A006530 -id:A006530 - cp's OEIS frontend

A335876 a(1) = 2, and for n > 1, a(n) = n + (n/p), where p is largest prime dividing n, A006530(n).

2, 3, 4, 6, 6, 8, 8, 12, 12, 12, 12, 16, 14, 16, 18, 24, 18, 24, 20, 24, 24, 24, 24, 32, 30, 28, 36, 32, 30, 36, 32, 48, 36, 36, 40, 48, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 64, 56, 60, 54, 56, 54, 72, 60, 64, 60, 60, 60, 72, 62, 64, 72, 96, 70, 72, 68, 72, 72, 80, 72, 96, 74, 76, 90, 80, 84, 84, 80, 96, 108, 84, 84

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

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

Cf. A006530, A052126, A171462, A331410, A334097, A335431 (positions of two's powers > 2).

Array[# (1 + 1/FactorInteger[#][[-1, 1]]) &, 83] (* Michael De Vlieger, Jul 08 2020 *)

A335876(n) = if(1==n,2,n + (n/vecmax(factor(n)[, 1])));

a(n) = n + A052126(n).

A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Aug 10 2021

nonn

This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).

			The terms together with their prime indices begin:
     1: {}          31: {11}            71: {20}
     2: {1}         32: {1,1,1,1,1}     73: {21}
     3: {2}         37: {12}            76: {1,1,8}
     5: {3}         41: {13}            79: {22}
     7: {4}         43: {14}            80: {1,1,1,1,3}
     8: {1,1,1}     44: {1,1,5}         83: {23}
    11: {5}         45: {2,2,3}         89: {24}
    12: {1,1,2}     47: {15}            92: {1,1,9}
    13: {6}         48: {1,1,1,1,2}     97: {25}
    17: {7}         52: {1,1,6}         99: {2,2,5}
    19: {8}         53: {16}           101: {26}
    20: {1,1,3}     59: {17}           103: {27}
    23: {9}         61: {18}           107: {28}
    27: {2,2,2}     63: {2,2,4}        108: {1,1,2,2,2}
    28: {1,1,4}     67: {19}           109: {29}
    29: {10}        68: {1,1,7}        112: {1,1,1,1,4}

Removing 1 gives a subset of A026424.
The unsorted even version is A129597.
The unsorted version is A342768(n) = A342767(n,n).
Except the first term, the even version is 2*a(n).
A000290 lists squares.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A056239 adds up prime indices, row sums of A112798.
A209281 = odd bisection sum of standard compositions (even: A346633).
A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A346697 = odd bisection sum of prime indices (weights of A346703).
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.

filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Nov 26 2022

sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]

isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ Michel Marcus, Aug 12 2021

a(n) = A129597(n)/2 for n > 1.

A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

Original entry on oeis.org

9, 18, 25, 27, 36, 45, 49, 50, 54, 55, 63, 72, 75, 81, 90, 98, 99, 100, 108, 110, 117, 121, 125, 126, 135, 144, 147, 150, 162, 165, 169, 175, 180, 196, 198, 200, 216, 220, 225, 234, 242, 243, 245, 250, 252, 270, 275, 288, 289, 294, 300, 315, 324, 325, 330, 338, 343, 350, 360, 361, 363, 375, 385, 392, 396, 400

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

Provided there do not exist any odd perfect numbers, these are numbers k for which A347240(k) >= A006530(k), as for any odd perfect number x, A347240(x) = -1 by its escape clause.
If k is included as a term, then 2*k is also present.
Not all odd squares of primes are present. For example, 67^2 and 79^2 are not included. See also A091490, which seems to be a subsequence of those exceptions.
Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.

			For n = 55 = 5*11, on the first iteration we get A000593(55) = 72 = 2^3 * 3^2, but both 2 and 3 are less than 11; therefore we iterate a second time to get A000593(72) = 13, which is the first value whose largest prime factor is larger than that of 55 (13 > 11), thus 55 is included in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).

