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.

A347885 Odd numbers k such that sigma(k^2) has an odd number of prime factors when counted with multiplicity.

Original entry on oeis.org

3, 5, 17, 21, 27, 33, 35, 37, 39, 41, 45, 49, 55, 57, 59, 61, 65, 69, 71, 75, 87, 89, 93, 95, 101, 107, 109, 115, 119, 125, 129, 131, 137, 139, 141, 145, 149, 151, 153, 155, 159, 167, 169, 173, 181, 187, 189, 193, 201, 215, 219, 221, 229, 231, 235, 237, 249, 255, 259, 265, 269, 273, 283, 287, 289, 291, 293, 297, 307
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Equally, odd numbers k such that A003415(sigma(k^2)) is odd, i.e., k^2 is in A347877. See A235991.
A square root of any hypothetical odd square x present in A005820 (triperfect numbers) would be a member of this sequence, because bigomega(x) would be even, and bigomega(3*x) would be odd. See also A347887.

Crossrefs

Cf. A000203, A001222, A003415, A235991, A347870, A347877, A347882, A347886 (complement among A005408), A347887 (subsequence).

Programs

  • Mathematica
    Select[Range[1, 300, 2], OddQ[PrimeOmega[DivisorSigma[1, #^2]]] &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    isA347885(n) = ((n%2)&&(bigomega(sigma(n^2))%2));

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)));

A347881 Odd numbers k for which A003415(sigma(k^2)) > k^2, where A003415 is the arithmetic derivative.

Original entry on oeis.org

105, 201, 231, 237, 259, 273, 315, 333, 399, 429, 455, 469, 483, 525, 553, 555, 585, 603, 627, 651, 665, 693, 711, 741, 763, 777, 819, 855, 871, 903, 975, 1001, 1005, 1027, 1057, 1071, 1085, 1113, 1119, 1141, 1155, 1185, 1197, 1209, 1221, 1235, 1273, 1281, 1287, 1295, 1351, 1365, 1395, 1407, 1443, 1449, 1463, 1467, 1501
Offset: 1

Views

Author

Antti Karttunen, Sep 18 2021

Keywords

Comments

Odd numbers k such that A342926(k^2) is strictly positive.
Square roots of odd squares present in A343218.

Crossrefs

Cf. A000203, A003415, A343218, A342926, A347882 (subsequence).

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 1501, 2], ad[DivisorSigma[1, #^2]] > #^2 &] (* Amiram Eldar, Sep 18 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347881(n) = ((n%2)&&(A003415(sigma(n^2))>(n^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 1-5 of 5 results.

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