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.

A351574 Terms of A228058 missing from A347874.

Original entry on oeis.org

117, 245, 333, 425, 549, 605, 637, 657, 725, 833, 845, 873, 981, 1025, 1053, 1325, 1413, 1421, 1445, 1629, 1737, 1805, 1813, 2009, 2057, 2061, 2169, 2225, 2493, 2525, 2597, 2645, 2817, 2825, 2873, 2925, 2989, 2997, 3033, 3141, 3357, 3425, 3509, 3573, 3577, 3681, 3725, 3789, 3897, 4113, 4205, 4325, 4361, 4693, 4753
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Comments

Numbers that satisfy Euler's criterion for the odd perfect numbers (A228058), but do not satisfy the criterion specified in A347874.

Links

Crossrefs

Setwise difference A228058 \ A347874, also setwise difference A228058 \ A386429.
Cf. A000203, A003415.

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));
isA351574(n) = (isA228058(n) && !isA347874(n));

A347872 Numbers k such that k and A003415(sigma(k)) have the same parity.

Original entry on oeis.org

5, 6, 8, 9, 12, 13, 14, 17, 18, 22, 24, 25, 28, 29, 30, 36, 37, 38, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 60, 61, 62, 66, 70, 73, 76, 78, 84, 86, 88, 89, 92, 94, 96, 97, 100, 101, 102, 108, 109, 110, 112, 113, 114, 117, 118, 120, 124, 126, 130, 132, 134, 137, 138, 140, 142, 144, 149, 150, 152, 153, 154, 156, 157
Offset: 1

Views

Author

Antti Karttunen, Sep 17 2021

Keywords

Comments

Numbers k for which A347870(k) is equal to A000035(k).

Links

Crossrefs

Cf. A000035, A000203, A003415, A342925, A347870.
Positions of zeros in A347871. Cf. A347873 for complement.
Cf. A000396, A342922, A347874 (subsequences).
Cf. also A347883.

Programs

  • Maple
    ader:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[2]*n/t[1], t= F)
end proc:
filter:= n -> n - ader(numtheory:-sigma(n)) mod 2 = 0:
select(filter, [$1..1000]); # Robert Israel, Sep 11 2024
  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[157], Mod[ad[DivisorSigma[1, #]], 2] == Mod[#, 2] &] (* Amiram Eldar, Sep 18 2021 *)

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));

A351562 Odd composites k such that A342926(2*k) is a multiple of 3.

Original entry on oeis.org

15, 33, 45, 51, 69, 87, 99, 105, 123, 135, 141, 147, 153, 159, 165, 175, 177, 195, 207, 213, 231, 249, 255, 261, 267, 285, 297, 303, 315, 321, 325, 339, 345, 357, 369, 375, 393, 405, 411, 423, 429, 435, 441, 447, 459, 465, 475, 477, 483, 495, 501, 507, 519, 531, 537, 555, 561, 573, 585, 591, 609, 615, 621, 627, 639
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A003415, A342925, A342926, A347880.
Cf. A347874 (the intersection of this sequence and A347872).

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);
isA351562(n) = ((n%2)&&!isprime(n)&&!(A342926(2*n)%3));
Showing 1-4 of 4 results.

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