A061228 -id:A061228 - cp's OEIS frontend

A325961 a(n) is the least k >= A061228(n)-1 such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

1, 3, 0, 5, 0, 7, 0, 9, 11, 11, 0, 13, 0, 15, 20, 17, 0, 19, 0, 21, 24, 23, 0, 25, 29, 27, 30, 29, 0, 31, 0, 33, 38, 35, 40, 37, 0, 39, 42, 41, 0, 43, 0, 45, 50, 47, 0, 49, 55, 51, 58, 53, 0, 55, 60, 57, 60, 59, 0, 61, 0, 63, 66, 65, 70, 67, 0, 69, 74, 71, 0, 73, 0, 75, 78, 77, 84, 79, 0, 81, 83, 83, 0, 85, 90, 87, 92, 89, 0, 91, 100

Offset: 1

Antti Karttunen, May 29 2019

nonn

a(n) attains the value of A325818(n) only with n = 1, 2 and the even terms of A000396. Note that A000203(n) > ((n+A020639(n))-1) with composite n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A061228, A324213, A325817, A325818, A325960, A325962, A325965, A325966.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325961(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); (0); };

a(n) = 0 if and only if n is either an odd prime or an odd perfect number, but if n is neither, then a(n) = 2n - A325962(n).

A334245 Let S = A061228, T = A070229, and F = S o T (where o denotes composition). Then a(n) is the least number in the intersection of the sequences F^k(n) and F^k(S(n)), k>=0, or the intersection of the sequences T(F^k(n)) and T(F^k(S(n))), k>=0, or 0 if these intersections are empty.

Original entry on oeis.org

12, 12, 12, 15, 12, 30, 15, 21, 15, 77, 21, 77, 30, 77, 30, 77, 30, 57, 77, 30, 77, 91, 77, 77, 77, 77, 35, 105, 77, 133, 77, 91, 77, 60, 77, 111, 57, 77, 77, 133, 77, 165, 91, 65, 91, 147, 60, 105, 65, 70, 65, 165, 77, 105, 105, 91, 105, 180, 133, 231, 133, 91, 105, 105, 105, 234, 105, 147, 91

Offset: 2

Ali Sada and Robert Israel, Apr 19 2020

60420 >= a(n) > 0 for n <= 6042, but either a(6043) = 0 or a(6043) > 10^30.
If p is prime, a(p) = a(2*p).
From n = 5, 55, 2, 24, 245, ... begin successive strings of exactly 1, 2, 3, 4, 5, ... identical merging points that are: 15, 105, 12, 77, 713, ... . - Bernard Schott, Jul 04 2020

			For n=5, 15 = T(F(5)) = T(S(5)) so a(5) = 15.
For n=6, 12 = F(6) = F(S(6)) so a(6) = 12.
For n=10, 15 = T(10) = T(F(10)) so a(10) = 15.

Robert Israel, Table of n, a(n) for n = 2..6042
A. Sada et al., Will this pattern continue, Discussion in SeqFan mailing list, Apr 2020.
Bernard and Blandine Schott, Network with terms < 100.

Cf. A061228, A070229, A335859.

S:= x -> x + min(numtheory:-factorset(x)):
T:= x -> x + max(numtheory:-factorset(x)):
f:= proc(n) local s,t,i,j,ct;
  s:= S(n); i:= 0; t:= T(n); j:= 1;
  for ct from 1 to 1000 do
    if s = t and i=j then return s fi;
    if s <= t then
      if i = 0 then s:= T(s); i:= 1
      else s:= S(s); i:= 0
      fi
    elif j = 0 then t:= T(t); j:= 1
    else t:= S(t); j:= 0
    fi
  od;
  0
end proc:
map(f, [$2..100]);

S[x_] := x + FactorInteger[x][[1, 1]];
T[x_] := x + FactorInteger[x][[-1, 1]];
f[n_] := Module[{s = S[n], t = T[n], i = 0, j = 1, ct}, For[ct = 1, ct <= 1000, ct++, If[s == t && i == j, Return[s]]; If[s <= t, If[i == 0, s = T[s]; i = 1, s = S[s]; i = 0], If[j == 0, t = T[t]; j = 1, t = S[t]; j = 0]]]; 0];
f /@ Range[2, 100] (* Jean-François Alcover, Aug 15 2020, after Maple *)

a(2^m+2) = a(2^m). - Bernard Schott, Jun 27 2020

A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

Original entry on oeis.org

2, 4, 6, 6, 10, 9, 14, 10, 12, 15, 22, 15, 26, 21, 20, 18, 34, 21, 38, 25, 28, 33, 46, 27, 30, 39, 30, 35, 58, 35, 62, 34, 44, 51, 42, 39, 74, 57, 52, 45, 82, 49, 86, 55, 50, 69, 94, 51, 56, 55, 68, 65, 106, 57, 66, 63, 76, 87, 118, 65, 122