Positions of nonzero terms in A347245.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
isA347242(n) = A347244(n);

A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Aug 16 2022

Number of primes p such that gpf(n+p) = p (such p must be prime factors of n).
Number of distinct prime factors p of n such that n+p is p-smooth.
Clearly we have a(n) <= omega(n) for all n, omega = A001221. The differences are given by A354527.
Is this sequence unbounded? Note that 4 does not appear until a(1660577).

			a(78) = 2 since the prime factors of 78 are 2,3,13, and we have gpf(78+3) = 3 and gpf(78+13) = 13, so the solutions to m - gpf(m) = 78 are m = 78+3 = 81 or m = 78+13 = 91. Note that gpf(78+2) != 2.
a(12) = 0 since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A006530, A076563, A001221, A354516 (indices of first occurrence of each number), A354527.

Cf. A354514 (0 together with indices of positive terms), A354515 (indices of 0), A354516, A354525 (indices n for which a(n) reaches omega(n)), A354526 (indices n for which a(n) is smaller than omega(n)).

gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i])

A064920 a(n) = n/gpf(n) + gpf(n) - 1, where gpf = A006530 = greatest prime factor.

Original entry on oeis.org

2, 3, 3, 5, 4, 7, 5, 5, 6, 11, 6, 13, 8, 7, 9, 17, 8, 19, 8, 9, 12, 23, 10, 9, 14, 11, 10, 29, 10, 31, 17, 13, 18, 11, 14, 37, 20, 15, 12, 41, 12, 43, 14, 13, 24, 47, 18, 13, 14, 19, 16, 53, 20, 15, 14, 21, 30, 59, 16, 61, 32, 15, 33, 17, 16, 67, 20, 25, 16, 71, 26, 73, 38, 19, 22

Offset: 2

Reinhard Zumkeller, Oct 14 2001

a(n) = A052126(n) + A006530(n) - 1; a(n) <= n and for n > 1: a(n) = n iff n is prime.

			a(18) = 18/2 + 2 - 1 = 10;
a(19) = 19/19 + 19 - 1 = 19.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A064916, A064921, A064922, A006530, A052126, A064923, A064924.

a[n_] := (g = FactorInteger[n][[-1, 1]]; n/g + g - 1); a /@ Range[2, 76] (* Jean-François Alcover, Apr 06 2011 *)

gpf(n)= { local(f); f=factor(n)~; return(f[1, length(f)]) } { for (n=2, 1000, g=gpf(n); a=n / g + g - 1; write("b064920.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 29 2009

a(n) = my(p = vecmax(factor(n)[,1])); n/p + p - 1; \\ Michel Marcus, Jun 19 2018

A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37

Offset: 1

Reinhard Zumkeller, Oct 04 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

See A180101 for a variant.

Differences[NestList[#+FactorInteger[#][[-1,1]]&,1,100]] (* Paolo Xausa, Dec 09 2023 *)

a(n) = A076271(n+1) - A076271(n) for all n;

a(A076273(k)+j) = A008578(k) for k>0 and 0 <= j < A075527(k-1).

A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 90, 61, 899, 63, 64, 65

Offset: 1

Antti Karttunen, May 31 2014

nonn

This self-inverse permutation (involution) of positive integers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
It also preserves the exponent of the largest prime factor (A071178), from which follows that the sequence A102750 is closed with respect to this permutation, i.e., for all n in A102750, a(n) is either same n or some other term of A102750.
Considered as an operation on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements a self-inverse bijection which is a composition of the effects of A242419 and A225891. (Or equally: A105119 and A242419). For details, please see the respective Comments sections and/or Example section of this entry.

			For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
First we apply A242419, which reverses the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, so that the first stair from the top (one unit wide and three units high) is moved to the last position, the second one (two units wide and two units high) stays in the middle, and the original bottom step (two units wide and one unit high) will be the new topmost step, thus we get the following Young diagram:
   _ _
  |   |_ _
  |       |
  |       |_
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (2,4,4,5,5,5), encoded in A112798 by p_2 * p_4^2 * p_5^3 = 3 * 7^2 * 11^3 = 195657.
Then we apply A225891, which rotates the exponents of distinct primes in the factorization of n one left, in this context the vertical line segments one step up, with the top-one going to the bottomost, and so we get:
   _ _
  |   |
  |   |_ _
  |       |
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,4,4,4,5), encoded in A112798 by p_2^2 * p_4^3 * p_5 = 3^2 * 7^3 * 11 = 33957, thus a(2200) = 33957.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A243068.

Cf. A112798, A242419, A105119, A225891, A006530, A071178, A102750, A243057, A243059.

