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 1-8 of 8 results.

A333617 Numbers that are divisible by the sum of the digits of all their divisors (A034690).

Original entry on oeis.org

1, 15, 52, 444, 495, 688, 810, 1782, 1891, 1950, 2028, 2058, 2295, 2970, 3007, 3312, 3510, 4092, 4284, 4681, 4687, 4824, 4992, 5143, 5307, 5356, 5487, 5742, 5775, 5829, 6724, 6750, 6900, 6913, 6972, 7141, 7471, 7560, 7650, 7722, 7783, 7807, 8280, 8325, 8700, 8721
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2020

Keywords

Comments

The corresponding quotients, k/A034690(k), are 1, 1, 2, 6, 5, 8, 6, 9, 61, ...

Examples

			15 is a term since its divisors are {1, 3, 5, 15}, and their sum of sums of digits is 1 + 3 + 5 + (1 + 5) = 15 which is a divisor of 15.

Links

Crossrefs

Cf. A007953, A034690, A093705, A337230, A005349.

Programs

  • Mathematica
    divDigSum[n_] := DivisorSum[n, Plus @@ IntegerDigits[#] &]; Select[Range[10^4], Divisible[#, divDigSum[#]] &]
  • PARI
    isok(k) = k % sumdiv(k, d, sumdigits(d)) == 0; \\ Michel Marcus, Mar 30 2020
  • Python
    from sympy import divisors
def sd(n): return sum(map(int, str(n)))
def ok(n): return n%sum(sd(d) for d in divisors(n)) == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(8721)) # Michael S. Branicky, Jan 15 2021

A333619 Numbers that are divisible by the total number of 1's in the Zeckendorf representations of all their divisors (A300837).

Original entry on oeis.org

1, 2, 4, 10, 15, 18, 20, 25, 44, 55, 56, 63, 70, 78, 80, 96, 108, 126, 128, 190, 275, 324, 338, 341, 416, 442, 451, 484, 494, 517, 520, 550, 637, 682, 720, 726, 736, 760, 780, 781, 803, 816, 845, 946, 990, 1088, 1111, 1113, 1199, 1235, 1239, 1311, 1426, 1441
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2020

Keywords

Examples

			4 is a term since its divisors are {1, 2, 4}, their Zeckendorf representations (A014417) are {1, 10, 101}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) = 4 which is a divisor of 4.

Links

Crossrefs

Cf. A007895, A014417, A093705, A300837, A328208, A333617, A333620.

Programs

  • Mathematica
    zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];
Select[Range[10^3], Divisible[#, zeckDivDigSum[#]] &]

A333620 Numbers that are divisible by the total number of 1's in the dual Zeckendorf representations of all their divisors (A333618).

Original entry on oeis.org

1, 2, 3, 4, 12, 28, 33, 68, 104, 126, 130, 143, 147, 220, 231, 248, 297, 336, 390, 391, 408, 416, 429, 442, 518, 575, 741, 752, 779, 812, 825, 1161, 1170, 1197, 1295, 1323, 1364, 1440, 1462, 1566, 1652, 1677, 1680, 1692, 1701, 1720, 1806, 1817, 1872, 1909, 2210
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2020

Keywords

Examples

			4 is a term since its divisors are {1, 2, 4}, their dual Zeckendorf representations (A104326) are {1, 10, 101}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) = 4 which is a divisor of 4.

Crossrefs

Cf. A093705, A104326, A112310, A328212, A333617, A333618, A333619.

Programs

  • Mathematica
    fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeckSum[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]];
dualZeckDivDigSum[n_] := DivisorSum[n, dualZeckSum[#] &];
Select[Range[10^3], Divisible[#, dualZeckDivDigSum[#]] &]

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 *)

A333622 Numbers k such that k is divisible by the sum of digits of all the divisors of k in factorial base (A319712).

Original entry on oeis.org

1, 2, 3, 4, 14, 22, 24, 27, 33, 36, 52, 72, 91, 92, 100, 135, 150, 187, 221, 231, 310, 323, 448, 481, 493, 494, 589, 663, 708, 754, 816, 884, 893, 897, 946, 1080, 1155, 1159, 1178, 1200, 1357, 1462, 1475, 1518, 1530, 1536, 1550, 1702, 1710, 1836, 1972, 1978, 2231
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2020

Keywords

Examples

			14 is a term since its divisors are {1, 2, 7, 14}, their representations in factorial base (A007623) are {1, 10, 101, 210}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) + (2 + 1 + 0) = 7 which is a divisor of 14.

