A349749 -id:A349749 - cp's OEIS frontend

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A349752 Odd numbers k for which the sigma(k) == -k (mod 3) and sigma(k) preserves the 3-adic valuation of k.

Original entry on oeis.org

7, 13, 15, 19, 31, 33, 37, 43, 61, 67, 69, 73, 79, 87, 97, 103, 105, 109, 123, 127, 139, 141, 147, 151, 153, 157, 163, 175, 177, 181, 193, 195, 199, 211, 223, 229, 231, 241, 249, 271, 277, 283, 285, 303, 307, 313, 325, 331, 337, 339, 349, 367, 373, 375, 379, 393, 397, 409, 411, 421, 429, 433, 439, 447, 457, 463

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Incidentally, of the 37 known terms of A228059, all of which are multiples of three, only 15 (less than half) satisfy this condition.

Intersection of A349749 and A349751.

Cf. A000203, A007949, A010872, A074941, A228059, A329963, A349750.

Select[Range[1, 463, 2], Divisible[(s = DivisorSigma[1, #]) + #, 3] && IntegerExponent[s, 3] == IntegerExponent[#, 3] &] (* Amiram Eldar, Dec 01 2021 *)

isA349752(n) = ((n%2) && (0==(sigma(n)+n)%3) && valuation(sigma(n), 3)==valuation(n, 3));

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

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));

A349750 Odd numbers k such that sigma(k) == k (mod 3), where sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 25, 33, 45, 51, 69, 87, 91, 99, 105, 121, 123, 133, 135, 141, 147, 153, 159, 165, 177, 195, 207, 213, 217, 231, 247, 249, 255, 259, 261, 267, 285, 289, 297, 301, 303, 315, 321, 339, 343, 345, 357, 369, 375, 393, 403, 405, 411, 423, 427, 429, 435, 441, 447, 459, 465, 469, 477, 481, 483, 495, 501, 507, 511, 519

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Odd numbers k such that A010872(k) is equal to A074941(k).

Odd numbers k for which A001065(k) is a multiple of 3.

Index entries for sequences related to sigma(n)

Cf. A000203, A001065, A010872, A074941, A349749, A349751.

Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #] - #, 3] &] (* Amiram Eldar, Dec 01 2021 *)

isA349750(n) = ((n%2)&&0==(sigma(n)-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)

A351536 Odd numbers for which sigma(k) is congruent to 2 modulo 4 and the 3-adic valuations of k and sigma(k) are equal.

Original entry on oeis.org

13, 37, 61, 73, 97, 109, 153, 157, 181, 193, 229, 241, 277, 313, 325, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 801, 829, 853, 877, 925, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297, 1321, 1381, 1429, 1453, 1489, 1525, 1549

Offset: 1

Antti Karttunen, Feb 13 2022

nonn

Index entries for sequences related to sigma(n)

Intersection of A191218 and A349749. Cf. A349755 (subsequence).

Cf. A000203, A007949, A351534.

Select[Range[1, 1500, 2], Mod[(s = DivisorSigma[1, #]), 4] == 2 && Equal @@ IntegerExponent[{#, s}, 3] &] (* Amiram Eldar, Feb 13 2022 *)

isA351536(n) = if(!(n%2),0,my(s=sigma(n)); (2 == (s%4)) && (valuation(n,3) == valuation(s,3)));

cp's OEIS Frontend

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A349752 Odd numbers k for which the sigma(k) == -k (mod 3) and sigma(k) preserves the 3-adic valuation of k.

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A386425 Odd composites k such that sigma(k) has the same powerful part as 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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A349750 Odd numbers k such that sigma(k) == k (mod 3), where sigma is the sum of divisors function.

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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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A351536 Odd numbers for which sigma(k) is congruent to 2 modulo 4 and the 3-adic valuations of k and sigma(k) are equal.

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