A046666 -id:A046666 - cp's OEIS frontend

A325962 a(1) = 1; for n > 1, a(n) is the largest k <= 1+A046666(n) such that n-k and n-(sigma(n)-k) are relatively prime, or -1 if no such nonnegative k exists.

1, 1, 0, 3, 0, 5, 0, 7, 7, 9, 0, 11, 0, 13, 10, 15, 0, 17, 0, 19, 18, 21, 0, 23, 21, 25, 24, 27, 0, 29, 0, 31, 28, 33, 30, 35, 0, 37, 36, 39, 0, 41, 0, 43, 40, 45, 0, 47, 43, 49, 44, 51, 0, 53, 50, 55, 54, 57, 0, 59, 0, 61, 60, 63, 60, 65, 0, 67, 64, 69, 0, 71, 0, 73, 72, 75, 70, 77, 0, 79, 79, 81, 0, 83, 80, 85, 82, 87, 0, 89, 82

Offset: 1

Antti Karttunen, May 29 2019

nonn

a(n) is equal to A325817(n) only with odd primes and the even terms of A000396. a(n) = -1 only on odd perfect numbers, if such numbers exist. Otherwise a(n) = 2n - A325961(n).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A046666, A065091, A324213, A325817, A325818, A325960, A325961, A325965, A325966.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325962(n) = { my(s=sigma(n)); forstep(i=1+n-A020639(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); (-1); };

For all n, a(A065091(n)) = 0.

A046667 a(n) = A046666(n)/2.

Original entry on oeis.org

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

Offset: 2

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..1000

Cf. A046666.

Table[(1/2)(n - First@(First/@FactorInteger[n])), {n, 2, 100}] (* Vincenzo Librandi, Apr 08 2020 *)

a(n)={if(n==1, 0, my(f=factor(n)[,1]); (n-f[1])/2)} \\ Andrew Howroyd, Mar 07 2020

A061228 a(1) = 2, a(n) = smallest number greater than n that is not coprime to n.

Original entry on oeis.org

2, 4, 6, 6, 10, 8, 14, 10, 12, 12, 22, 14, 26, 16, 18, 18, 34, 20, 38, 22, 24, 24, 46, 26, 30, 28, 30, 30, 58, 32, 62, 34, 36, 36, 40, 38, 74, 40, 42, 42, 82, 44, 86, 46, 48, 48, 94, 50, 56, 52, 54, 54, 106, 56, 60, 58, 60, 60, 118, 62, 122, 64, 66, 66, 70, 68, 134, 70, 72, 72

Offset: 1

Amarnath Murthy, Apr 23 2001

nonn

			a(9) = 12 as 10 and 11 are coprime to 9.
a(11) = 22 as 11 is a prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020639, A046666, A064413, A087349.

Cf. A070229, A256393.

a061228 n = n + a020639 n  -- Reinhard Zumkeller, May 06 2015

for n from 1 to 150 do if n=1 then printf(`%d,`,2); fi: for k from n+1 to 2*n do if igcd(n,k)>1 then printf(`%d,`,k); break; fi: od: od:
# alternative:
2, seq(t + min(numtheory:-factorset(t)), t = 2..1000); # Robert Israel, Oct 21 2015

Table[n+First@(First/@FactorInteger[n]),{n,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
nxt[{n_,a_}]:=Module[{c=n+2},While[CoprimeQ[n+1,c],c++];{n+1,c}]; NestList[nxt,{1,2},70][[;;,2]] (* Harvey P. Dale, May 21 2025 *)

a(n) = n + if(n == 1, 1, factor(n)[1,1]); \\ Amiram Eldar, Apr 10 2025

a(n) = A020639(n) + n.
a(2m) = 2m+2, a(p) = 2p if p is a prime.
a(n) = n + the smallest divisor of n that is larger than 1, for n >= 2.
a(p^k) = p^k + p if p is prime. - Robert Israel, Oct 21 2015
a(n) = A087349(n-1) + 1 for n >= 2. - Amiram Eldar, Apr 10 2025

More terms from James Sellers, Apr 24 2001

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

A326148 Odd numbers > 1, not powers of primes, for which A326147(n) is equal to abs(A326146(n)).

Original entry on oeis.org

15, 91, 207, 703, 847, 1023, 1891, 2701, 2725, 5551, 12403, 15043, 18721, 19359, 38503, 49141, 79003, 88831, 104653, 146611, 148951, 188191, 218791, 226801, 269011, 286903, 346957, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1207359, 1314631, 1345873, 1373653, 1537381, 1755001

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Odd numbers > 1, not powers of primes, for which A326146(n) [= (sigma(n)-A020639(n)-n)] is not zero and divides n-A020639(n).
Question: Are any of these terms present also in A326064 and A326074? None of the first 519 terms are. If such intersections are empty, then there are no odd perfect numbers.
Of the first 519 terms, 485 are semiprimes.

Cf. A020639, A046666, A326064, A326074, A326146, A326147.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326146(n) = (sigma(n)-A020639(n)-n);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);
isA326148(n) = if((n>1)&&(n%2)&&!isprimepower(n), my(s=factor(n)[1, 1], t=n-s, u=sigma(n)-s-n); (u && !(t%u)), 0);

A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 1, 18, 2, 2, 4, 22, 2, 1, 2, 2, 26, 28, 4, 30, 1, 6, 2, 2, 1, 36, 4, 2, 2, 40, 4, 42, 2, 6, 4, 46, 2, 1, 1, 6, 2, 52, 4, 2, 2, 2, 2, 58, 2, 60, 4, 2, 1, 2, 4, 66, 2, 6, 4, 70, 1, 72, 2, 2, 2, 2, 4, 78, 26, 1, 2, 82, 2, 2, 4, 6, 2, 88, 2, 14, 2, 2, 4, 10, 2, 96, 1, 6, 1, 100, 4, 102, 2, 6

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A020639, A046666, A326146, A326148.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326073, A326129, A326130, A326140, A326144.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);

a(n) = gcd(n-A020639(n), A000203(n)-A020639(n)-n).