Links

Crossrefs

Cf. A007623, A034968, A093705, A319712, A333617, A333619, A333620, A333623.

Programs

  • Mathematica
    fctDigSum[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; fctDivDigDum[n_] := DivisorSum[n, fctDigSum[#] &]; Select[Range[10^3], Divisible[#, fctDivDigDum[#]] &] (* after Jean-François Alcover at A034968 *)

A333623 Numbers k such that k is divisible by the sum of digits of all the divisors of k in primorial base (A319715).

Original entry on oeis.org

1, 2, 3, 4, 14, 22, 40, 64, 90, 104, 120, 160, 169, 175, 182, 220, 272, 275, 338, 360, 500, 550, 640, 646, 752, 775, 792, 858, 928, 930, 1120, 1230, 1280, 1332, 1496, 1710, 2050, 2204, 2303, 2368, 2475, 2584, 2632, 2640, 2806, 2838, 2886, 2898, 3002, 3174, 3192
Offset: 1

Views

Author

Amiram Eldar, Mar 29 2020

Keywords

Links

Crossrefs

Cf. A049345, A093705, A276150, A319715, A333617, A333619, A333620, A333622.

Programs

  • Mathematica
    max = 5; bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; primDigSum[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; primDivDigDum[n_] := DivisorSum[n, primDigSum[#] &]; Select[Range[nmax], Divisible[#, primDivDigDum[#]] &]

Formula

14 is a term since its divisors are {1, 2, 7, 14}, their representations in primorial base (A049345) are {1, 10, 101, 210}, and their sum of sums of digits is 1 + (1 + 0) + (1 + 0 + 1) + (2 + 1 + 0) = 7 which is a divisor of 14.

A338515 Starts of runs of 3 consecutive numbers that are divisible by the total binary weight of their divisors (A093653).

Original entry on oeis.org

1, 348515, 8612344, 29638764, 30625110, 32039808, 32130600, 32481682, 43664313, 55318282, 55503719, 59671714, 69254000, 73152296, 93470904, 100366594, 103640097, 105026790, 109038462, 109212287, 122519464, 126667271, 147208982, 162007166, 169237545, 173392238
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2020

Keywords

Examples

			1 is a term since 1, 2 and 3 are terms of A093705.

Links

Crossrefs

Subsequence of A338514.
Cf. A000120, A093653, A093705.
Similar sequences: A154701, A330932, A334346, A338453.

Programs

  • Mathematica
    divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; div = divQ /@ Range[3]; Reap[Do[If[And @@ div, Sow[k - 3]]; div = Join[Rest[div], {divQ[k]}], {k, 4, 10^7}]][[2, 1]]

A338516 Starts of runs of 4 consecutive numbers that are divisible by the total binary weight of their divisors (A093653).

Original entry on oeis.org

1377595575, 4275143301, 13616091683, 13640596128, 15016388244, 15176619135, 21361749754, 23605084359, 24794290167, 28025464183, 29639590888, 30739547718, 33924433023, 35259630279, 38008366692, 38670247670, 38681191672, 40210059079, 40507412213, 49759198333, 52555068607
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2020

Keywords

Comments

Can 5 consecutive numbers be divisible by the total binary weight of their divisors? If they exist, then they are larger than 10^11.

Examples

			1377595575 is a term since the 4 consecutive numbers from 1377595575 to 1377595578 are all terms of A093705.

Crossrefs

Subsequence of A338514 and A338515.
Cf. A000120, A093653, A093705.
Similar sequences: A141769, A330933, A334372, A338454.

Programs

  • Mathematica
    divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; div = divQ /@ Range[4]; Reap[Do[If[And @@ div, Sow[k - 4]]; div = Join[Rest[div], {divQ[k]}], {k, 5, 5*10^9}]][[2, 1]]
SequencePosition[Table[If[Mod[n,Total[Flatten[IntegerDigits[#,2]&/@Divisors[n]]]]==0,1,0],{n,526*10^8}],{1,1,1,1}][[;;,1]] (* The program will take a long time to run. *) (* Harvey P. Dale, May 28 2023 *)
Showing 1-8 of 8 results.

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