A324898 -id:A324898

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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).

+10
64

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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)

MATHEMATICA

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 *)

PROG

(Haskell)

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

(PARI)

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(k<up_to, n++; if(isA228058(n), k++; v[k] = n)); (v); };

v228058 = A228058list(up_to);

A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

CROSSREFS

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 and also A324898 (subsequence if it does not contain any prime powers).

Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

KEYWORD

nonn

AUTHOR

T. D. Noe, Aug 13 2013

EXTENSIONS

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

STATUS

approved

A191218

Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

+10
26

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.

See A228058 for the terms where m > 1. - Antti Karttunen, Apr 22 2019

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n)

EXAMPLE

For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4

MAPLE

with(numtheory): genodd := proc(b) local n, s, d; for n from 1 to b by 2 do s := sigma(n);

if modp(s, 4)=2 then print(n); fi; od; end;

MATHEMATICA

Select[Range[1, 501, 2], Mod[DivisorSigma[1, #], 4]==2&] (* Harvey P. Dale, Nov 12 2017 *)

PROG

(PARI) forstep(n=1, 10^3, 2, if(2==(sigma(n)%4), print1(n, ", "))) \\ Joerg Arndt, May 27 2011

(PARI) list(lim)=my(v=List()); forstep(e=1, logint(lim\=1, 5), 4, forprimestep(p=5, sqrtnint(lim, e), 4, my(pe=p^e); forstep(m=1, sqrtint(lim\pe), 2, if(m%p, listput(v, pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022

CROSSREFS

Subsequence of A191217.

Cf. A228058, A324898 (subsequences).

Cf. A000203, A191219, A324647, A324718, A324719.

KEYWORD

nonn,easy

AUTHOR

Luis H. Gallardo, May 26 2011

STATUS

approved

A324647

Odd numbers k such that 2k is equal to A318468(k) (bitwise-AND of 2*k and sigma(k)).

+10
11

1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.

First 22 terms factored:

1116225 = 3^2 * 5^2 * 11^2 * 41

1245825 = 3^2 * 5^2 * 7^2 * 113

1380825 = 3^2 * 5^2 * 19^2 * 17 [Here the unitary prime is not the largest]

2127825 = 3^2 * 5^2 * 7^2 * 193

10046025 = 3^4 * 5^2 * 11^2 * 41

16813125 = 3^2 * 5^4 * 7^2 * 61

203753025 = 3^2 * 5^2 * 7^2 * 18481

252880425 = 3^2 * 5^2 * 7^2 * 22937

408553425 = 3^2 * 5^2 * 7^2 * 37057

415433025 = 3^2 * 5^2 * 7^4 * 769

740361825 = 3^2 * 5^2 * 7^2 * 67153

969523425 = 3^4 * 5^2 * 13^2 * 2833

1369580625 = 3^2 * 5^4 * 7^2 * 4969

1612924425 = 3^2 * 5^2 * 7^2 * 146297

1763305425 = 3^2 * 5^2 * 7^2 * 159937

2018027025 = 3^2 * 5^2 * 7^2 * 183041

2048985225 = 3^2 * 5^2 * 7^2 * 185849

2286684225 = 3^2 * 5^2 * 7^2 * 207409

3341556225 = 3^2 * 5^2 * 7^2 * 303089

3915517725 = 3^4 * 5^2 * 7^2 * 39461

3985769025 = 3^4 * 5^2 * 7^2 * 40169

4051698525 = 3^2 * 5^2 * 7^2 * 367501.

Compare the above factorizations to the various constraints listed for odd perfect numbers in the Wikipedia article. However, this is NOT a subsequence of A191218 (A228058), see below.

