A324213 -id:A324213 - 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.

A325970 a(n) is the largest k <= A066503(n) such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

Original entry on oeis.org

0, 0, -1, 2, -1, -1, -1, 6, 6, -1, -1, 5, -1, -1, -1, 14, -1, 11, -1, 9, -1, -1, -1, 17, 20, -1, 23, 1, -1, -1, -1, 30, -1, -1, -1, 30, -1, -1, -1, 29, -1, -1, -1, 21, 29, -1, -1, 41, 42, 40, -1, 25, -1, 47, -1, 41, -1, -1, -1, 29, -1, -1, 41, 62, -1, -1, -1, 33, -1, -1, -1, 65, -1, -1, 59, 37, -1, -1, -1, 69, 78

Offset: 1

Antti Karttunen, May 31 2019

sign

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

Cf. A000203, A005117, A007947, A033879, A066503, A324213, A325826, A325960, A325967, A325971, A325972.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325970(n) = { my(s=sigma(n)); forstep(k=n-A007947(n), -oo, -1, if(1==gcd(k, n+n-sigma(n)), return(k))); };
\\ Or alternatively:
A325970(n) = { my(s=sigma(n)); for(i=A007947(n), s, if(1==gcd(n-i, n-(s-i)), return(n-i))); };

a(n) = n - A325971(n).

For n >= 3, a(A005117(n)) = -1.

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.

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).

A325972 a(n) is the largest k <= sigma(n)-A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 5, 0, 13, 10, 7, 0, 21, 0, 9, 8, 29, 0, 32, 0, 31, 10, 13, 0, 53, 26, 15, 36, 29, 0, 41, 0, 61, 14, 19, 12, 85, 0, 21, 16, 79, 0, 53, 0, 61, 62, 25, 0, 117, 50, 83, 20, 71, 0, 113, 16, 105, 22, 31, 0, 137, 0, 33, 82, 125, 18, 77, 0, 91, 26, 73, 0, 188, 0, 39, 108, 101, 18, 89, 0, 175, 118, 43, 0, 181, 22

Offset: 1

Antti Karttunen, May 31 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A007947, A324213, A325818, A325962, A325966, A325970, A325971.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A325972(n) = { my(s=sigma(n)); forstep(i=s-A007947(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325971(n).

A325816 Number of ways to form the sum sigma(n) = x+y, so that n-x and n-y are not coprime, with x and y in range 0..sigma(n).

Original entry on oeis.org

0, 0, 2, 0, 3, 11, 5, 0, 2, 10, 7, 15, 9, 13, 17, 0, 9, 14, 13, 22, 19, 19, 13, 41, 2, 27, 23, 55, 17, 49, 23, 0, 33, 31, 27, 4, 25, 31, 31, 55, 25, 65, 31, 43, 53, 45, 25, 71, 2, 14, 53, 67, 29, 81, 37, 61, 43, 49, 31, 113, 45, 55, 57, 0, 43, 97, 47, 77, 69, 73, 47, 74, 49, 61, 67, 95, 49, 113, 55, 101, 2, 67, 43, 129, 55, 81, 81, 91, 49

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, A324213, A325815.

A325816(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(n) = 1 + A000203(n) - A324213(n).

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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A325970 a(n) is the largest k <= A066503(n) such that k and (2n-sigma(n)) [= A033879(n)] are relatively prime.

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

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A325972 a(n) is the largest k <= sigma(n)-A007947(n) such that n-k and n-(sigma(n)-k) are relatively prime.

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A325816 Number of ways to form the sum sigma(n) = x+y, so that n-x and n-y are not coprime, with x and y in range 0..sigma(n).

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