(define (A242420 n) (* (expt (A006530 n) (- (A071178 n) 1)) (A243057 n)))

a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).
For all k in A102750, a(k) = A243057(k) = A243059(k).
By composing related permutations:
a(n) = A225891(A242419(n)) = A242419(A105119(n)).

A339466 Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

71, 101, 131, 151, 191, 197, 211, 239, 251, 281, 311, 331, 401, 419, 421, 431, 443, 461, 463, 491, 521, 547, 571, 599, 601, 617, 631, 647, 659, 661, 677, 683, 691, 701, 727, 743, 751, 761, 821, 827, 859, 881, 883, 911, 941, 947, 953, 967, 971, 991, 1013, 1021

Offset: 1

Bernard Schott, Dec 10 2020

nonn

Paul Erdős asked if there are infinitely many primes p such that (p-1)/gpf(p-1) = 2^k or = 2^q * 3^r (see Richard K. Guy reference). This sequence lists the primes p that do not satisfy these two previous relations.

Replacing in the definition gpf by lpf (A020639) leads to A122259. In fact this sequence is a subsequence of A122259. - Peter Luschny, Dec 13 2020

			71 is prime, 70/7 = 10 = 2*5 hence 71 is a term.
101 is prime, 100/5 = 20 = 2^2*5 hence 101 is a term.
151 is prime, 150/5 = 30 = 2*3*5 hence 151 is a term.
The first few quotients obtained are: 10, 20, 10, 30, 10, 28, 30, 14, 50, 40, ...

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Robert Israel, Table of n, a(n) for n = 1..10000
P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), p. 311-321. [alternate link]

Cf. A006093, A006530, A052126, A339464.
Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r), A339463 (ratio=2^q*5^r).
Cf. A122259.

s:=func; [p:p in PrimesInInterval(3,1100)|( not 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 2) or ( 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 3) where a is (p-1) div s(p-1)]; // Marius A. Burtea, Dec 10 2020

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and gpf((n-1)/gpf(n-1)) > 3:
select(is_a, [$5..1021]); # Peter Luschny, Dec 13 2020

q[n_] := Module[{f = FactorInteger[n - 1]}, (Length[f] == 1 && f[[1, 1]] == 2) || (Length[f] == 2 && f[[1, 1]] == 2 && f[[2, 2]] == 1) || (Length[f] == 2 && f[[2, 1]] == 3 && f[[2, 2]] > 1) || (Length[f] == 3 && f[[2, 1]] == 3 && f[[3, 2]] == 1)]; Select[Range[3, 1000], PrimeQ[#] && ! q[#] &] (* Amiram Eldar, Dec 10 2020 *)

is(n) = {if(!isprime(n) || n==2, return(0)); my(pm1 = n-1, f = factor(pm1)[,1]); (pm1 /= (f[#f]*1<1} \\ David A. Corneth, Dec 13 2020

More terms from Amiram Eldar, Dec 11 2020

A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

Original entry on oeis.org

3, 7, 5, 5, 13, 7, 3, 7, 31, 11, 7, 5, 11, 11, 13, 7, 11, 13, 13, 19, 17, 11, 13, 13, 11, 17, 61, 19, 31, 13, 31, 17, 61, 19, 307, 23, 13, 11, 17, 13, 19, 17, 23, 127, 29, 7, 31, 71, 19, 19, 23, 29, 29, 79, 31, 13, 13, 11, 2801, 23, 61, 29, 181, 31, 67, 37, 5, 17, 31, 19, 3221, 29, 307, 31, 53, 37, 331, 41

