A326137 -id:A326137 - cp's OEIS frontend

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).

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

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

Original entry on oeis.org

45, 495, 585, 765, 855, 1305, 18837, 21525, 31635, 38295, 45315, 50445, 51255, 60435, 63495, 68085, 77265, 96615, 1403115, 2446353, 3411975, 3999465, 4091745, 4233537, 4287255, 4631319, 10813425, 10967085, 11490345, 15578199, 16143309, 16329645, 16633071, 17179515, 17311203, 17355915, 21159075, 21933975, 22579725

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977 and A325978 are never zero on same n, these are odd composite numbers n such that A325977(n) is not zero and divides A325978(n).
Based on the first 147 terms it seems that this sequence is a subsequence of A072357, that is each term has exactly one prime factor with exponent 2, with one or more other prime factors that are all unitary (i.e., each term satisfies A001222(x) - A001221(x) = 1). On the other hand, it has been proved that no odd perfect number, if such numbers exist at all, can have such a factorization (see A326137 and a link to P. P. Nielsen's paper there).
Nineteen initial terms factorize as:
= 3^2 * 5^1,
= 3^2 * 5^1 * 11^1,
= 3^2 * 5^1 * 13^1,
= 3^2 * 5^1 * 17^1,
= 3^2 * 5^1 * 19^1,
= 3^2 * 5^1 * 29^1,
= 3^2 * 7^1 * 13^1 * 23^1,
= 3^1 * 5^2 * 7^1 * 41^1,
= 3^2 * 5^1 * 19^1 * 37^1,
= 3^2 * 5^1 * 23^1 * 37^1,
= 3^2 * 5^1 * 19^1 * 53^1,
= 3^2 * 5^1 * 19^1 * 59^1,
= 3^2 * 5^1 * 17^1 * 67^1,
= 3^2 * 5^1 * 17^1 * 79^1,
= 3^2 * 5^1 * 17^1 * 83^1,
= 3^2 * 5^1 * 17^1 * 89^1,
= 3^2 * 5^1 * 17^1 * 101^1,
= 3^2 * 5^1 * 19^1 * 113^1,
  1403115 = 3^1 * 5^1 * 7^2 * 23^1 * 83^1,
and the 62nd term as a(62) = 2919199437 = 3^2 * 7^1 * 11^1 * 43^1 * 163^1 * 601^1.
If we select a subsequence of terms for which the quotient A325978(n)/A325977(n) is positive, then we are left with the following numbers: 495, 585, 31635, 38295, 45315, 51255, 60435, 63495, 1403115, 3999465, etc. which is a subsequence of A326070.

Antti Karttunen (terms 1-62) & Giovanni Resta, Table of n, a(n) for n = 1..147
Index entries for sequences where any odd perfect numbers must occur

Cf. A072357, A228058, A325973, A325974, A325975, A325977, A325978, A325979, A326064, A326070, A326071, A326137.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325981(n) = ((n%2)&&!isprime(n)&&(A325975(n)==abs(A325977(n))));
\\ Or alternatively as:
isA325981(n) = if(!(n%2)||isprime(n),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(x&&!(y%x)));

A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

Original entry on oeis.org

105, 195, 4785, 22515, 56865, 228285, 237315, 484245, 671853, 1838145, 1946955, 3446895, 4522695, 12955245, 37730865, 52475055, 53568885, 87612975

Offset: 1

Antti Karttunen, Jun 09 2019

Not a subsequence of A036798, even though many terms are members.

Questions: Are all terms multiples of three? Multiples of 3^(2k+1) but not of 3^(2k)? Are any of the terms included in A228058, A326137?

Index entries for sequences where any odd perfect numbers must occur

Cf. A033879, A083254, A036798, A228058, A318878, A318879, A326137, A326140.

Cf. also A325981, A326131.

isA326141(n) = if(!(n%2),0, my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); (gcd(t,u)==u));

A326063 Composite numbers n such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

4, 6, 9, 25, 28, 49, 117, 121, 169, 289, 361, 496, 529, 775, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 8128, 9409, 10201, 10309, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Composite numbers n such that A318505(n) [sum of divisors of n excluding n itself and the second largest of them, A032742(n)] divides A060681(n) [the largest difference between consecutive divisors of n, = n - A032742(n)].
Numbers k such that A326062(k) = A318505(k).
Question: Is it possible that this sequence could contain a term with more than one non-unitary prime factor? If not, then there are no odd perfect numbers. (See e.g., A326137).

			For n = 9 = 3*3, its divisors are [1, 3, 9], thus A318505(9) = 1 and A060681(9) = 9-3 = 6, and 1 divides 6, so 9 is included, like all squares of primes.
For n = 117 = 3^2 * 13,its divisors are [1, 3, 9, 13, 39, 117], thus A318505(117) = 1+3+9+13 = 26 and A060681(117) = (117-39) = 78, which is a multiple of 26, thus 117 is included in the sequence.

Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.

Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
isA326063(n) = (gcd((sigma(n)-A032742(n))-n, n-A032742(n)) == (sigma(n)-A032742(n))-n);

A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A326063 Composite numbers n such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI