A333620 -id:A333620 - cp's OEIS frontend

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

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

Amiram Eldar, Mar 29 2020

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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[#]] &]

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

Amiram Eldar, Mar 29 2020

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Amiram Eldar, Mar 29 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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[#]] &]

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.

A333621 Numbers that are divisible by the total number of 1's in both the Zeckendorf and the dual Zeckendorf representations of all their divisors (A300837 and A333618).

Original entry on oeis.org

1, 2, 4, 126, 416, 442, 3025, 4588, 9243, 10428, 11900, 15070, 18176, 19436, 20532, 26956, 28582, 32108, 33028, 35278, 35929, 37634, 47678, 50386, 61952, 69254, 74578, 88984, 93534, 95120, 96334, 100326, 102297, 142894, 144039, 145768, 147664, 152817, 163125, 183002

Offset: 1

Amiram Eldar, Mar 29 2020

			126 is a term since A300837(126) = 21 and A333618(126) = 7 are both divisors of 126.

Intersection of A333619 and A333620.

Cf. A007895, A112310, A300837, A330711, A333617, A333618.

zeckDigSum[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]];
zeckDivDigSum[n_] := DivisorSum[n, zeckDigSum[#] &];
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[10^4], Divisible[#, zeckDivDigSum[#]] && Divisible[#, dualZeckDivDigSum[#]] &]

cp's OEIS Frontend

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

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

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

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A333621 Numbers that are divisible by the total number of 1's in both the Zeckendorf and the dual Zeckendorf representations of all their divisors (A300837 and A333618).

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