A309892 -id:A309892 - cp's OEIS frontend

A356438 Numbers k such that A309892(k) = k/gpf(k), where gpf = A006530.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85

Offset: 1

Jianing Song, Aug 07 2022

Note that A309892(k) <= k/gpf(k); these sequence lists k such that the equality holds.

For k >= 2, k is a term if and only if k/gpf(k) < nextprime(gpf(k)), where nextprime = A151800.

			15 is a term since the number of steps needed to reach 0 of the iteration x -> x - gpf(x) starting at 15 is 3: 15 -> 10 -> 5 -> 0, and 3 = 15/gpf(15).

Jianing Song, Table of n, a(n) for n = 1..36902 (all terms <= 50000)

Other than 1, indices of 1 in A356428.
Includes A000040 and A001358 as subsequences.
Complement of A356441.
Cf. A309892, A006530 (gpf), A151800, A076563.

isA356438(n) = if(n>1, my(p=vecmax(factor(n)[, 1])); n/p

A356441 Numbers k such that A309892(k) < k/gpf(k), where gpf = A006530; complement of A356438.

Original entry on oeis.org

8, 16, 18, 24, 27, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 189, 192, 196, 198, 200, 210, 216, 220, 224, 225, 231, 234, 240, 242, 243

Offset: 1

Jianing Song, Aug 07 2022

k is a term if and only if k/gpf(k) > nextprime(gpf(k)), where nextprime = A151800.

			8 is a term since the number of steps needed to reach 0 of the iteration x -> x - gpf(x) starting at 8 is 3: 8 -> 6 -> 3 -> 0, and 3 < 8/gpf(8).

Jianing Song, Table of n, a(n) for n = 1..13098 (all terms <= 50000)

Cf. A356438, A309892, A006530 (gpf), A151800, A076563.

isA356441(n) = if(n>1, my(p=vecmax(factor(n)[, 1])); n/p>nextprime(p+1), 0)

A175126 a(0) = a(1) = 0, for n >= 2, a(n) = number of steps of iteration of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 4, 4, 5, 1, 6, 1, 7, 7, 8, 1, 9, 1, 10, 10, 11, 1, 12, 11, 13, 13, 14, 1, 15, 1, 16, 16, 17, 16, 18, 1, 19, 19, 20, 1, 21, 1, 22, 22, 23, 1, 24, 22, 25, 25, 26, 1, 27, 26, 28, 28, 29, 1, 30, 1, 31, 31, 32, 31, 33, 1, 34, 34, 35, 1, 36, 1, 37, 37, 38, 36, 39, 1, 40, 40

Offset: 0

Jaroslav Krizek, Feb 15 2010

See A005843 and A175127 for the smallest and greatest numbers m such that a(m) = k for k >= 2.

			Example (a(6)=3): 6-2=4, 4-2=2, 2-2=0; iterations has 3 steps.
a(25) = 11, as we have 25 -> 20 -> 18 -> 16 -> 14 -> 12 -> 10 -> 8 -> 6 -> 4 -> 2 -> 0, in total eleven steps to reach zero. - _Antti Karttunen_, Aug 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A020639, A046666, A309892.

From a(2) on, one more than A046667.

Contribution from R. J. Mathar, Mar 11 2010: (Start)
A020639 := proc(n) min(op(numtheory[factorset](n))) ; end proc:
A046666 := proc(n) n-A020639(n) ; end proc:
A175126 := proc(n) local a; if n = 1 then 0; elif n = 0 then 0; else 1+procname(A046666(n)) ; end if; end proc:
seq(A175126(n),n=1..100) ; (End)

stps[n_]:=Length[NestWhileList[#-FactorInteger[#][[1,1]]&,n,#>0&]]-1; Join[{0},Rest[Array[stps,90]]] (* Harvey P. Dale, Aug 15 2012 *)

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A175126(n) = if(n<2,0,1+A175126(n-A020639(n))); \\ Antti Karttunen, Aug 22 2019

a(n) = if(n>1, (n-factor(n)[1, 1])/2 + 1, 0) \\ Jianing Song, Aug 07 2022

a(2n) = n >= 2; a(p) = 1 for p = prime.
a(n) = 0 if n<=1, else a(n) = 1+a(A046666(n)). - R. J. Mathar, Mar 11 2010
a(n) = (n-lpf(n))/2 + 1 for n > 1, lpf = A020639. - Jianing Song, Aug 07 2022

Corrected A-number typo in the comment - R. J. Mathar, Mar 11 2010
Extended beyond a(30) by R. J. Mathar, Mar 11 2010
Term a(0) = 0 prepended by Antti Karttunen, Aug 22 2019

A356428 a(0) = a(1) = 0; for n > 1, a(n) is the number of distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at n and ending at 0, where gpf = A006530.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1

Offset: 0

Jianing Song, Aug 07 2022

nonn

Conjecture: sequence is unbounded. Since a(n) - a(n-gpf(n)) = 0 or 1 (see the formula below), this would imply that every number occurs in this sequence. But it seems that the bigger terms appear rather late: 6 does not appear until a(6664), and 7 does not appear until a(135450) (see A356429).

The last largest prime p in this iteration is found when p^2 > x in this iteration. - David A. Corneth, Aug 09 2022

			In the following examples the numbers produced by the iterations are listed together with their GPFs.
48 (3) -> 45 (5) -> 40 (5) -> 35 (7) -> ... -> 7 (7) -> 0, the distinct gpf(x)'s are 3, 5, and 7, so a(48) = 3.
96 (3) -> 93 (31) -> 62 (31) -> 31 (31) -> 0, the distinct gpf(x)'s are 3 and 31, so a(96) = 2.
320 (5) -> 315 (7) -> 308 (11) -> 297 (11) -> 286 (13) -> 273 (13) -> 260 (13) -> 247 (19) -> ... -> 19 (19) -> 0, the distinct gpf(x)'s are 5, 7, 11, 13, and 19, so a(320) = 5.
In the above computation for a(320) the calculation can stop at 247 (19) as all largest prime factors in positive x are 19. - _David A. Corneth_, Aug 09 2022

Jianing Song, Table of n, a(n) for n = 0..10000

Cf. A006530 (gpf), A076563, A309892, A356429.

gpf(n) = vecmax(factor(n)[, 1]);
a(n) = if(n>1, my(s=n, k=0, p); while(s, p=gpf(s); s-=p; k+=(s==0)||(gpf(s)>p)); k, 0)

a(n) = {if(n <= 1, return(0)); my(cn = n, maxpr, pr = List()); while(cn > 1, maxpr = h(cn); listput(pr, maxpr); cn-=maxpr; if(maxpr^2 > cn, return(#Set(pr)))); #Set(pr)}
h(n) = {my(f = factor(n)); f[#f~, 1]} \\ David A. Corneth, Aug 08 2022

from sympy import factorint
def gpf(n): return 1 if n == 1 else max(factorint(n))
def a(n):
    s = set()
    while n != 0: g = gpf(n); s.add(g); n = n - g
    return len(s - {1})
print([a(n) for n in range(92)]) # Michael S. Branicky, Aug 08 2022

For n > 1, let p = gpf(n), then a(n) = 1+a(n-p) if p = n or gpf(n-p) > p; otherwise a(n) = a(n-p).

A356427 a(0) = 0, a(1) = 1; for n > 1, a(n) is the last step before reaching 0 of the iterations x -> x - gpf(x) starting at n, where gpf = A006530.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 3, 7, 3, 3, 5, 11, 3, 13, 7, 5, 7, 17, 5, 19, 5, 7, 11, 23, 7, 5, 13, 7, 7, 29, 5, 31, 5, 11, 17, 7, 11, 37, 19, 13, 7, 41, 7, 43, 11, 7, 23, 47, 7, 7, 7, 17, 13, 53, 17, 11, 7, 19, 29, 59, 11, 61, 31, 7, 31, 13, 11, 67, 17, 23, 7, 71, 23, 73, 37, 7, 19, 11

Offset: 0

Jianing Song, Aug 07 2022

nonn

For n > 1, a(n) is the unique prime in the iterations x -> x - gpf(x) starting at n and ending at 0.

