A333617 -id:A333617 - 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[#]] &]

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

Amiram Eldar, Mar 29 2020

			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.

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

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[#]] &]

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.

A337230 Numbers that are a divisor of the product of the digits of its divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 54, 56, 63, 64, 72, 75, 84, 96, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 224, 225, 252, 256, 288, 294, 336, 375, 378, 384, 441, 448, 486, 512, 567, 576, 588, 625, 672, 675, 729

Offset: 1

Scott R. Shannon, Aug 20 2020

In the first 130 million numbers there are 117 terms, the last being 234375. It is possible no more terms exist, although this is unknown.
The number with the largest ratio of it divisors' digit product to the number itself is 5376, with a ratio of ~2.2*10^43.
From David A. Corneth, Aug 22 2020: (Start)
The product of the digits of divisors must be nonzero. For example, 10 isn't a term as the product of digits of divisors is 0.
Terms are 7-smooth.
The largest term is 234375. Proof:
2^k1 has a 0 for k1 = 10, 3^k2 has a 0 for k2 = 10, 5^k3 has a 0 for k3 = 8 and 7^k4 has a 0 for k4 = 4.
Checking all numbers of the form 2^e1 * 3^e2 * 5^e3 * 7^e4 with ei bounded by the respective ki gives the 117 terms mentioned above with the largest of them being 234375. (End)

			a(4) = 4 is a term as the divisors of 4 are 1,2,4 and 1*2*4 = 8 which is a divisible by 4.
a(10) = 12 is a term as the divisors of 12 are 1,2,3,4,6,12 and 1*2*3*4*6*1*2 = 288 which is divisible by 12.
a(19) = 28 is a term as the divisors of 28 are 1,2,4,7,14,28 and 1*2*4*7*1*4*2*8 = 3584 which is divisible by 28.

Scott R. Shannon, Table of n, a(n) for n = 1..117

Cf. A002473, A027750, A000005, A007954, A007602, A005349, A038186, A333617.

Select[Range[3^6], (prod = Times @@ (Times @@@ IntegerDigits @ Divisors[#])) > 0 && Divisible[prod, #] &] (* Amiram Eldar, Aug 22 2020 *)

is(n) = {if(n < 10, return(n > 0)); f = factor(n); if(f[#f~, 1] > 7, return(0)); my(d = divisors(n), p = 1); for(i = 2, #d, dd = digits(d[i]); for(j = 1, #dd, p *= dd[j]); if(p == 0, return(0))); p % n == 0} \\ David A. Corneth, Aug 22 2020

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

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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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A337230 Numbers that are a divisor of the product of the digits of its divisors.

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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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