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 1-10 of 16 results. Next

A380846 Numbers k such that A380845(k) = 2*k.

Original entry on oeis.org

18, 42, 90, 186, 196, 306, 378, 420, 534, 618, 654, 690, 762, 834, 868, 906, 1062, 1110, 1194, 1242, 1326, 1362, 1422, 1458, 1530, 1698, 1764, 1818, 1866, 2118, 2214, 2262, 2324, 2346, 2490, 2598, 2670, 2706, 2730, 2778, 2838, 2862, 2884, 2922, 2958, 2994, 3066, 3138
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2025

Keywords

Comments

Analogous to perfect numbers (A000396) with A380845 instead of A000203.
All the terms are abundant numbers (A005101), since A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).
Are there odd terms in this sequence? There are none below 10^11.

Examples

			18 is a term since A380845(18) = 36 = 2 * 18.

Links

Crossrefs

Cf. A000203, A000396, A005100, A380845, A380847, A380848.
Subsequence of A005101.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 2*k]; Select[Range[3200], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 2*k;}

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

Examples

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

Links

Crossrefs

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A380847 Numbers k such that A380845(k) = 3*k.

Original entry on oeis.org

1800, 3720, 7560, 15240, 20832, 30600, 42336, 61320, 85344, 109320, 116040, 122760, 171360, 218760, 238920, 245640, 343392, 346440, 395880, 437640, 462600, 484680, 491400, 580680, 687456, 854760, 875400, 896520, 917880, 925320, 950520, 954120, 976200, 982920, 1011720
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2025

Keywords

Comments

Analogous to triperfect numbers (A005820) with A380845 instead of A000203.
All the terms are 3-abundant numbers (A068403), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).

Examples

			1800 is a term since A380845(18) = 5400 = 3 * 1800.

Links

Crossrefs

Cf. A000203, A005100, A005820, A380845, A380846, A380848.
Subsequence of A068403.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 3*k]; Select[Range[10^6], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 3*k;}

A380848 Numbers k such that A380845(k) = 4*k.

Original entry on oeis.org

123832800, 247695840, 268337160, 495421920, 536707080, 990874080, 1073446920, 1981778400, 2146926600, 3963587040, 4293885960, 7927204320, 8587804680, 15854438880, 17175642120, 31708908000, 34351317000, 63417846240, 68702666760, 124884879840, 126713795040, 126835722720
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2025

Keywords

Comments

Analogous to 4-perfect numbers (A027687) with A380845 instead of A000203.
All the terms are 4-abundant numbers (A068404), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).
Are there numbers k such that A380845(k) = m*k for integers m >= 5? There are none below 1.6*10^11.

Crossrefs

Cf. A000203, A005100, A027687, A380845, A380846, A380847.
Subsequence of A068404.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 4*k]; Select[Range[3*10^8], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 4*k;}

Extensions

a(19)-a(22) from Jinyuan Wang, Feb 12 2025

A380931 Numbers k such that A380845(k) > 4*k.

Original entry on oeis.org

5155920, 7733880, 10311840, 15467760, 20623680, 30935520, 41247360, 46403280, 61871040, 61901280, 75546240, 82494720, 87693480, 92806560, 103168800, 103194000, 113513400, 123742080, 123802560, 134152200, 140540400, 151092480, 151351200, 162162000, 164989440, 175386960
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to 4-abundant numbers (A068404) with A380845 instead of A000203.

Examples

			5155920 is a term since A380845(5155920) = 21067042 > 4 * 5155920 = 20623680.

Links

Crossrefs

Cf. A000203, A380845, A380846, A380847, A380848.
Subsequence of A068404, A380929 and A380931.
Similar sequences: A307114, A340110.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 4*k]; Select[Range[10^8], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 4*k;}

A380849 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A380845(k) - k is the sum of aliquot divisors of k that have the same binary weight as k.

Original entry on oeis.org

27940, 112420, 150368, 156840, 225060, 450340, 569376, 925920, 1102200, 1211232, 1802020, 2196592, 2423648, 3377640, 3604260, 4612644, 4874400, 4949160, 5092440, 6375336, 6632808, 6786340, 7155940, 7208740, 7626900, 7685128, 9443060, 9569780, 9643400, 9678020
Offset: 1

Views

Author

Amiram Eldar, Feb 05 2025

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with A380845 instead of A000203.
The larger counterparts are in A380850.

Examples

			27940 is a term since A380845(27940) - 27940 = 36068 > 27940 and A380845(36068) - 36068 = 27940.

Links

Crossrefs

Cf. A000203, A002025, A002046, A380845, A380846, A380850.

