cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Antti Karttunen, Nov 30 2021

Keywords

Comments

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

Links

Crossrefs

Cf. A000203, A007949, A329963, A349750, A349751.
Cf. A349169, A349752, A349755 (subsequences).

Programs

  • Mathematica
    Select[Range[1, 400, 2], IntegerExponent[DivisorSigma[1, #], 3] == IntegerExponent[#, 3] &] (* Amiram Eldar, Dec 01 2021 *)
  • PARI
    isA349749(n) = ((n%2)&&valuation(sigma(n),3)==valuation(n,3));

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

Views

Author

Antti Karttunen, Nov 30 2021

Keywords

Comments

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

Links

Crossrefs

Intersection of A349749 and A349751.
Cf. A000203, A007949, A010872, A074941, A228059, A329963, A349750.

Programs

  • Mathematica
    Select[Range[1, 463, 2], Divisible[(s = DivisorSigma[1, #]) + #, 3] && IntegerExponent[s, 3] == IntegerExponent[#, 3] &] (* Amiram Eldar, Dec 01 2021 *)
  • PARI
    isA349752(n) = ((n%2) && (0==(sigma(n)+n)%3) && valuation(sigma(n), 3)==valuation(n, 3));

A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

45, 153, 261, 325, 369, 405, 477, 801, 909, 925, 1017, 1233, 1341, 1377, 1525, 1557, 1573, 1773, 1825, 2097, 2205, 2313, 2349, 2421, 2425, 2529, 2637, 2725, 2853, 3177, 3321, 3501, 3609, 3645, 3757, 3825, 3925, 4041, 4149, 4293, 4477, 4525, 4581, 4689, 4825, 5013, 5121, 5337, 5445, 5553, 5725, 5733, 5769, 5877, 6025
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2025

Keywords

Comments

Sequence contains also some terms of A386428: 28125, 253125, 1378125, 2278125, 3341637, 3403125, 4753125, etc.

Links

Crossrefs

Intersection of A228058 and A347874.
Conjectured to be also the intersection of A228058 and A349751.
Setwise difference A228058 \ A351574.
Cf. also A349755, A387162.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
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));
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
isA386429(n) = (isA228058(n) && isA347874(n));

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

Views

Author

Antti Karttunen, Nov 30 2021

Keywords

Comments

Odd numbers k such that A010872(k) is equal to A074941(k).
Odd numbers k for which A001065(k) is a multiple of 3.

Links

Crossrefs

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

Programs

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

Views

Author

Antti Karttunen, Jul 21 2025

Keywords

Comments

Question: Is 2205 the only term also in A228058?
If it exists, a(25) > 1275068416.

Links

Crossrefs

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

Programs

  • PARI
    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)
Showing 1-5 of 5 results.

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