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.

A347877 Numbers k for which A003415(sigma(k)) is odd.

Original entry on oeis.org

2, 4, 5, 9, 10, 13, 16, 17, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 52, 53, 58, 61, 64, 68, 72, 73, 74, 80, 82, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 128, 136, 137, 146, 148, 149, 153, 157, 160, 162, 164, 173, 178, 180, 181, 193, 194, 196, 197, 200, 202, 208, 212, 218, 226, 229, 232, 233, 234
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Numbers k such that A342925(k) == 1 (mod 2).
Squares present in this sequence are terms of A347885 squared. (There are no even squares present, see A235991 for the explanation).

Links

Crossrefs

Cf. A000203, A003415, A023194 (subsequence), A235991, A342925, A347872, A347873, A347885, A347878 (complement).
Cf. A347870 (characteristic function), A349909 (its partial sums).

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[234], OddQ[ad[DivisorSigma[1, #]]] &] (* Amiram Eldar, Sep 19 2021 *)

Formula

For all n >= 1, A349909(a(n)) = n. - Antti Karttunen, Feb 23 2022

A347887 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and even. Here A003415 is the arithmetic derivative.

Original entry on oeis.org

201, 231, 237, 259, 273, 315, 333, 399, 429, 455, 483, 525, 555, 585, 627, 651, 665, 741, 763, 855, 903, 975, 1057, 1071, 1085, 1113, 1209, 1235, 1351, 1395, 1407, 1505, 1533, 1635, 1659, 1677, 1767, 1785, 1935, 2037, 2079, 2163, 2211, 2265, 2317, 2331, 2345, 2451, 2457, 2479, 2541, 2555, 2583, 2607, 2611, 2613
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

A square root of any hypothetical odd term x in A005820 (triperfect numbers) would be a member of this sequence, because such x should be a term of A342923 [Numbers x such that A342925(x)-x = 3*A003415(x)], and as the right hand side would then certainly be even (A235992 contains all odd squares), the left hand side should also be even. See also comments in A347870 and in A347391.

Crossrefs

Cf. A000203, A003415, A005820, A097023, A235991, A235992, A342923, A342925, A342926, A347383, A347391.
Subsequence of A347881 and of A347885. The intersection with A347882 gives A347888.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 3000, 2], (d = ad[DivisorSigma[1, #^2]] - #^2) > 0 && EvenQ[d] &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347887(n) = if(!(n%2),0,my(u=(A003415(sigma(n^2))-(n^2))); ((u>0)&&!(u%2)));

A347886 Odd numbers k such that sigma(k^2) has an even number of prime factors when counted with multiplicity.

Original entry on oeis.org

1, 7, 9, 11, 13, 15, 19, 23, 25, 29, 31, 43, 47, 51, 53, 63, 67, 73, 77, 79, 81, 83, 85, 91, 97, 99, 103, 105, 111, 113, 117, 121, 123, 127, 133, 135, 143, 147, 157, 161, 163, 165, 171, 175, 177, 179, 183, 185, 191, 195, 197, 199, 203, 205, 207, 209, 211, 213, 217, 223, 225, 227, 233, 239, 241, 243, 245, 247, 251, 253
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Equally, odd numbers k such that A003415(sigma(k^2)) is even, i.e., k^2 is in A347878. See A235991.

Links

Crossrefs

Cf. A000203, A001222, A003415, A005820, A235991, A342923, A342925, A347391, A347870, A347878, A347882, A347885 (complement among A005408).

Programs

  • Mathematica
    Select[Range[1, 250, 2], EvenQ[PrimeOmega[DivisorSigma[1, #^2]]] &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    isA347886(n) = ((n%2)&&!(bigomega(sigma(n^2))%2));

A347888 Odd numbers k for which A003415(sigma(k^2))-(k^2) is strictly positive and a multiple of six. Here A003415 is the arithmetic derivative.

Original entry on oeis.org

273, 399, 651, 741, 903, 1209, 1407, 1533, 1659, 1677, 1767, 2037, 2163, 2331, 2451, 2457, 2613, 2667, 2847, 3003, 3081, 3297, 3423, 3591, 3685, 3783, 3819, 3843, 3885, 3999, 4017, 4095, 4161, 4179, 4329, 4345, 4389, 4431, 4503, 4683, 4953, 5061, 5187, 5529, 5691, 5817, 5859, 5871, 5985, 6123, 6231, 6279, 6327, 6357
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

A square root of any hypothetical odd term x (if such numbers exist) in A005820 (triperfect numbers) should be a member of this sequence. See comments in A347882, A347887 and also in A347870 and in A347391.
Of the first 200 terms of A097023, 44 appear also in this sequence, the first ones being 50281, 73535, 379953, etc.

Crossrefs

Intersection of A347882 and A347887. Subsequence of A347881 and of A347885.
Cf. A000203, A003415, A005820, A097023, A235991, A235992, A342923, A342925, A342926, A347383, A347391.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 6500, 2], (d = ad[DivisorSigma[1, #^2]] - #^2) > 0 && Divisible[d, 6] &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347888(n) = if(!(n%2),0,my(u=(A003415(sigma(n^2))-(n^2))); ((u>0)&&!(u%6)));
Showing 1-4 of 4 results.

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