Programs

  • Mathematica
    f[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, # < n && DigitCount[#, 2, 1] == h &]];
seq[lim_] := Module[{s = {}, m}, Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, n]], {n, 1, lim}]; s]; seq[10^6]
  • PARI
    f(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (d < n && hammingweight(d) == h));}
list(lim) = {my(m); for(n = 1, lim, m = f(n); if(m > n && f(m) == n, print1(n, ", ")));}

A380850 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A380845(k) - k is the sum of aliquot divisors of k that have the same binary weight as k.

Original entry on oeis.org

36068, 145124, 153670, 294075, 290532, 581348, 593100, 1099530, 2066625, 1237830, 2326244, 2338832, 2476870, 6393390, 4652772, 4883976, 6854625, 9279675, 9548325, 6514464, 11725857, 8760548, 9237668, 9305828, 9457356, 8717912, 12190132, 12353716, 10607740, 12493444
Offset: 1

Views

Author

Amiram Eldar, Feb 06 2025

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with A380845 instead of A000203.
The terms are ordered according to their lesser counterparts (A380849).

Examples

			36068 is a term since A380845(36068) - 36068 = 27940 < 36068 and A380845(27940) - 27940 = 36068.

Links

Crossrefs

Cf. A000203, A002025, A002046, A380845, A380846, A380849.

Programs

  • Mathematica
    f[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, # < n && DigitCount[#, 2, 1] == h &]];
seq[lim_] := Module[{s = {}, m}, Do[m = f[n]; If[m > n && f[m] == n, AppendTo[s, m]], {n, 1, lim}]; s]; seq[10^6]
  • PARI
    f(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (d < n && hammingweight(d) == h));}
list(lim) = {my(m); for(n = 1, lim, m = f(n); if(m > n && f(m) == n, print1(m, ", ")));}

A380930 Numbers k such that A380845(k) > 3*k.

Original entry on oeis.org

1080, 2160, 3600, 4320, 7200, 7440, 8640, 11340, 13608, 14400, 14880, 15120, 17280, 18600, 22680, 22860, 27216, 28800, 29760, 30240, 30480, 31752, 33264, 34020, 34560, 37200, 41664, 45360, 45720, 45900, 51408, 53340, 54432, 57600, 59520, 60480, 60960, 61200, 63504
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to 3-abundant numbers (A068403) with A380845 instead of A000203.

Links

Crossrefs

Cf. A000203, A380845, A380846, A380847.
Subsequence of A068403 and A380929.
Subsequences: A380848, A380931.
Similar sequences: A285615, A293187, A300664, A328135, A340109.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 3*k]; Select[Range[64000], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 3*k;}

Formula

1080 is a term since A380845(1080) = 3330 > 3 * 1080 = 3240.

A380932 Odd numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

322245, 590205, 874665, 966735, 1934415, 2900205, 3224025, 3378375, 3869775, 4729725, 6081075, 6449625, 6818175, 7740495, 8783775, 8906625, 9029475, 9889425, 10135125, 10961685, 11609325, 11821425, 12900825, 13378365, 14189175, 15049125, 15481935, 15909075, 16253055
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

The odd terms in A380929.
Analogous to odd abundant numbers (A005231) with A380845 instead of A000203.

Examples

			322245 is a term since it is odd, and A380845(322245) = 679582 > 2 * 322245 = 644490.

Links

Crossrefs

Cf. A000203, A380845.
Intersection of A005408 and A380929.
Subsequence of A005231.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526, A379031.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1,10^6,2], q]
  • PARI
    isok(k) = if(!(k % 2), 0, my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k);

A381070 Numbers k such that A380845(k)/k > A380845(m)/m for all m < k.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 540, 1080, 2160, 4320, 8640, 17280, 34560, 45360, 68040, 90720, 106680, 136080, 213360, 272160, 320040, 640080, 1280160, 2560320, 2577960, 5155920, 10311840, 15467760, 30935520, 61871040, 123742080, 247484160, 494968320, 681080400
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

Analogous to superabundant numbers (A004394) with A380845 instead of A000203.
The least number k for which A380845(k)/k >= 2 is k = a(6) = A380846(1) = 18.
The least number k for which A380845(k)/k >= 3 is k = a(12) = A380930(1) = 1080.
The least number k for which A380845(k)/k >= 4 is k = a(30) = A380931(1) = 5155920.
It seems that A380845(k)/k is unbounded (see the plot in the links section). What is the least number k for which A380845(k)/k >= 5?

Links

Crossrefs

Cf. A000203, A004394, A380845, A380846, A380929, A380930, A380931, A381069.

Programs

  • Mathematica
    r[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &]/n]; seq[lim_] := Module[{s = {}, rm = 0, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]
  • PARI
    r(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (hammingweight(d) == h)) / n;}
list(lim) = {my(rm = 0, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }
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