Offset: 1

Antti Karttunen, Feb 16 2021

			The top left corner of the array:
   n=   1     2   3     4   5     6   7        8     9    10  11    12  13    14
  2n=   2     4   6     8  10    12  14       16    18    20  22    24  26    28
-----+---------------------------------------------------------------------------
   1 |  3,    7,  5,    3,  7,    7, 11,      31,   13,    7, 13,    5, 17,   11,
   2 |  5,   13,  7,    5, 11,   13, 13,      11,   31,   13, 17,    7, 19,   13,
   3 |  7,   31, 11,   13, 13,   31, 17,      71,   11,   31, 19,   13, 23,   31,
   4 | 11,   11, 13,   11, 17,   13, 19,    2801,   19,   17, 23,   13, 29,   19,
   5 | 13,   19, 17,   61, 19,   19, 23,    3221,   61,   19, 29,   61, 31,   23,
   6 | 17,   61, 19,   17, 23,   61, 29,   30941,  307,   61, 31,   19, 37,   61,
   7 | 19,  307, 23,   29, 29,  307, 31,   88741,  127,  307, 37,   29, 41,  307,
   8 | 23,  127, 29,  181, 31,  127, 37,     911,   79,  127, 41,  181, 43,  127,
   9 | 29,   79, 31,   53, 37,   79, 41,  292561,   67,   79, 43,   53, 47,   79,
  10 | 31,   67, 37,  421, 41,   67, 43,  732541,  331,   67, 47,  421, 53,   67,
  11 | 37,  331, 41,   37, 43,  331, 47,   17351,   67,  331, 53,   41, 59,  331,
  12 | 41,   67, 43,  137, 47,   67, 53,    4271, 1723,   67, 59,  137, 61,   67,
  13 | 43, 1723, 47,   43, 53, 1723, 59,  579281,  631, 1723, 61,   47, 67, 1723,
  14 | 47,  631, 53,   47, 59,  631, 61, 3500201,   61,  631, 67,   53, 71,  631,
  15 | 53,   61, 59,   53, 61,   61, 67,   14621,  409,   61, 71,   59, 73,   67,
  16 | 59,  409, 61,  281, 67,  409, 71,    5581, 3541,  409, 73,  281, 79,  409,
  17 | 61, 3541, 67, 1741, 71, 3541, 73,     181,   97, 3541, 79, 1741, 83, 3541,
  18 | 67,   97, 71, 1861, 73,   97, 79,   21491,   71,   97, 83, 1861, 89,   97,
  19 | 71,   71, 73,  449, 79,   73, 83,   26881, 5113,   79, 89,  449, 97,   83,

Cf. A006530, A246278, A341527, A341627.

Cf. also A341607.

up_to = 105;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341527(n) = denominator(A341528(n) / A341529(n));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A341628sq(row,col) = A006530(A341527(A246278sq(row,col)));
A341628list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341628sq(col,(a-(col-1))))); (v); };
v341628 = A341628list(up_to);
A341628(n) = v341628[n];

A(n,k) = A006530(A341627(n,k)) = A006530(A341527(A246278(n,k))).

A347241 a(1) = 1, and for n > 1, a(n) is the largest prime factor (A006530) of all terms encountered when iterating the map x -> A000593(x), when starting from x = n, and including the n itself. If 1 is never reached when starting from n, then a(n) = -1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 13, 5, 11, 3, 13, 7, 5, 2, 17, 13, 19, 5, 7, 11, 23, 3, 31, 13, 5, 7, 29, 5, 31, 2, 11, 17, 7, 13, 37, 19, 13, 5, 41, 7, 43, 11, 13, 23, 47, 3, 19, 31, 17, 13, 53, 5, 13, 7, 19, 29, 59, 5, 61, 31, 13, 2, 13, 11, 67, 17, 23, 7, 71, 13, 73, 37, 31, 19, 11, 13, 79, 5, 19, 41, 83, 7, 17, 43, 29

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

			For n = 17, the iteration proceeds as follows 17 -> 18 (= 2*3*3), 18 -> 13 (13 is a prime), 13 -> 14 (= 2*7), 14 -> 8 (= 2*2*2), 8 -> 1. The largest prime factor present (when including the starting term also) is 17, thus a(17) = 17.

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

Cf. A000593, A006530, A161942, A336361, A347240, A347242.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347241(n) = { my(m=1); while(n>1, m = max(m, A006530(n)); n = A000593(n)); (m); };

a(n) = max(A006530(n), A347240(n)).

cp's OEIS Frontend

A335876 a(1) = 2, and for n > 1, a(n) = n + (n/p), where p is largest prime dividing n, A006530(n).

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A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.

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A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

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PARI

A354512 Number of solutions m >= 2 to m - gpf(m) = n, gpf = A006530.

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A064920 a(n) = n/gpf(n) + gpf(n) - 1, where gpf = A006530 = greatest prime factor.

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Mathematica

PARI

PARI

A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

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A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

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Formula

A339466 Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.

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