Offset: 1

Reinhard Zumkeller, May 07 2002

Or, for n>1, n + (largest prime divisor of n). [Anne Robinson, daughter of Herman P. Robinson, Oct 08 1981]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981

Cf. A036441.
Cf. A006530, A061228, A256393.
Iterations: A076271 (start=1), A036441 (start=2), A123581 (start=3).

a070229 n = n + a006530 n
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

Table[n + Last@(First/@FactorInteger[n]), {n, 70}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

vector(100, n, if(n>1, vecmax(factor(n)[, 1]), 1) + n) \\ Altug Alkan, Nov 13 2015

a(n) = (n/lpf(n)+1)*lpf(n).

a(n) = n+lpf(n) where lpf=A006530 largest prime factor. For example, a(14)=14+7=21 (instead of ((14/7)+1)*7). - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Jun 14 2007

A046666 a(n) = n - (smallest prime dividing n).

Original entry on oeis.org

0, 0, 2, 0, 4, 0, 6, 6, 8, 0, 10, 0, 12, 12, 14, 0, 16, 0, 18, 18, 20, 0, 22, 20, 24, 24, 26, 0, 28, 0, 30, 30, 32, 30, 34, 0, 36, 36, 38, 0, 40, 0, 42, 42, 44, 0, 46, 42, 48, 48, 50, 0, 52, 50, 54, 54, 56, 0, 58, 0, 60, 60, 62, 60, 64, 0, 66, 66, 68, 0, 70, 0

Offset: 2

N. J. A. Sloane

nonn

From Reinhard Zumkeller, Jul 08 2010: (Start)
a(n)=0 iff n is not composite;
for composite n: a(n) = max(m: m < n and gcd(m,n) > 1). (End)

Antti Karttunen, Table of n, a(n) for n = 2..16385

Cf. A020639, A061228.

with(numtheory):
a:= n-> n-min(factorset(n)[]):
seq(a(n), n=2..100);  # Alois P. Heinz, Jan 07 2015

Table[n-First@(First/@FactorInteger[n]),{n,2,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

a(n) = n - factor(n)[1,1]; \\ Michel Marcus, Jan 07 2015

a(n) = n - A020639(n). - Michel Marcus, Jan 07 2015

A256393 Start from a(1) = 2, then alternately add either the largest (if n is even), or the smallest (if n is odd) prime factor of the preceding term a(n-1) to get a(n).

Original entry on oeis.org

2, 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 35, 40, 45, 48, 51, 54, 57, 60, 65, 70, 77, 84, 91, 98, 105, 108, 111, 114, 133, 140, 147, 150, 155, 160, 165, 168, 175, 180, 185, 190, 209, 220, 231, 234, 247, 260, 273, 276, 299, 312, 325, 330, 341, 352, 363, 366, 427

Offset: 1

Jan Guichelaar, Mar 28 2015

nonn

After the initial term, each even-indexed term equals the preceding term plus its largest prime factor, and each odd-indexed term equals the preceding term plus its smallest prime factor.
See also sequence A076271 where a(n+1) = a(n) + lpf(a(n)).
Each term shares exactly one prime factor with the immediately preceding term, and because the sequence is strictly increasing, all the terms after 2 are composite. - Antti Karttunen, Apr 19 2015
From a(3) onward, the terms are alternately even and odd. - Jan Guichelaar, Apr 24 2015
a(2*n) = A070229(a(2*n-1)); a(2*n+1) = A061228(a(2*n)). - Reinhard Zumkeller, May 06 2015
For prime p let [p] denote the sequence with a(1)=p, and generated as for the terms of the current sequence (which according to this notation is then the same as [2]). It so happens that the sequence [p] (for any p?) merges with [2] sooner or later, taking the form of a "tree" as shown in the attached image (Including prime starts up to p=67). Is this pattern of merging bounded or not? Is there just one tree or are there many? Interesting to speculate. The numbers corresponding to the arrival points in [2] of [p] is the sequence 2,6,15,21,51,57,77,84.... The sequence of ("excluded") numbers which do not arise in [p] for any prime p starts as 8,16,20,25,28,32,36,44... Other sequences may refer to the number of iterations required to merge [p] into [2]. See tree picture. - David James Sycamore, Aug 25 2016
In this picture, one could also include some [c] sequences, with composite c, see A276269. - Michel Marcus, Aug 26 2016

Antti Karttunen, Table of n, a(n) for n = 1..4096
David Sycamore, Tree showing what happens if the initial 2 is replaced by a different prime (Corrected Aug 29 2016)

