A324213 -id:A324213 - cp's OEIS frontend

A325815 a(n) = A324213(n) - A034444(n).

1, 2, 1, 6, 2, -2, 2, 14, 10, 5, 4, 10, 4, 8, 4, 30, 8, 22, 6, 17, 10, 14, 10, 16, 28, 12, 16, -2, 12, 16, 8, 62, 12, 20, 18, 84, 12, 26, 22, 32, 16, 24, 12, 38, 22, 24, 22, 50, 54, 76, 16, 28, 24, 36, 32, 56, 34, 38, 28, 48, 16, 38, 44, 126, 38, 40, 20, 46, 24, 64, 24, 118, 24, 50, 54, 42, 44, 48, 24, 82, 118, 56, 40, 88, 50, 48, 36, 86, 40

Offset: 1

Antti Karttunen, May 29 2019

sign

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

Cf. A000396, A034444, A324213, A325816.

A034444(n) = (2^omega(n));
A324213(n) = { my(s=sigma(n)); sum(i=0, s, (1==gcd(n-i, n-(s-i)))); };
A325815(n) = (A324213(n)-A034444(n));

a(n) = A324213(n) - A034444(n).

a(A000396(n)) < 0 for all n.

A325819 a(n) = A324213(A228058(n)).

Original entry on oeis.org

26, 84, 78, 168, 118, 146, 242, 182, 208, 276, 200, 398, 396, 322, 438, 344, 390, 412, 536, 628, 432, 338, 582, 472, 558, 840, 512, 824, 640, 726, 1022, 852, 914, 628, 744, 616, 1178, 1018, 858, 1140, 856, 760, 990, 936, 1490, 1014, 1564, 1482, 1104, 1096, 1196, 1138, 1008, 1550, 1556, 1180, 1474, 1158, 1508, 858, 2020

Offset: 1

Antti Karttunen, May 29 2019

nonn

If a(n) > 2 for all n, then there are no odd perfect numbers. See also the conjectures in A324213.

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A228058, A324213, A325809.

up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 29 2019
A324213(n) = { my(s=sigma(n)); sum(i=0, s, (1==gcd(n-i, n-(s-i)))); };
A325819(n) = A324213(A228058(n));

a(n) = A324213(A228058(n)).

A325980 Lexicographically earliest sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A324213(i) = A324213(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 2, 6, 7, 8, 9, 10, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 14, 28, 23, 29, 30, 31, 26, 32, 33, 34, 35, 36, 26, 37, 15, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 35, 50, 53, 54, 50, 55, 56, 57, 38, 58, 45, 59, 45, 60, 61, 62, 63, 64, 45, 65, 66, 67, 68, 69, 60, 70, 71, 72, 68, 73, 49, 74, 75, 76, 71

Offset: 1

Antti Karttunen, May 30 2019

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A324213(n)].

For all i, j: a(i) = a(j) => A325815(i) = A325815(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A324213, A325815.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A324213(n) = { my(s=sigma(n)); sum(i=0, s, (1==gcd(n-i, n-(s-i)))); };
v325980 = rgs_transform(vector(up_to,n,[A046523(n), A324213(n)]));
A325980(n) = v325980[n];

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

Original entry on oeis.org

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A325817 a(n) is the least k >= 0 such that n-k and n-(sigma(n)-k) are relatively prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 27, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 0, 0, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 5, 0, 0, 2, 0, 0, 1, 0, 1, 2

Offset: 1

Antti Karttunen, May 29 2019

nonn

a(n) is the least k >= 0 such that -n + k and (n-sigma(n))+k are coprime.

			For n=15, gcd(15-0, 15-(24-0)) = 3, gcd(15-1, 15-(24-1)) = 2 and gcd(15-2, 15-(24-2)) = 1, thus a(15) = 2.

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

Cf. A000203, A000396, A001065, A009194, A014567 (positions of zeros), A324213, A325818, A325826, A325962, A325965, A325967, A325976.

lk[n_]:=Module[{k=0},While[!CoprimeQ[n-k,n-(DivisorSigma[1,n]-k)],k++];k]; Array[lk,110] (* Harvey P. Dale, Nov 24 2024 *)

A325817(n) = { my(s=sigma(n)); for(k=0, s, if(1==gcd(-n + k, (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))); };

a(n) = A000203(n) - A325818(n) = A001065(n) - A325826(n) = n - A325976(n).
For all n:
a(A000396(n)) = A000396(n)-1.
a(n) <= n-1.
a(n) <= A325965(n).
a(n) <= A325967(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).

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).

A325807 Number of ways to partition the divisors of n into complementary subsets x and y for which gcd(n-Sum(x), n-Sum(y)) = 1. (Here only distinct unordered pairs of such subsets are counted.)

