A325817 -id:A325817 - cp's OEIS frontend

A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

2, 4, 3, 8, 4, 2, 4, 16, 12, 9, 6, 14, 6, 12, 8, 32, 10, 26, 8, 21, 14, 18, 12, 20, 30, 16, 18, 2, 14, 24, 10, 64, 16, 24, 22, 88, 14, 30, 26, 36, 18, 32, 14, 42, 26, 28, 24, 54, 56, 80, 20, 32, 26, 40, 36, 60, 38, 42, 30, 56, 18, 42, 48, 128, 42, 48, 22, 50, 28, 72, 26, 122, 26, 54, 58, 46, 48, 56, 26, 86, 120, 60, 42, 96, 54

Offset: 1

Antti Karttunen and David A. Corneth, May 26 2019, with better name from Charlie Neder, Jun 02 2019

nonn

Number of ways to form the sum sigma(n) = x+y so that n-x and n-y are coprime, with x and y in range 0..sigma(n).
From Antti Karttunen, May 28 - Jun 08 2019: (Start)
Empirically, it seems that a(n) >= A034444(n) and also that a(n) >= A034444(A000203(n)) unless n is in A000396.
Specifically, if it could be proved that a(n) >= A034444(n)/2 for n >= 2, which in turn would imply that a(n) >= A001221(n) for all n, then we would know that no odd perfect numbers could exist. Note that a(n) must be 2 on all perfect numbers, whether even or odd. See also A325819.
(End)

			For n=1, sigma(1) = 1, both gcd(1-0, 1-(1-0)) = gcd(1,0) = 1 and gcd(1-1, 1-(1-1)) = gcd(0,1) = 1, thus a(1) = 2.
--
For n=3, sigma(3) = 4, we have 5 cases to consider:
  gcd(3-0, 3-(4-0)) = 1 = gcd(3-4, 3-(4-4)),
  gcd(3-1, 3-(4-1)) = 2 = gcd(3-3, 3-(4-3)),
  gcd(3-2, 3-(4-2)) = 1,
of which three cases give 1 as a result, thus a(3) = 3.
--
For n=6, sigma(6) = 12, we have 13 cases to consider:
  gcd(6-0, 6-(12-0)) = 6 = gcd(6-12, 6-(12-12)),
  gcd(6-1, 6-(12-1)) = 5 = gcd(6-11, 6-(12-11)),
  gcd(6-2, 6-(12-2)) = 4 = gcd(6-10, 6-(12-10)),
  gcd(6-3, 6-(12-3)) = 3 = gcd(6-9, 6-(12-9)),
  gcd(6-4, 6-(12-4)) = 2 = gcd(6-8, 6-(12-8))
  gcd(6-5, 6-(12-5)) = 1 = gcd(6-7, 6-(12-7)),
  gcd(6-6, 6-(12-6)) = 0,
of which only two give 1 as a result, thus a(6) = 2.
--
For n=10, sigma(10) = 18, we have 19 cases to consider:
  gcd(10-0, 10-(18-0)) = 2 = gcd(10-18, 10-(18-18)),
  gcd(10-1, 10-(18-1)) = 1 = gcd(10-17, 10-(18-17)),
  gcd(10-2, 10-(18-2)) = 2 = gcd(10-16, 10-(18-16)),
  gcd(10-3, 10-(18-3)) = 1 = gcd(10-15, 10-(18-15)),
  gcd(10-4, 10-(18-4)) = 2 = gcd(10-14, 10-(18-14)),
  gcd(10-5, 10-(18-5)) = 1 = gcd(10-13, 10-(18-13)),
  gcd(10-6, 10-(18-6)) = 2 = gcd(10-12, 10-(18-12)),
  gcd(10-7, 10-(18-7)) = 1 = gcd(10-11, 10-(18-11)),
  gcd(10-8, 10-(18-8)) = 2 = gcd(10-10, 10-(18-10)),
  gcd(10-9, 10-(18-9)) = 1,
of which 9 cases give 1 as a result, thus a(10) = 9.
--
For n=15, sigma(15) = 24, we have 25 cases to consider:
  gcd(15-0, 15-(24-0)) = 3 = gcd(15-24, 15-(24-24)),
  gcd(15-1, 15-(24-1)) = 2 = gcd(15-23, 15-(24-23)),
  gcd(15-2, 15-(24-2)) = 1 = gcd(15-22, 15-(24-22)),
  gcd(15-3, 15-(24-3)) = 6 = gcd(15-21, 15-(24-21)),
  gcd(15-4, 15-(24-4)) = 1 = gcd(15-20, 15-(24-20)),
  gcd(15-5, 15-(24-5)) = 2 = gcd(15-19, 15-(24-19)),
  gcd(15-6, 15-(24-6)) = 3 = gcd(15-18, 15-(24-18)),
  gcd(15-7, 15-(24-7)) = 2 = gcd(15-17, 15-(24-17)),
  gcd(15-8, 15-(24-8)) = 1 = gcd(15-16, 15-(24-16)),
  gcd(15-9, 15-(24-9)) = 6 = gcd(15-15, 15-(24-15)),
  gcd(15-10, 15-(24-10)) = 1 = gcd(15-14, 15-(24-14)),
  gcd(15-11, 15-(24-11)) = 2 = gcd(15-13, 15-(24-13)),
  gcd(15-12, 15-(24-12)) = 3,
of which 2*4 = 8 cases give 1 as a result, thus a(15) = 8.