Cf. A006530 (greatest prime factor), A020639 (least prime factor), A076271.
Cf. A257244 (the first differences; the unique prime factors shared by each pair of successive terms), A257245, A257246 (their bisections), A257247 (numbers n such that GCD(a(2n-1),a(2n)) = GCD(a(2n),a(2n+1)), which is prime).
Cf. A061228, A070229.

a256393 n = a256393_list !! (n-1)
a256393_list = 2 : zipWith ($) (cycle [a070229, a061228]) a256393_list
-- Reinhard Zumkeller, May 06 2015

a[1]:= 2;
for n from 2 to 100 do
  if n::even then a[n]:= a[n-1] + max(numtheory:-factorset(a[n-1]))
  else a[n]:= a[n-1] + min(numtheory:-factorset(a[n-1]))
  fi
od:
seq(a[i],i=1..100); # Robert Israel, May 03 2015

f[n_] := Block[{pf = First /@ FactorInteger@ n}, If[EvenQ@ n, Max@ pf, Min@ pf]]; s = {2}; lmt = 58; For[k = 2, k <= lmt, k++, AppendTo[s, s[[k - 1]] + f@ s[[k - 1]]]]; s (* Michael De Vlieger, Apr 19 2015 *)
FoldList[Function[f, If[EvenQ@ #2, #1 + First@ f, #1 + Last@ f]][FactorInteger[#1][[All, 1]]] &, Range[2, 59]] (* Michael De Vlieger, Aug 26 2016 *)

lista(nn) = {print1(a = 2, ", "); for (n=2, nn, f = factor(a); if (n % 2, a += f[1, 1], a += f[#f~, 1]); print1(a, ", "););} \\ Michel Marcus, Apr 02 2015

;; With memoization-macro definec.
(definec (A256393 n) (cond ((= 1 n) 2) ((even? n) (+ (A256393 (- n 1)) (A006530 (A256393 (- n 1))))) (else (+ (A256393 (- n 1)) (A020639 (A256393 (- n 1))))))) ;; Antti Karttunen, Apr 18 2015

a(1) = 2; a(2n) = a(2n-1) + gpf(a(2n-1)), a(2n+1) = a(2n) + lpf(a(2n)), where gpf = greatest prime factor = A006530, lpf = least prime factor = A020639.

More terms from Michel Marcus, Apr 02 2015

Replaced the name with more succinct description, moved old name to comments - Antti Karttunen, Apr 18-19 2015

A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

Original entry on oeis.org

3, 6, 28, 221, 391, 496, 1189, 1421, 1961, 2419, 5429, 7811, 8128, 11659, 15049, 18871, 36581, 44461, 48689, 57721, 80851, 86519, 98431, 107869, 117739, 146171, 169511, 181829, 207761, 235421, 240199, 280151, 312131, 387349, 437669, 497951, 525991, 637981, 685801, 735349, 752249, 804101, 885119, 950821, 1009009

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Numbers n such that 1+(A001065(n)-A020639(n)) is not zero and divides 1+n-A020639(n).
Note that whenever n is even, then the above condition reduces to "(even) numbers n such that A048050(n) is not zero and divides n-1", which is a condition satisfied only by the even terms of A000396.
a(375) = 360866239 = 449 * 509 * 1579 is the first term with more than two distinct prime factors, the second is a(392) = 413733139 = 199 * 239 * 8699, and the third is a(485) = 718660177 = 41 * 853 * 20549.
Question: Are any of these terms present also in A326064 and A326148? None of the first 564 terms are. If such intersections are empty, then there are no odd perfect numbers.
If one selects only semiprimes from this sequence, one is left with 6, 221, 391, 1189, 1961, 2419, 5429, 7811, 11659, 15049, 18871, 36581, ... (555 terms out of the first 564 terms). Their smaller prime factors are: 2, 13, 17, 29, 37, 41, 61, 73, 89, 101, 113, 157, 173, 181, 197, 233, 241, 257, 269, 281, 313, ... while their larger prime factors are: 3, 17, 23, 41, 53, 59, 89, 107, 131, 149, 167, 233, 257, 269, 293, 347, 359, 383, 401, 419, 467, 503, 521, ..., and both sequences of primes seem to be monotonic.

Cf. A000203, A001065, A020639, A048050, A061228, A325960, A325961, A326064, A326073, A326146, A326148.

Cf. A000396 (a subsequence, the even terms of this sequence if there are no odd perfect numbers).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326073(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n);
A326146(n) = (sigma(n)-A020639(n)-n);
isA326074(n) = (A326073(n)==abs(1+A326146(n)));

A159475 a(1) = 1; for n >= 1, a(n) is the smallest number m > n such that gcd(n,m) > 1.

Original entry on oeis.org

1, 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

Jaroslav Krizek, Apr 13 2009

nonn

Essentially the same as A061228. [R. J. Mathar, Apr 16 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A000040, A020639, A061228.

