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 1-6 of 6 results.

A338453 Starts of runs of 3 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 242, 243, 1837, 2361, 3693, 3728, 6061, 6457, 9782, 11181, 11721, 13855, 15177, 20017, 22591, 28021, 31461, 31887, 33098, 33993, 38137, 52016, 52112, 60321, 76897, 78542, 78745, 80461, 108394, 116017, 119541, 124453, 125493, 127117, 127417, 145369, 151805, 154113
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2020

Keywords

Comments

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2).

Examples

			3 is a term since A093653(3) = A093653(4) = A093653(5) = 3.

Links

Crossrefs

Cf. A093653.
Subsequence of A338452.
Similar sequences: A005238, A006073, A045939.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 155000}]; s
SequencePosition[Table[Total[DigitCount[Divisors[n],2,1]],{n,160000}],{x_,x_,x_}][[All,1]] (* Harvey P. Dale, Feb 04 2023 *)

A338454 Starts of runs of 4 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

242, 947767, 1041607, 2545015, 3275463, 8170983, 15720871, 21532430, 23752181, 25135885, 25595913, 27981703, 28226983, 30505142, 30962767, 33364805, 37264493, 49002661, 49766629, 52910454, 53408456, 57917191, 57952016, 58331576, 59230454, 60014053, 60723111, 63378005
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2020

Keywords

Comments

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2) = A093653(k+3).

Examples

			242 is a term since A093653(242) = A093653(243) = A093653(244) = A093653(245) = 18.

Links

Crossrefs

Cf. A093653.
Subsequence of A338452 and A338453.
Similar sequences: A006601, A045932, A045940.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 4; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^7}]; s

A338514 Numbers k such that k and k+1 are both divisible by the total binary weight of their divisors (A093653).

Original entry on oeis.org

1, 2, 54, 2119, 11100, 13727, 14382, 15799, 16399, 20159, 20950, 33421, 34617, 36328, 36396, 39400, 42198, 42438, 42650, 46253, 46873, 50370, 55368, 56600, 58793, 67013, 67320, 69023, 72325, 76057, 86393, 90781, 92906, 93216, 105909, 132088, 134028, 134823, 140466
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2020

Keywords

Comments

Numbers k such that k and k+1 are both in A093705, or, equivalently, k is divisible by A093653(k) and k+1 is divisible by A093653(k+1).

Examples

			1 is a term since 1 and 2 are both terms of A093705.

Links

Crossrefs

Cf. A000120, A093653, A093705.
Similar sequences: A330927, A330931, A334345, A338452.

Programs

  • Mathematica
    divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; q1 = divQ[1]; Reap[Do[q2 = divQ[n]; If[q1 && q2, Sow[n - 1]]; q1 = q2, {n, 2, 10^5}]][[2, 1]]
SequencePosition[Table[If[Divisible[n,Total[DigitCount[Divisors[n],2,1]]],1,0],{n,150000}],{1,1}][[All,1]] (* Harvey P. Dale, Jun 14 2022 *)

A339550 Numbers k such that A339549(k) = A339549(k+1).

Original entry on oeis.org

1, 9, 85, 697, 1285, 2605, 4573, 5845, 6001, 6241, 6613, 7141, 7453, 8005, 10897, 12453, 13141, 15445, 19789, 20345, 21445, 21913, 22873, 25957, 36565, 36601, 39597, 44761, 46405, 53677, 56137, 56593, 61013, 63445, 70094, 72913, 76977, 80913, 82405, 87085, 87601
Offset: 1

Views

Author

Amiram Eldar, Dec 08 2020

Keywords

Comments

Analogous to A338452 as A339549 is analogous to A093653.

Examples

			9 is a term since A339549(9) = A339549(10) = 4.

Links

Crossrefs

Cf. A000120, A093653, A338452, A339549.

Programs

  • Mathematica
    f[n_] := Times @@ (DigitCount[#, 2, 1] & /@ Divisors[n]); Select[Range[10000], f[#] == f[# + 1] &]

A360639 Numbers k such that k and k+2 are both A000120-perfect numbers (A175522).

Original entry on oeis.org

123, 219, 695, 1261, 1851, 1943, 3543, 5963, 7031, 7613, 7769, 7861, 10081, 11357, 11629, 12083, 13211, 13791, 14185, 15699, 15835, 15929, 16241, 18649, 20197, 20989, 22521, 23449, 23521, 23963, 24461, 27215, 27829, 28263, 28367, 29485, 29651, 30359, 30901, 31803
Offset: 1

Views

Author

Amiram Eldar, Feb 15 2023

Keywords

Comments

The smallest gap between two consecutive A000120-perfect numbers is 2.
All terms of this sequence are odd.

Examples

			123 is a term since 123 and 125 are both in A175522: A093653(123)/A000120(123) = A093653(125)/A000120(125) = 12/6 = 2.

Links

Crossrefs

Subsequence of A175522.
Cf. A000120, A093653, A338452, A360640.

Programs

  • Mathematica
    q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] == 2 * DigitCount[n, 2, 1]; seq[kmax_] := Module[{s = {}, k = 1, q1 = False, q2}, Do[q2 = q[k]; If[q1 && q2, AppendTo[s, k-2]]; q1 = q2, {k, 3, kmax, 2}]; s]; seq[32000]
  • PARI
    lista(kmax) = {my(is1 = 0, is2); forstep(k=1, kmax, 2, is2 = (sumdiv(k, d, hammingweight(d)) == 2*hammingweight(k)); if(is1 && is2, print1(k-2, ", ")); is1 = is2); }

A338455 Starts of runs of 5 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

1307029927, 2116078861, 2665774183, 2809370965, 4108623302, 4493733751, 5333670902, 5497285284, 5679049670, 8209799382, 9665369455, 9708528486, 10353426151, 10606564910, 12777118615, 12795699493, 13660293367, 13847206214, 14351020663, 15735895813, 17912257013
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2020

Keywords

Comments

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2) = A093653(k+3) = A093653(k+4).
Can 6 consecutive numbers have the same total binary weight of their divisors? If they exist, then they are larger than 10^11.

Examples

			1307029927 is a term since A093653(1307029927) = A093653(1307029928) = A093653(1307029929) = A093653(1307029930) = A093653(1307029931) = 72.

Crossrefs

Cf. A093653.
Subsequence of A338452, A338453 and A338454.
Similar sequences: A045933, A045941, A049051.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 5; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^7}]; s
Showing 1-6 of 6 results.

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