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

Cf. A000010, A000203, A000396, A001221, A014567, A034444, A058062, A062401, A228058, A325807, A325815, A325816, A325817, A325818, A325819, A325960, A325961, A325962, A325965, A325966, A325967, A325968, A325970, A325971, A325972.

Array[Sum[Boole[1 == GCD[#1 - i, #1 - (#2 - i)]], {i, 0, #2}] & @@ {#, DivisorSigma[1, #]} &, 85] (* Michael De Vlieger, Jun 09 2019 *)

A324213(n) = { my(s=sigma(n)); sum(i=0,s,(1==gcd(n-i,n-(s-i)))); };

a(n) = Sum_{i=0..sigma(n)} [1 == gcd(n-i,n-(sigma(n)-i))], where [ ] is the Iverson bracket and sigma(n) is A000203(n).
a(A000396(n)) = 2.
a(n) = A325815(n) + A034444(n).
a(n) = 1+A000203(n) - A325816(n).
a(A228058(n)) = A325819(n).

A325818 a(n) is the largest k <= sigma(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

1, 3, 4, 7, 6, 7, 8, 15, 13, 17, 12, 27, 14, 23, 22, 31, 18, 38, 20, 41, 32, 35, 24, 59, 31, 39, 40, 29, 30, 71, 32, 63, 46, 53, 48, 91, 38, 59, 56, 89, 42, 95, 44, 83, 76, 69, 48, 123, 57, 93, 70, 95, 54, 119, 72, 119, 80, 89, 60, 167, 62, 95, 104, 127, 84, 143, 68, 125, 94, 143, 72, 194, 74, 113, 124, 137, 96

Offset: 1

Antti Karttunen, May 29 2019

nonn

a(n) is the largest k <= sigma(n) such that (-n + k) and (n-sigma(n))+k are coprime.

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

Cf. A000203, A000396, A324213, A325817, A325826, A325961, A325966, A325968.

A325818(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(s-i))); };

a(n) = A000203(n) - A325817(n).
a(n) = n + A325826(n).
For all n:
a(A000396(n)) = A000396(n)+1.
a(n) >= A325961(n).
a(n) >= A325966(n).
a(n) >= A325968(n).

A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 7, 4, 7, 1, 15, 1, 9, 7, 15, 1, 20, 1, 21, 11, 13, 1, 35, 6, 13, 13, 1, 1, 41, 1, 31, 13, 19, 13, 55, 1, 21, 17, 49, 1, 53, 1, 39, 31, 23, 1, 75, 8, 43, 19, 43, 1, 65, 17, 63, 23, 31, 1, 107, 1, 33, 41, 63, 19, 77, 1, 57, 25, 73, 1, 122, 1, 39, 49, 61, 19, 89, 1, 105, 40, 43, 1, 139, 23, 43, 31, 91, 1, 143, 19, 75

Offset: 1

Antti Karttunen, May 29 2019

nonn

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. A000040, A000203, A000396, A001065, A033879, A324213, A325817, A325818, A325969, A325970, A325976.