[1] cat [n + PrimeDivisors(n)[1]: n in [2..100]]; // Vincenzo Librandi, Dec 02 2018

f[n_]:= n + FactorInteger[n][[1, 1]]; Join[{1}, Array[f, 70, 2]] (* Vincenzo Librandi, Dec 02 2018 *)

a(n) = if (n==1, 1, n+factor(n)[1,1]); \\ Michel Marcus, Dec 02 2018

If n = Product (p_i^e_i), where p_1 < p_2 < ... < p_k, a(n) = n + p_1. For p = primes (A000040), a(p) = 2*p.

a(n) = n + A020639(n), for n > 1. - Michel Marcus, Dec 02 2018

More terms from Michel Marcus, Dec 02 2018

A177980 Iterate (n + lpf(n)) / 2 until a prime is reached, where lpf equals the least prime factor. a(n) is that terminating prime.

Original entry on oeis.org

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

Offset: 2

Grant Garcia, Dec 16 2010

nonn

The function (n + lpf(n)) / 2 reduces the input according to its lowest prime factor if it is composite or simply returns the input if it is prime.

Sequence contains only prime numbers (and every prime number).

			7 is prime, so (7 + lpf(7)) / 2 = (7 + 7) / 2 = 7.
15 is composite: (15 + 3) / 2 = 9, (9 + 3) / 2 = 6, (6 + 2) / 2 = 4, (4 + 2) / 2 = 3.

Dumitru Damian, Table of n, a(n) for n = 2..30000

Cf. A020639, A061228.

g[n_] := (n + FactorInteger[n][[1, 1]])/2; f[n_] := Last@ NestWhileList[g, n, !PrimeQ@ # &]; Array[f, 73, 2]

from sympy import factorint, isprime
def a177980(n):
    while True:
        if isprime(n): return n
        else: n=int((n+A020639(n))/2)
[a177980(n) for n in range(2, 160)] # Dumitru Damian, Dec 15 2021

A326146 a(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, 5, 1, 6, -10, 14, -12, 8, 6, 13, -16, 19, -18, 20, 8, 12, -22, 34, 1, 14, 10, 26, -28, 40, -30, 29, 12, 18, 8, 53, -36, 20, 14, 48, -40, 52, -42, 38, 30, 24, -46, 74, 1, 41, 18, 44, -52, 64, 12, 62, 20, 30, -58, 106, -60, 32, 38, 61, 14, 76, -66, 56, 24, 72, -70, 121, -72, 38, 46, 62, 12, 88

Offset: 1

Antti Karttunen, Jun 10 2019

sign

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

Cf. A000203, A001065, A020639, A061228, A105086, A326074, A326147, A326148.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326146(n) = (sigma(n)-A020639(n)-n);

a(n) = A000203(n) - A020639(n) - n = A000203(n) - A061228(n).
a(n) = A001065(n) - A020639(n).
For n > 1, a(n) = A105086(n) - n.

A087349 a(n) = n + (smallest prime-factor of n+1).

Original entry on oeis.org

3, 5, 5, 9, 7, 13, 9, 11, 11, 21, 13, 25, 15, 17, 17, 33, 19, 37, 21, 23, 23, 45, 25, 29, 27, 29, 29, 57, 31, 61, 33, 35, 35, 39, 37, 73, 39, 41, 41, 81, 43, 85, 45, 47, 47, 93, 49, 55, 51, 53, 53, 105, 55, 59, 57, 59, 59, 117, 61, 121, 63, 65, 65, 69, 67, 133, 69, 71, 71

Offset: 1

Reinhard Zumkeller, Oct 20 2003

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

Cf. A020639, A061228.

a087349 n = a020639 (n + 1) + n  -- Reinhard Zumkeller, Mar 31 2014

a[n_] := n + FactorInteger[n+1][[1,1]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

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

a(n) = n + A020639(n+1).

a(n) = A061228(n+1) - 1.

cp's OEIS Frontend

A325961 a(n) is the least k >= A061228(n)-1 such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

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A334245 Let S = A061228, T = A070229, and F = S o T (where o denotes composition). Then a(n) is the least number in the intersection of the sequences F^k(n) and F^k(S(n)), k>=0, or the intersection of the sequences T(F^k(n)) and T(F^k(S(n))), k>=0, or 0 if these intersections are empty.

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A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

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A046666 a(n) = n - (smallest prime dividing n).

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A256393 Start from a(1) = 2, then alternately add either the largest (if n is even), or the smallest (if n is odd) prime factor of the preceding term a(n-1) to get a(n).

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A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

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A159475 a(1) = 1; for n >= 1, a(n) is the smallest number m > n such that gcd(n,m) > 1.

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Magma