A228057 -id:A228057 - 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

nonn

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

A228056 Numbers of the form p * m^2, where p is prime and m > 1.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 32, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 125, 128, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192, 200, 207, 208, 212, 236, 242, 243, 244, 245, 252, 261, 268, 272, 275, 279, 284

Offset: 1

T. D. Noe, Aug 13 2013

nonn

This sequence is the first step toward candidates for odd perfect numbers, A228058.

T. D. Noe, Table of n, a(n) for n = 1..10000
Raghavendra Bhat, Distribution of Square Prime Numbers, arXiv:2109.10238 [math.NT], 2021.
Raghavendra Bhat, An Algebraic Structure for Square-Prime Numbers, arXiv:2303.14296 [math.GM], 2023.
Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, Filtered rays over iterated absolute differences on layers of integers, arXiv:2309.03922 [math.NT], 2023. See 3.1 p. 9.

Cf. A228057, A228058.

Cf. A124010.

import Data.List (partition)
a228056 n = a228056_list !! (n-1)
a228056_list = filter f [1..] where
   f x = length us == 1 && (head us > 1 || not (null vs)) where
         (us,vs) = partition odd $ a124010_row x
-- Reinhard Zumkeller, Aug 14 2013

nn = 300; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[PrimePi[nn/4]]]}, {n, 2, Sqrt[nn/2]}]], # < nn &]]

list(lim)=my(v=List()); forfactored(n=2, lim\1, my(e=n[2][, 2]); if(vecsum(e%2)==1 && e!=[1]~, listput(v, n[1]))); Vec(v); \\ Charles R Greathouse IV, Oct 01 2021

from math import isqrt
from sympy import primepi
def A228056(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//y**2) for y in range(2,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jun 06 2025

Bhat proves there are ~ (Pi^2/6-1)*x/log x members of this sequence up to x, so a(n) ~ kn log n with k = 6/(Pi^2-6) = 1.550546.... - Charles R Greathouse IV, Oct 01 2021

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).

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