of value 4). I offer a prize of 100 liters of Pilsner Urquell to the discoverer of value of first occurrence of value 5. See A175071 (natural numbers m with result 1) and A175072 (natural numbers m with result 2). See A175077 = results 1 or 2 under iterations of {r mod (max prime p < r)} starting at r = n.
Essentially the same as A121561. [R. J. Mathar, Jan 28 2010]
The function r mod (max prime p < r), which appears in the definition, equals r - (max prime p < r) = A049711(r), because p < r < 2*p by Bertrand's postulate, where p is the largest prime less than r. - Pontus von Brömssen, Jul 31 2022

			a(123) = 3; iteration procedure for n = 123: 123 mod 113 = 10, 10 mod 7 = 3, 3 mod 2 = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A049711, A064722, A121559, A121561, A175071, A175072, A175077.

Array[-1 + Length@ NestWhileList[Mod[#, NextPrime[#, -1]] &, #, Not[1 <= # <= 2] &, 1, 120] &, 105] (* Michael De Vlieger, Oct 30 2017 *)

A175078(n) = if(n<=2,0,1+A175078(n%precprime(n-1))); \\ Antti Karttunen, Oct 30 2017

a(n) = A121561(n-1) for n >= 2, because the functions that are iterated (A049711 here, A064722 in A121561) satisfies A049711(r) = A064722(r-1) + 1. - Pontus von Brömssen, Jul 31 2022

Name shortened by Antti Karttunen, Oct 30 2017

A175089 Numbers m with result 0 under iterations of {r mod (max prime p <= r)} starting at r = m.

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100

Offset: 1

Jaroslav Krizek, Jan 28 2010

nonn

Complement of A175088.
Union of A000040 (primes) and A175090. [Jaroslav Krizek, Feb 05 2010]
Numbers m such that A121559(m) = 0. - Michel Marcus, Aug 22 2014

			Iteration procedure for a(3) = 5: 5 mod 5 = 0.
Iteration procedure for a(5) = 9: 9 mod 7 = 2, 2 mod 2 = 0.

Cf. A007917 and A064722 (both for the iterations).

More terms from Michel Marcus, Aug 22 2014

A135543 Record number of steps under iterations of "map n to n - (largest prime <= n)" (A064722) until reaching the limiting value 0 or 1. Also, places where A121561 reaches a new record.

Original entry on oeis.org

1, 2, 9, 122, 1357323

Offset: 0

Sergio Pimentel, Feb 22 2008

a(5) must be very large (> 100000000). Can anyone extend the sequence?
Conjecture: there exist positive values of n for which a(n) != A175079(n) - 1. - Jaroslav Krizek, Feb 05 2010
From Thomas R. Nicely's data (see link) it seems that the smallest known prime with following prime gap of length a(4)+1 or more is 90823#/510510 - 1065962 (39279 digits), so a(5) = A104138(a(4)) + a(4) <= 90823#/510510 - 1065962 + 1357323 = A002110(8787)/510510 + 291361. (The bounding primes of this prime gap are only known to be probable primes, but if either of them were not prime, the gap would only be larger and the bound on a(5) would still hold.) - Pontus von Brömssen, Jul 31 2022

			a(4) = 1357323 because after iterating n - (largest prime <= n) we get:
  1357323 - 1357201 = 122 =>
  122 - 113 = 9 =>
  9 - 7 = 2 =>
  2 - 2 = 0,
which takes 4 steps.

Thomas R. Nicely, First known occurrence prime gaps (1000000 to 999999998).

Cf. A002110, A064722, A104138, A121559, A121560, A121561, A175078, A175079.

LrgstPrm[n_] := Block[{k = n}, While[ !PrimeQ@ k, k-- ]; k]; f[n_] := Block[{c = 0, d = n}, While[d > 1, d = d - LrgstPrm@d; c++ ]; c]; lst = {}; record = -1; Do[ a = f@n; If[a > record, record = a; AppendTo[lst, a]; Print@ n], {n, 100}] (* Robert G. Wilson v *)

from sympy import prevprime
from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def f(n): return 0 if n == 0 or n == 1 else 1 + f(n - prevprime(n+1))
def agen(record=-1):
    for k in count(1):
        if f(k) > record: record = f(k); yield k
print(list(islice(agen(), 4))) # Michael S. Branicky, Jul 26 2022

Iterate n - (largest prime <= n) until reaching 0 or 1. Count the iterations required to reach 0 or 1 and determine if it is a new record.
From Pontus von Brömssen, Jul 31 2022: (Start)
a(n) = A104138(a(n-1)) + a(n-1) for n >= 2.
A121561(a(n)) = n.
a(n) = A175079(n) - 1 for n >= 1, i.e., the conjecture in the Comments is false. This follows from the result that A175078(n) = A121561(n-1) for n >= 2.
(End)

A175088 Numbers m with result 1 under iterations of {r mod (max prime p <= r)} starting at r = m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 18, 20, 24, 27, 30, 32, 35, 38, 42, 44, 48, 51, 54, 57, 60, 62, 65, 68, 72, 74, 77, 80, 84, 87, 90, 93, 95, 98, 102, 104, 108, 110, 114, 117, 119, 121, 125, 128, 132, 135, 138, 140, 143, 145, 147, 150, 152, 155, 158, 161, 164, 168, 171

Offset: 1

Jaroslav Krizek, Jan 28 2010

nonn

Terms are composites for all n >= 2.
Complement of A175089. [Jaroslav Krizek, Feb 05 2010]
Numbers m such that A121559(m) = 1. - Michel Marcus, Aug 22 2014

			Iteration procedure for a(6) = 14: 14 mod 13 = 1.
Iteration procedure for a(10) = 27: 27 mod 23 = 4, 4 mod 3 = 1.

Cf. A007917 and A064722 (both for the iterations).

(x /. Solve[Fold[Mod[#1, #2] &, x, Reverse[Prime /@ Range[40]]] == 1,
    x, Integers]) /. C[1] -> 0 (* Morgan L. Owens, Jun 22 2016 *)

More terms from Michel Marcus, Aug 22 2014

cp's OEIS Frontend

A121560 Lengths of blocks of zeros in sequence A121559.

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A200947 Sequence A007924 expressed in decimal.

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A121561 The number of iterations of "subtract the largest prime less than or equal to the current value" to go from n to the limiting value 0 or 1.

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A175077 Final number reached by iterating r -> A049711(r) starting at r = n.

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A175090 Composites c with result 0 under iterations of {r mod (max prime p <= r)} starting at r = c.

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A175078 Number of iterations of {r mod (max prime p < r)} needed to reach 1 or 2 starting at r = n.

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A175089 Numbers m with result 0 under iterations of {r mod (max prime p <= r)} starting at r = m.

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A135543 Record number of steps under iterations of "map n to n - (largest prime <= n)" (A064722) until reaching the limiting value 0 or 1. Also, places where A121561 reaches a new record.

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