A325961 -id:A325961 - 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).

A325966 a(n) is the largest i <= sigma(n)-A020639(n) such that n-i and n-(sigma(n)-i) are relatively prime.

Original entry on oeis.org

0, 1, 0, 5, 0, 7, 0, 13, 10, 15, 0, 25, 0, 21, 20, 29, 0, 37, 0, 39, 28, 33, 0, 55, 26, 39, 36, 29, 0, 67, 0, 61, 44, 51, 42, 89, 0, 57, 52, 87, 0, 91, 0, 81, 74, 69, 0, 121, 50, 91, 68, 95, 0, 115, 66, 117, 76, 87, 0, 163, 0, 93, 100, 125, 78, 139, 0, 121, 92, 141, 0, 193, 0, 111, 120, 137, 88, 163, 0, 183, 118, 123, 0

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. A000203, A000396, A020639, A324213, A325818, A325961, A325962, A325965.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325966(n) = { my(s=sigma(n)); forstep(i=s-A020639(n), 0, -1, if(1==gcd(n-i, n-(s-i)), return(i))); };

a(n) = A000203(n) - A325965(n).
For all n:
a(A000396(n)) = A000396(n)+1.
a(n) <= A325818(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.

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.

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)));

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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A325966 a(n) is the largest i <= sigma(n)-A020639(n) such that n-i and n-(sigma(n)-i) 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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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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A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

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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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