A349752 -id:A349752 - cp's OEIS frontend

A228059 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 that are closer to being perfect than previous terms.

45, 405, 2205, 26325, 236925, 1380825, 1660725, 35698725, 3138290325, 29891138805, 73846750725, 194401220013, 194509436121, 194581580193, 194689796301, 194798012409, 194906228517, 194942300553, 195230876841, 195339092949, 195447309057, 195699813309

Offset: 1

T. D. Noe, Aug 14 2013

nonn

A number x is perfect if sigma(x) = 2x, where sigma is the sum of divisors of x. See A228058 for numbers of the form p^(1+4k) * r^2. This sequence ends when the first odd perfect number occurs.
The first two papers by Dris listed below are for information only; this sequence in independent of the papers. In the second paper, Dris attempts to prove that the exponent of p above is 1 for odd perfect numbers. Coincidently, the first 9 numbers in this sequence have exponent 1.
a(38) > 10^12. - Giovanni Resta, Aug 16 2018
a(38) <= 283665529390725 = 15349 * (3^3 * 5 * 19 * 53)^2. - Giovanni Resta, Aug 23 2018
a(39) <= 3116918388785625 = 37993 * (3^2 * 5^2 * 19 * 67)^2. - Alexander Violette, Mar 05 2022
The first 37 terms are all multiples of 3, as well as the two additional terms given above. See also comments in A349752. - Antti Karttunen, Jan 04 2025

			           45 =   5 * 3^2.
          405 =   5 * 3^4.
         2205 =   5 * (3 * 7)^2.
        26325 =  13 * (3^2 * 5)^2.
       236925 =  13 * (3^3 * 5)^2.
      1380825 =  17 * (3 * 5 * 19)^2.
      1660725 =  61 * (3 * 5 * 11)^2.
     35698725 =  61 * (3^2 * 5 * 17)^2.
   3138290325 =  53 * (3^4 * 5 * 19)^2.
  29891138805 =   5 * (3^2 * 11^2 * 71)^2.
  73846750725 = 509 * (3 * 5 * 11 * 73)^2.

Giovanni Resta, Table of n, a(n) for n = 1..37
Jose Arnaldo B. Dris, The abundancy index of divisors of odd perfect numbers, J. Integer Sequences, 15 (2012), Article 12.4.4.
Jose Arnaldo B. Dris, A short "proof" for Sorli's conjecture on odd perfect numbers, arxiv 1308.2156 [math.NT], 2013-2015.
Jose Arnaldo B. Dris, Euclid-Euler Heuristics for (Odd) Perfect Numbers, arXiv preprint arXiv:1310.5616 [math.NT], 2013-2017.
Jose Arnaldo B. Dris, A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1311.6803 [math.NT], 2013-2015.
Jose Arnaldo Bebita Dris, Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379, A349752.

Cf. also A171929.

nn = 7; f[n_] := Abs[DivisorSigma[1, n]/n - 2]; n = 45; t = {n}; lastF = f[n]; cnt = 1; While[cnt < 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 && f[n] < lastF, cnt++; lastF = f[n]; Print[{n, lastF}]; AppendTo[t, n]]]; t

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));
m=-1; n=0; while(m!=0, n++; if(isA228058(n), if((m<0) || abs((sigma(n)/n)-2)Antti Karttunen, Apr 22 2019

a(10) (as communicated by T. D. Noe) added by Jose Arnaldo Bebita Dris, Aug 16 2018

a(11)-a(22) from Giovanni Resta, Aug 16 2018

A349749 Odd numbers k for which the 3-adic valuation of sigma(k) is equal to the 3-adic valuation of k, where sigma is the sum of divisors function.

Original entry on oeis.org

1, 7, 13, 15, 19, 25, 31, 33, 37, 43, 61, 67, 69, 73, 79, 87, 91, 97, 103, 105, 109, 121, 123, 127, 133, 139, 141, 147, 151, 153, 157, 163, 175, 177, 181, 193, 195, 199, 211, 217, 223, 229, 231, 241, 247, 249, 259, 271, 277, 283, 285, 289, 301, 303, 307, 313, 325, 331, 337, 339, 343, 349, 367, 373, 375, 379, 393, 397

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Odd numbers for which sigma (A000203) preserves the 3-adic valuation (A007949).

Cf. A000203, A007949, A329963, A349750, A349751.

Cf. A349169, A349752, A349755 (subsequences).

Select[Range[1, 400, 2], IntegerExponent[DivisorSigma[1, #], 3] == IntegerExponent[#, 3] &] (* Amiram Eldar, Dec 01 2021 *)

isA349749(n) = ((n%2)&&valuation(sigma(n),3)==valuation(n,3));

A349751 Odd numbers k such that sigma(k) == -k (mod 3), where sigma is the sum of divisors function.