For n > 1, a(n) = gcd(A046666(n), A326146(n)).

A175127 a(n) = the largest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m .

Original entry on oeis.org

4, 6, 9, 10, 12, 15, 16, 18, 21, 25, 24, 27, 28, 30, 35, 34, 36, 39, 40, 42, 49, 46, 48, 51, 55, 54, 57, 58, 60, 65, 64, 66, 69, 70, 77, 75, 76, 78, 81, 85, 84, 91, 88, 90, 95, 94, 96, 99, 100, 102, 105, 106, 108, 111, 121, 119, 117, 118, 120, 125, 124, 126, 133, 130, 132

Offset: 2

Jaroslav Krizek, Feb 15 2010

nonn

A175126(a(n)) = A175126(A005843(n)) = n. [From Jaroslav Krizek, May 12 2010]

			Example (a(4)=9): 9-3=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 9 is the largest number with such result.

Cf. A175126, A046666, A005843.

Edited by R. J. Mathar, Mar 11 2010

A383777 a(n) is the number of steps that n requires to reach 0 under the map: x -> 2*x + 1 if x is even; 0 if x = 1; x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 0 is never reached.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 2, 1, 2, 3, 4, 1, 4, 1, 2, 5, 4, 1, 2, 1, 2, 3, 10, 1, 10, 3, 2, 11, 4, 1, 2, 1, 10, 3, 12, 3, 2, 1, 6, 3, 4, 1, 8, 1, 2, 9, 4, 1, 2, 9, 2, 3, 6, 1, 2, 3, 2, 3, 4, 1, 8, 1, 4, 9, 10, 9, 10, 1, 2, 11, 4, 1, 4, 1, 2, 5, 10, 5, 2, 1, 6, 3, 6, 1

Offset: 0

Ya-Ping Lu, May 17 2025

nonn

Conjecture: a(n) != -1.

			a(10) = 4 because it takes 4 steps for 10 to reach 1 by iterating the map: 10 -> 2*10+1=21 -> 21-3=18 -> 2*18+1=37 -> 37-37=0.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A005408, A006577, A046666.

A383777[n_] := Length[NestWhileList[If[OddQ[#], # - FactorInteger[#][[1,1]], 2*# + 1] &, n, # >0 &]] - 1;
Array[A383777, 100, 0] (* Paolo Xausa, May 22 2025 *)

from sympy import primefactors; mp = lambda x: (0 if x ==1 else x - min(primefactors(x)) if x%2 else 2*x+1)
def A383777(n, c = 0):
    while n != 0: n = mp(n); c += 1
    return c

A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 3, 0, 4, 0, 3, 4, 5, 0, 6, 0, 4, 6, 7, 0, 6, 8, 0, 5, 8, 9, 0, 10, 0, 6, 8, 9, 10, 11, 0, 12, 0, 7, 12, 13, 0, 10, 12, 14, 0, 8, 12, 14, 15, 0, 16, 0, 9, 12, 15, 16, 17, 0, 18, 0, 10, 15, 16, 18, 19, 0, 14, 18, 20, 0, 11, 20, 21, 0, 22

Offset: 1

Rémy Sigrist, Feb 02 2025

			Table T(n, k) begins:
  n   n-th row
  --  ------------------
   1  0
   2  0, 1
   3  0, 2
   4  0, 2, 3
   5  0, 4
   6  0, 3, 4, 5
   7  0, 6
   8  0, 4, 6, 7
   9  0, 6, 8
  10  0, 5, 8, 9
  11  0, 10
  12  0, 6, 8, 9, 10, 11
  13  0, 12
  14  0, 7, 12, 13

Index entries for sequences related to divisors

Cf. A000005, A027750, A046666, A060681, A072513, A094471 (row sums), A258324.

Table[Map[n*(# - 1)/# &, Divisors[n]], {n, 23}] // Flatten (* Michael De Vlieger, Feb 03 2025 *)

row(n) = apply (d -> n*(d-1)/d, divisors(n))

T(n, k) = n * (A027750(n, k) - 1) / A027750(n, k).
Sum_{k = 1..A000005(n)} T(n, k) = A094471(n).
Product_{k = 2..A000005(n)} T(n, k) = A072513(n).
LCM{k = 2..A000005(n)} T(n, k) = A258324(n).
T(n, 1) = 0.
T(n, 2) = A060681(n) for any n > 1. - Michel Marcus, Feb 03 2025
T(n, A000005(n)-1) = A046666(n) for any n > 1.
T(n, A000005(n)) = n-1.

cp's OEIS Frontend

A325962 a(1) = 1; for n > 1, a(n) is the largest k <= 1+A046666(n) such that n-k and n-(sigma(n)-k) are relatively prime, or -1 if no such nonnegative k exists.

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A046667 a(n) = A046666(n)/2.

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A061228 a(1) = 2, a(n) = smallest number greater than n that is not coprime to n.

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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.

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A326148 Odd numbers > 1, not powers of primes, for which A326147(n) is equal to abs(A326146(n)).

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A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

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A175127 a(n) = the largest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m .

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A383777 a(n) is the number of steps that n requires to reach 0 under the map: x -> 2*x + 1 if x is even; 0 if x = 1; x - lpf(x) otherwise where lpf(x) is the least prime factor of x. a(n) = -1 if 0 is never reached.

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