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 1, 8, 3, 4, 1, 16, 1, 4, 2, 16, 1, 16, 1, 16, 4, 4, 1, 40, 3, 3, 4, 1, 1, 40, 1, 32, 2, 4, 4, 244, 1, 4, 4, 48, 1, 40, 1, 16, 8, 3, 1, 220, 3, 27, 2, 10, 1, 32, 4, 64, 4, 4, 1, 672, 1, 4, 14, 64, 4, 40, 1, 13, 2, 64, 1, 1205, 1, 4, 16, 10, 4, 40, 1, 236, 15, 4, 1, 864, 4, 3, 2, 64, 1, 640, 2, 16, 4, 4, 2, 537, 1, 26, 8, 241, 1, 40, 1, 64, 40

Offset: 1

Antti Karttunen, May 24 2019

nonn

			For n = 1, its divisor set [1] can be partitioned only to an empty set [] and set [1], with sums 0 and 1 respectively, and gcd(1-0,1-1) = gcd(1,0) = 1, thus this partitioning is included, and a(1) = 1.
For n = 3, its divisor set [1, 3] can be partitioned as [] and [1,3] (sums 0 and 4, thus gcd(3-0,3-4) = 1), [1] and [3] (sums 1 and 3, thus gcd(3-1,3-3) = 2), thus a(3) = 1, and similarly a(p) = 1 for any other odd prime p as well.
For n = 6, its divisor set [1, 2, 3, 6] can be partitioned in eight ways as:
  [] and [1, 2, 3, 6] (sums 0 and 12, gcd(6-0, 6-12) = 6),
  [1, 2] and [3, 6]   (sums 3 and 9,  gcd(6-3, 6-9) = 3),
  [1, 3] and [2, 6]   (sums 4 and 8,  gcd(6-4, 6-8) = 2),
  [2] and [1, 3, 6]   (sums 2 and 10, gcd(6-2, 6-10) = 4),
  [3] and [1, 2, 6]   (sums 3 and 9,  gcd(6-3, 6-9) = 3),
  [6] and [1, 2, 3]   (sums 6 and 6,  gcd(6-6, 6-6) = 0),
  [1] and [2, 3, 6]   (sums 1 and 11, gcd(6-1, 6-11) = 5),
  [1, 6] and [2, 3]   (sums 7 and 5,  gcd(6-7, 6-5) = 1),
with only the last partitioning satisfying the required condition, thus a(6) = 1.
For n = 10, its divisor set [1, 2, 5, 10] can be partitioned in eight ways as:
  [] and [1, 2, 5, 10] (sums 0 and 18, gcd(10-0, 10-18) = 2),
  [1, 2] and [5, 10]   (sums 3 and 15, gcd(10-3, 10-15) = 1),
  [1, 5] and [2, 10]   (sums 6 and 12, gcd(10-6, 10-12) = 2),
  [2] and [1, 5, 10]   (sums 2 and 16, gcd(10-2, 10-16) = 2),
  [5] and [1, 2, 10]   (sums 5 and 13, gcd(10-5, 10-13) = 1),
  [10] and [1, 2, 5]   (sums 10 and 8, gcd(10-10, 10-8) = 2),
  [1] and [2, 5, 10]   (sums 1 and 17, gcd(10-1, 10-17) = 1),
  [1, 10] and [2, 5]   (sums 11 and 7, gcd(10-11, 10-7) = 1),
of which four satisfy the required condition, thus a(10) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..1079

Cf. A000203, A000396, A065091, A100577, A324213, A325806, A325809.

Table[Function[d, Count[DeleteDuplicates[Sort /@ Map[{#, Complement[d, #]} &, Subsets@ d]], ?(CoprimeQ @@ (n - Total /@ #) &)]]@ Divisors@ n, {n, 105}] (* _Michael De Vlieger, May 27 2019 *)

A325807(n) = { my(divs=divisors(n), s=sigma(n),r); sum(b=0,(2^(-1+length(divs)))-1,r=sumbybits(divs,2*b);(1==gcd(n-(s-r),n-r))); };
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); };

For all n >= 1:
a(n) <= A100577(n).
a(A065091(n)) = 1, a(A000396(n)) = 1.
a(A228058(n)) = A325809(n).

cp's OEIS Frontend

A325815 a(n) = A324213(n) - A034444(n).

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A325819 a(n) = A324213(A228058(n)).

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A325980 Lexicographically earliest sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A324213(i) = A324213(j) for all i, j.

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A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

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A325817 a(n) is the least k >= 0 such that n-k and n-(sigma(n)-k) 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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