Original entry on oeis.org

7, 13, 15, 19, 31, 33, 37, 43, 45, 51, 61, 67, 69, 73, 79, 87, 97, 99, 103, 105, 109, 123, 127, 135, 139, 141, 147, 151, 153, 157, 159, 163, 165, 175, 177, 181, 193, 195, 199, 207, 211, 213, 223, 229, 231, 241, 249, 255, 261, 267, 271, 277, 283, 285, 297, 303, 307, 313, 315, 321, 325, 331, 337, 339, 345, 349, 357

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Odd numbers k for which A155085(k) is a multiple of 3.

			7 is present as 7 mod 3 = +1, while sigma(7) = 8, and 8 mod 3 = 2, i.e., -1.
45 is present as 45 mod 3 = 0, while sigma(45) = 78, and 78 mod 3 = 0 as well.

Cf. A000203, A010872, A074941, A155085, A349750.

Cf. A349752 (intersection with A349749).

Select[Range[1, 360, 2], Divisible[DivisorSigma[1, #] + #, 3] &] (* Amiram Eldar, Dec 01 2021 *)

isA349751(n) = ((n%2)&&0==(sigma(n)+n)%3);

A349755 Numbers k for which the 3-adic valuations of k and sigma(k) are equal, and that also satisfy Euler's criterion for odd perfect numbers (see A228058).

Original entry on oeis.org

153, 325, 801, 925, 1525, 1573, 1773, 1825, 2097, 2205, 2425, 2725, 3757, 3825, 3925, 4041, 4477, 4525, 4689, 4825, 5013, 5725, 6025, 6877, 6925, 6957, 7381, 7605, 7825, 7929, 8125, 8425, 8577, 8725, 8833, 9325, 9549, 9873, 9925, 10225, 10525, 10693, 10825, 10933, 11425, 11493, 11737, 12789, 13189, 13437, 13525

Offset: 1

Antti Karttunen, Dec 02 2021

Obviously, all odd perfect numbers x, if such numbers exist at all, have to satisfy not only the famous condition given by Euler (see A228058), but also valuation(sigma(x), p) = valuation(x, p) for all odd primes p = 3, 5, 7, 11, etc. See also comments in A349752.

a(109), a(283), a(440) = 31213, 88837, 146461, are the first terms not occurring in A387162. - Antti Karttunen, Aug 27 2025

Intersection of A228058 and A349749.
Cf. A387162 (subsequence).
Cf. A000203, A007949, A349752.

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));
isA349755(n) = (isA228058(n)&&valuation(sigma(n), 3)==valuation(n, 3));

A386420 Odd numbers k that are closer to being perfect than previous terms and also satisfy the conditions that sigma(k) preserves the 3-adic valuation of k, and that sigma(k) == -k (mod 3).

Original entry on oeis.org

7, 15, 105, 495, 1365, 2205, 9405, 26145, 31815, 497835, 654675, 1984455, 7188885, 9018009, 9338595, 9958905, 13777785, 13800465, 14571585, 47020995, 78867495, 132884115, 210124665, 363860775

Offset: 1

Antti Karttunen, Jul 21 2025

Question: Is 2205 the only term also in A228058?

If it exists, a(25) > 1275068416.

Index entries for sequences where (the least) odd perfect number must occur

Subsequence of A349752, thus also of A349749 and of A349751.
Cf. A000203.
Cf. also A171929, A228059, A386419, A386421, A386422.

isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
m=-1; n=0; k=0; while(m!=0, n++; if(!(n%(2^25)),print1("("n")")); if(isA349752(n), if((m<0) || abs((sigma(n)/n)-2)

A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

Original entry on oeis.org

Offset: 1

Antti Karttunen, Aug 27 2025

Intersection of A228058 and A349752.
Subsequence of A349755 from which this differs for the first time at n=109, with a(109) = 31225, while A349755(109) = 31213.
Probably the intersection of A349755 and A386429.

nn=275;n=1;a228058={};While[Length[a228058 ] < 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[a228058,n]]];lim=a228058[[-1]];a349752=Select[Range[1,lim,2],Divisible[(s=DivisorSigma[1,#])+#,3] && IntegerExponent[s,3]==IntegerExponent[#,3]&];Intersection[a228058,a349752] (* James C. McMahon, Aug 27 2025 *)

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));
isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
isA387162(n) = (isA349752(n) && isA228058(n));

cp's OEIS Frontend

A228059 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 that are closer to being perfect than previous terms.

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A349749 Odd numbers k for which the 3-adic valuation of sigma(k) is equal to the 3-adic valuation of k, where sigma is the sum of divisors function.

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A349751 Odd numbers k such that sigma(k) == -k (mod 3), where sigma is the sum of divisors function.

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A349755 Numbers k for which the 3-adic valuations of k and sigma(k) are equal, and that also satisfy Euler's criterion for odd perfect numbers (see A228058).

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A386420 Odd numbers k that are closer to being perfect than previous terms and also satisfy the conditions that sigma(k) preserves the 3-adic valuation of k, and that sigma(k) == -k (mod 3).

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A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

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