A325826(n) = { my(s=sigma(n)); forstep(k=s-n, 0, -1, if(1==gcd((n+n-sigma(n)), k), return(k))); };

A325818(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(s-i))); };
A325826(n) = (A325818(n) - n);

a(n) = A325818(n) - n = A001065(n) - A325817(n) = A325976(n) - A033879(n).
a(A000040(n)) = a(A000396(n)) = 1.
a(n) >= A325969(n).
gcd(a(n), A325976(n)) = 1.

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.

Original entry on oeis.org

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).

A325965 a(n) is the least k >= A020639(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

1, 2, 4, 2, 6, 5, 8, 2, 3, 3, 12, 3, 14, 3, 4, 2, 18, 2, 20, 3, 4, 3, 24, 5, 5, 3, 4, 27, 30, 5, 32, 2, 4, 3, 6, 2, 38, 3, 4, 3, 42, 5, 44, 3, 4, 3, 48, 3, 7, 2, 4, 3, 54, 5, 6, 3, 4, 3, 60, 5, 62, 3, 4, 2, 6, 5, 68, 5, 4, 3, 72, 2, 74, 3, 4, 3, 8, 5, 80, 3, 3, 3, 84, 3, 6, 3, 4, 3, 90, 5, 8, 3, 4, 3, 6, 5, 98, 2, 4, 2

Offset: 1

Antti Karttunen, May 29 2019

nonn

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

Cf. A000203, A000396, A020639, A325817, A325966.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325965(n) = { my(s=sigma(n)); for(i=A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325966(n).
For all n:
a(A000396(n)) = A000396(n)-1.
a(n) >= A325817(n).

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.

Original entry on oeis.org

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.

A325967 a(n) is the minimum sum of all such subsets of divisors of n for which n-s and (sigma(n)-s)-n are relatively prime, where s is the sum of the subset.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 27, 0, 1, 0, 0, 4, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 3, 0, 1, 0, 0, 4, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 4, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 4, 1, 0, 1, 8, 1, 0, 1, 6, 5, 0, 0, 4, 0, 0, 1, 0, 1, 4

Offset: 1

Antti Karttunen, May 29 2019

nonn

Partition the divisors of n in all possible ways into such complementary subsets x and y for which gcd(n-Sum(x), n-Sum(y)) = 1. a(n) is the minimal value of min(Sum(x), Sum(y)) attained over all such pairs of subsets x and y.
Records 0, 5, 27, 495, 8127, 8289, 10359, 11049, 13809, 15189, 15879, ... occur at 1, 6, 28, 496, 8128, 33148, 41428, 44188, 55228, 60748, 63508, ...
Equivalently, the least k expressible as a sum of distinct divisors of n such that gcd(n-k,A033879(n)) = 1, with the convention that gcd(k,0) = k. - Charlie Neder, Jun 09 2019

			For n=15, its divisors are [1, 3, 5, 15]. If we take an empty set [] and its complement [1, 3, 5, 15], their sums are 0 and 24, but gcd(15-0, 24-15) = gcd(15, 9) = 3 > 1. If we take subsets [1] and [3, 5, 15], then their sums are 1 and 23, but gcd(15-1, 23-15) = gcd(14,8) = 2 > 1. If we take subsets [3] and [1, 5, 15], their sums are 3 and 21, but gcd(15-3, 21-15) = gcd(12, 6) = 6 > 1. Only when we take the subset with a next larger sum, [1, 3] and its complement [5, 15], we get such sums 4 and 20 for which gcd(15-4, 20-15) = gcd(11, 5) = 1. Thus a(15) = 4, the size of the subset with lesser sum.

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

Cf. A000203, A000396, A009194, A014567 (positions of zeros), A325807, A325817, A325968, A325969.

\\ Probably not the most optimal algorithm, but at least faster than the implementation using sumbybits (below):
A325967aux(n, ds, s, ms, divs, from=1) = if(1==gcd((s-ds)-n,n-ds), return(ds), for(i=from, #divs, if(ds+divs[i] >= ms, return(ms), ms = min(ms,A325967aux(n, ds+divs[i], s, ms, divs, i+1)))); (ms));
A325967(n) = if(1==gcd(n, sigma(n)), 0, my(divs = List(divisors(n)), s=sigma(n), ms=2*s); fordiv(n,d, if(d>=ms, return(ms), listpop(divs,1); ms = min(ms,A325967aux(n, d, s, ms, divs)))); (ms));

A325967(n) = { my(divs=divisors(n), s=sigma(n),r,ms=-1); for(b=0,(2^(length(divs)))-1,r=sumbybits(divs,b);if(1==gcd(n-(s-r),n-r),if(ms<0||r0,s += (b%2)*v[i]; i++; b >>= 1); (s); };

a(n) = A000203(n) - A325968(n) = A001065(n) - A325969(n).
For all n, a(A000396(n)) = A000396(n)-1.
For all n, a(n) >= A325817(n).

