A228056 -id:A228056 - cp's OEIS frontend

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Chris Boyd, Sep 14 2013

nonn

No term is the product of two other terms.
Squares of terms and pairwise products of distinct terms form a subsequence of A028260.
Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016
Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Chris Boyd, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Subsequence of A026424.
Cf. A028260, A162642, A229153.
Cf. A007913, A013661.
Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

from math import isqrt
from sympy import primepi
def A229125(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(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020

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

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

A228057 Odd numbers of the form p * m^2, where p is prime and m > 1.

Original entry on oeis.org

27, 45, 63, 75, 99, 117, 125, 147, 153, 171, 175, 207, 243, 245, 261, 275, 279, 325, 333, 343, 363, 369, 387, 405, 423, 425, 475, 477, 507, 531, 539, 549, 567, 575, 603, 605, 637, 639, 657, 675, 711, 725, 747, 775, 801, 833, 845, 847, 867, 873, 891, 909, 925

Offset: 1

T. D. Noe, Aug 13 2013

nonn

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A228056, A228058.

a228057 n = a228057_list !! (n-1)
a228057_list = filter odd a228056_list -- Reinhard Zumkeller, Aug 14 2013

nn = 1000; Union[Select[Flatten[Table[p*n^2, {p, Prime[Range[2, PrimePi[nn/4]]]}, {n, 3, Sqrt[nn/2], 2}]], # < nn &]]

A290340 Numbers m such that each of the four consecutive integers m, m+1, m+2, m+3 has squarefree rank 1.

Original entry on oeis.org

17, 241, 242, 1249, 4049, 4799, 17297, 120049, 206081, 281249, 388961, 470447, 538721, 1462049, 1566449, 1808801, 1916881, 3302449, 3302450, 3693761, 3959297, 5385761, 5664976, 6118001, 6986321, 9305297, 10479041, 14268481, 16831601, 20110481, 22997761, 27661922, 28140001

Offset: 1

Jason Kimberley, Jul 27 2017

nonn

A162642(k) is the squarefree rank of k.
Numbers that are the first of four consecutive terms of A228056 form a subsequence: 242, 3302450, 22997761, 27661922, 28140001, 64866050, ... consisting of those numbers m in this sequence such that m, m+1, m+2, and m+3 are all composite. - Charles R Greathouse IV, Sep 30 2021
One of for positive integer m, m+1, m+2, m+3 is of the form 4*k + 2 = 2*(2*k + 1). As 2 has an odd exponent the exponents in the prime factorization and 2*k + 1 is odd, the number of odd exponents in the prime factorization of 2*k + 1 must be 0 i.e., 2*k + 1 is a perfect square and so one of m, m+1, m+2, m+3 is of the form 2*t^2 where t is an odd square. - David A. Corneth, Nov 09 2023

			m = 17 is in the sequence as the number of odd prime exponents of each of the numbers m = 17 through m + 3 = 20 is 1. - _David A. Corneth_, Nov 06 2023

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 79 terms from Jason Kimberley, a(n) for n = 80..347 from Charles R Greathouse IV)
David A. Corneth, PARI program

Cf. A162642, A229125, A277225, A228056.

A162642:=func;
c:=func;
c(c(c([n:n in[1..10^6]|A162642(n)eq 1])));

list(lim)=my(u=vectorsmall(4),v=List(),s,i); forfactored(n=2,lim\1+3, if(i++>4,i=1); s-=u[i]; s+=u[i]=(vecsum(n[2][,2]%2)==1); if(s==4, listput(v,n[1]-3))); Vec(v); \\ Charles R Greathouse IV, Sep 30 2021

PARI
```
\\ See PARI link
```

cp's OEIS Frontend

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

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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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A228057 Odd numbers of the form p * m^2, where p is prime and m > 1.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Haskell

Mathematica

A290340 Numbers m such that each of the four consecutive integers m, m+1, m+2, m+3 has squarefree rank 1.

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Author

Keywords

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Programs

Magma

PARI

PARI