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 1-4 of 4 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));

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));

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

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

Views

Author

Antti Karttunen, Aug 27 2025

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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));

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

Views

Author

Antti Karttunen, Feb 13 2022

Keywords

Links

Crossrefs

Intersection of A191218 and A349749. Cf. A349755 (subsequence).
Cf. A000203, A007949, A351534.

Programs

  • Mathematica
    Select[Range[1, 1500, 2], Mod[(s = DivisorSigma[1, #]), 4] == 2 && Equal @@ IntegerExponent[{#, s}, 3] &] (* Amiram Eldar, Feb 13 2022 *)
  • PARI
    isA351536(n) = if(!(n%2),0,my(s=sigma(n)); (2 == (s%4)) && (valuation(n,3) == valuation(s,3)));
Showing 1-4 of 4 results.

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