The first terms that do not belong to A191218 are 399736269009 = (3 * 7^2 * 11 * 17 * 23)^2 and 1013616036225 = (3^2 * 5 * 13 * 1721)^2, that both occur instead in A325311. The first terms with omega(n) <> 4 are 9315603297, 60452246925, 68923392525, and 112206463425. They factor as 3^2 * 7^2 * 11^2 * 13^2 * 1033, 3^2 * 5^2 * 7^2 * 17^2 * 18973, 3^2 * 5^2 * 13^2 * 19^2 * 5021, 3^2 * 5^2 * 7^2 * 199^2 * 257. - Giovanni Resta, Apr 21 2019

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..500

Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number

Wikipedia, Perfect number: Odd perfect numbers

Index entries for sequences where any odd perfect numbers must occur

PROG

(PARI) for(n=1, oo, if((n%2)&&((2*n)==bitand(2*n, sigma(n))), print1(n, ", ")));

CROSSREFS

Odd terms of A324652.

Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 14 2019

EXTENSIONS

a(23)-a(24) from Giovanni Resta, Apr 21 2019

STATUS

approved

A324649

Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

+10
7

6, 20, 28, 36, 66, 72, 88, 100, 104, 114, 132, 150, 240, 258, 264, 272, 280, 304, 354, 368, 392, 402, 464, 496, 498, 516, 550, 552, 642, 644, 680, 708, 748, 770, 774, 784, 786, 834, 836, 840, 860, 978, 1026, 1032, 1040, 1044, 1056, 1062, 1064, 1068, 1074, 1092, 1104, 1120, 1184, 1232, 1266, 1312, 1362, 1376, 1410, 1504

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Positions of zeros in A324648. Fixed points of A318458, also positions of the records in the latter.

Intersection with A324652 gives A324643.

The odd terms are: 7425, 76545, 92565, ... (A324897).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to binary expansion of n

Index entries for sequences related to sigma(n)

MATHEMATICA

Select[Range@ 1600, BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Apr 21 2019, after Vincenzo Librandi at A318458 *)

PROG

(PARI) for(n=1, oo, if(bitand(n, sigma(n)-n)==n, print1(n, ", ")));

CROSSREFS

Cf. A318458, A324648, A324652.

Cf. A000396, A324643, A324897, A324898 (subsequences).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 14 2019

STATUS

approved

A324897

Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

+10
4

7425, 76545, 92565, 236925, 831105, 954765, 1401345, 2011905, 2048445, 2129985, 2253825, 2445345, 2621745, 2974725, 3283245, 3847725, 5709825, 6447105, 8422785, 8503425, 8945685, 10781505, 12488385, 13470345, 14322945, 15213825, 15340545, 19470465, 19502145, 20075265, 22749825, 25740225, 25756605, 26215245, 27009045

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.

The first 16 terms factored:

7425 = 3^3 * 5^2 * 11,

76545 = 3^7 * 5 * 7,

92565 = 3^2 * 5 * 11^2 * 17,

236925 = 3^6 * 5^2 * 13,

831105 = 3^2 * 5 * 11 * 23 * 73,

954765 = 3^2 * 5 * 7^2 * 433,

1401345 = 3^2 * 5 * 11 * 19 * 149,

2011905 = 3^3 * 5 * 7 * 2129,

2048445 = 3^2 * 5 * 7^2 * 929,

2129985 = 3^2 * 5 * 11 * 13 * 331,

2253825 = 3^5 * 5^2 * 7 * 53,

2445345 = 3^2 * 5 * 7^2 * 1109,

2621745 = 3^2 * 5 * 7^2 * 29 * 41,

2974725 = 3^4 * 5^2 * 13 * 113,

3283245 = 3^2 * 5 * 7^2 * 1489,

3847725 = 3^2 * 5^2 * 7^2 * 349.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..611

Index entries for sequences related to binary expansion of n

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences related to sigma(n)

MATHEMATICA

Select[Range[1, 10^7, 2], BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Jun 22 2019, after Vincenzo Librandi at A318458 *)

PROG

(PARI) isok(k) = (k%2) && (bitand(k, sigma(k)-k) == k); \\ Michel Marcus, Jul 18 2021

CROSSREFS

Subsequence of A324649.

Cf. A318458, A324647, A324898 (a subsequence).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 19 2019

STATUS

approved

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