A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 0, 1, 4, 1, 3, 1, 0, 1, 0, 1, 5, 1, 5, 1, 0, 1, 3, 1, 0, 1, 0, 1, 5, 1, 0, 1, 6, 1, 7, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 1, 3, 1, 5, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 7, 1, 0, 1, 2, 1, 0, 1, 5, 1, 5, 1, 0, 1, 9, 1, 3, 1, 9, 1, 0, 1, 5, 1, 0, 1, 0, 1, 5

Offset: 1

Antti Karttunen, May 29 2019

nonn

By definition, if n is neither an odd prime nor an odd perfect number, then a(n) >= (A020639(n)-1).

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

Cf. A000203, A000396, A020639, A324213, A325817, A325818, A325826, A325961, A325962, A325965, A325966, A325970.

Cf. A006005 (positions of zeros, provided no odd perfect numbers exist).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325960(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i-n))); (0); };

a(n) = (A325961(n) - A325962(n)) / 2, assuming no odd perfect numbers exist.

a(2n) = 1.

A325976 a(n) is the largest k <= n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 9, 11, 11, 13, 13, 13, 16, 17, 17, 19, 19, 21, 21, 23, 23, 25, 23, 27, 1, 29, 29, 31, 32, 31, 33, 35, 36, 37, 37, 39, 39, 41, 41, 43, 43, 43, 43, 47, 47, 49, 50, 49, 49, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 64, 65, 65, 67, 67, 67, 69, 71, 71, 73, 73, 75, 73, 77, 77, 79, 79, 81, 81, 83, 83, 85, 83

Offset: 1

Antti Karttunen, Jun 01 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A033879, A325817, A325818, A325826, A325959.

A325976(n) = { my(s=sigma(n)); forstep(k=n, 0, -1, if(1==gcd((n+n-s), k), return(k))); };

A325817(n) = { my(s=sigma(n)); for(i=0, s, if(1==gcd(n-i, n-(s-i)), return(i))); };
A325976(n) = (n - A325817(n));

a(n) = n - A325817(n) = A033879(n) + A325826(n).
a(n) >= A325959(n).
gcd(a(n), A325826(n)) = 1.

A325971 a(n) is the least k >= A007947(n) such that -n + k and (n-sigma(n))+k are relatively prime.

Original entry on oeis.org

1, 2, 4, 2, 6, 7, 8, 2, 3, 11, 12, 7, 14, 15, 16, 2, 18, 7, 20, 11, 22, 23, 24, 7, 5, 27, 4, 27, 30, 31, 32, 2, 34, 35, 36, 6, 38, 39, 40, 11, 42, 43, 44, 23, 16, 47, 48, 7, 7, 10, 52, 27, 54, 7, 56, 15, 58, 59, 60, 31, 62, 63, 22, 2, 66, 67, 68, 35, 70, 71, 72, 7, 74, 75, 16, 39, 78, 79, 80, 11, 3, 83, 84, 43, 86, 87, 88, 23, 90, 31

Offset: 1

Antti Karttunen, May 31 2019

nonn

a(n) is the least k >= A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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

Cf. A000203, A007947, A324213, A325817, A325961, A325965, A325970, A325972.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325971(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(i))); (0); };
A325971(n) = { my(s=sigma(n)); for(k=A007947(n), s, if(1==gcd(-n + k, (n-sigma(n))+k), return(k))); };

a(n) = A000203(n) - A325972(n).

a(n) = n - A325970(n).

cp's OEIS Frontend

A324213 Number of k with 0 <= k <= sigma(n) such that n-k and 2n-sigma(n) are relatively prime.

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A325818 a(n) is the largest k <= sigma(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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A325826 a(n) is the largest k <= sigma(n)-n such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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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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A325965 a(n) is the least k >= A020639(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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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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A325967 a(n) is the minimum sum of all such subsets of divisors of n for which n-s and (sigma(n)-s)-n are relatively prime, where s is the sum of the subset.

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PARI

PARI

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A325960 a(n) is k-n for the least k >= n+(A020639(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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