			In the following examples the numbers produced by the iterations are listed together with their GPFs.
48 (3) -> 45 (5) -> 40 (5) -> 35 (7) -> ... -> 7 (7) -> 0, so a(48) = 7.
96 (3) -> 93 (31) -> 62 (31) -> 31 (31) -> 0, so a(96) = 31.

Jianing Song, Table of n, a(n) for n = 0..10000

Cf. A006530 (gpf), A076563, A309892, A356438, A356441.

a(n) = if(n>1, my(s=n); while(!isprime(s), s=s-vecmax(factor(s)[, 1])); s, n)

For n > 0, a(n) = gpf(n) if n is in A356438; otherwise a(n) > gpf(n).

A356429 Smallest m such that A356428(m) = n, or -1 if there is no such m.

Original entry on oeis.org

2, 8, 48, 315, 320, 6664, 135450, 273000, 518661, 519440, 519622, 148830266, 558797841, 558797968, 24900609294

Offset: 1

Jianing Song, Aug 07 2022

a(n) is the smallest m such that there are exactly n distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at m and ending at 0, where gpf = A006530.
Conjecture: a(n) != -1 for all n. This would be true if A356428 is unbounded; otherwise, this sequence consists of entirely -1's after some point.
Since A356428(n) - A356428(n-gpf(n)) = 0 or 1, sequence is strictly increasing if no term equals -1.
If a(m) > -1 for m >= 15 then a(m) > 10^9. - David A. Corneth, Aug 09 2022

			In the following examples the numbers produced by the iterations are listed together with their GPFs.
320 (5) -> 315 (7) -> 308 (11) -> 297 (11) -> 286 (13) -> 273 (13) -> 260 (13) -> 247 (19) -> ... -> 19 (19) -> 0, the distinct gpf(x)'s are 5, 7, 11, 13, and 19. 320 is the smallest number such that the distinct gpf(x)'s in the iterations is 5, so a(5) = 320.
6664 (17) -> 6647 (23) -> 6624 (23) -> 6601 (41) -> 6560 (41) -> 6519 (53) -> 6466 (53) -> 6413 (53) -> 6360 (53) -> 6307 (53) -> 6254 (59) -> 6195 (59) -> 6136 (59) -> 6077 (103) -> ... -> 103 (103) -> 0, the distinct gpf(x)'s are 17, 23, 41, 53, 59, and 103. 6664 is the smallest number such that the distinct gpf(x)'s in the iterations is 6, so a(6) = 6664.

Cf. A006530 (gpf), A076563, A309892, A356428.

a(12) from Michael S. Branicky, Aug 08 2022
a(13)-a(14) from David A. Corneth, Aug 09 2022
a(15) from Jinyuan Wang, Jul 07 2025

cp's OEIS Frontend

A356438 Numbers k such that A309892(k) = k/gpf(k), where gpf = A006530.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A356441 Numbers k such that A309892(k) < k/gpf(k), where gpf = A006530; complement of A356438.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A175126 a(0) = a(1) = 0, for n >= 2, a(n) = number of steps of iteration of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A356428 a(0) = a(1) = 0; for n > 1, a(n) is the number of distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at n and ending at 0, where gpf = A006530.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Formula

A356427 a(0) = 0, a(1) = 1; for n > 1, a(n) is the last step before reaching 0 of the iterations x -> x - gpf(x) starting at n, where gpf = A006530.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A356429 Smallest m such that A356428(m) = n, or -1 if there is no such m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions