A100974
Integers that are Rhonda numbers to base 15.
Original entry on oeis.org
2392, 2472, 11468, 15873, 17424, 18126, 19152, 20079, 24388, 30758, 31150, 33004, 33550, 37925, 39483, 42550, 44714, 58870, 59605, 66950, 70182, 71485, 71709, 85557, 85848, 86241, 86591, 92150, 110334, 112671, 113300, 116270, 120414
Offset: 1
Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004
The product of the base 15 digits of 2392 is 10*9*7=630. The sum of the prime factors of 2392 is 3*2+13+23=42 and 630=15*42. So 2392 is a Rhonda number to base 15.
Cf. Rhonda numbers to other bases:
A100968 (base 4),
A100969 (base 6),
A100970 (base 8),
A100973 (base 9),
A099542 (base 10),
A100971 (base 12),
A100972 (base 14),
A100975 (base 16),
A255735 (base 18),
A255732 (base 20),
A255736 (base 30),
A255731 (base 60).
-
a100974 n = a100974_list !! (n-1)
a100974_list = filter (rhonda 15) $ iterate z 1 where
z x = 1 + if r < 14 then x else 15 * z x' where (x', r) = divMod x 15
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 07 2015
A255732
Rhonda numbers in vigesimal number system.
Original entry on oeis.org
1815, 11050, 15295, 21165, 22165, 30702, 34510, 34645, 42292, 44165, 52059, 53416, 65945, 78430, 80712, 84251, 84835, 86591, 112608, 146055, 148144, 156284, 175419, 178350, 194590, 200655, 201825, 202664, 204085, 209095, 209550, 211250, 212346, 212850
Offset: 1
a(1) = 1815 = 4*20^2 + 10*20^1 + 15*20^0 = 3*5*11*11,
with 4 * 10 * 15 = 20 * (3+5+11+11) = 600;
a(10) = 44165 = 5*20^3 + 10*20^2 + 8*20^1 + 5*20^0 = 5*11*11*73,
with 5 * 10 * 8 * 5 = 20 * (5+11+11+73) = 2000.
Cf. Rhonda numbers to other bases:
A100968 (base 4),
A100969 (base 6),
A100970 (base 8),
A100973 (base 9),
A099542 (base 10),
A100971 (base 12),
A100972 (base 14),
A100974 (base 15),
A100975 (base 16),
A255735 (base 18),
A255736 (base 30),
A255731 (base 60), see also
A255872.
A255735
Integers that are Rhonda numbers to base 18.
Original entry on oeis.org
1470, 3000, 8918, 17025, 19402, 20650, 21120, 22156, 26522, 36549, 38354, 43281, 46035, 48768, 54229, 54528, 56584, 58216, 58224, 62238, 68096, 68150, 73161, 74024, 74636, 87978, 94041, 114000, 124656, 132240, 133926, 135876, 153105, 153870, 156621, 159819
Offset: 1
a(1) = 1470 = 4*18^2 + 9*18^1 + 12*18^0 = 2*3*5*7*7,
with 4 * 9 * 12 = 18 * (2+3+5+7+7) = 432;
a(10) = 36549 = 6*18^3 + 4*18^2 + 14*18^1 + 9*18^0 = 3*3*31*131,
with 6 * 4 * 14 * 9 = 18 * (3+3+31+131) = 3024.
Cf. Rhonda numbers to other bases:
A100968 (base 4),
A100969 (base 6),
A100970 (base 8),
A100973 (base 9),
A099542 (base 10),
A100971 (base 12),
A100972 (base 14),
A100974 (base 15),
A100975 (base 16),
A255732 (base 20),
A255736 (base 30),
A255731 (base 60),
A255872.
A255736
Integers that are Rhonda numbers to base 30.
Original entry on oeis.org
3024, 3168, 5115, 5346, 5950, 6762, 7750, 7956, 8470, 9476, 9576, 9849, 10360, 11495, 13035, 13356, 16335, 22610, 22784, 23864, 37515, 38025, 40704, 40986, 49887, 52925, 59800, 60955, 61812, 67782, 68590, 74800, 78430, 85063, 90160, 90649, 90897, 91540
Offset: 1
a(1) = 3024 = 3 * 30^2 + 10 * 30^1 + 24 * 30^0 = 2*2*2*2*3*3*3*7,
with 3 * 10 * 24 = 30 * (2+2+2+2+3+3+3+7) = 720;
a(10) = 9476 = 10 * 30^2 + 15 * 30^1 + 26 * 30^0 = 2*2*23*103,
with 10 * 15 * 26 = 30 * (2+2+23+103) = 3900.
Cf. Rhonda numbers to other bases:
A100968 (base 4),
A100969 (base 6),
A100970 (base 8),
A100973 (base 9),
A099542 (base 10),
A100971 (base 12),
A100972 (base 14),
A100974 (base 15),
A100975 (base 16),
A255735 (base 18),
A255732 (base 20),
A255731 (base 60), see also
A255872.
A255731
Rhonda numbers in sexagesimal number system.
Original entry on oeis.org
3348, 3510, 6750, 17430, 18750, 18876, 18944, 19475, 20564, 21312, 26550, 28280, 37230, 38396, 43940, 48042, 77770, 88270, 91224, 97470, 108882, 111403, 120046, 123630, 181996, 182646, 235467, 253460, 260429, 264735, 278675, 289161, 295960, 296055, 306642
Offset: 1
a(1) = 3348 = 55 * 60^1 + 48 * 60^0 = 2*2*3*3*3*31,
with 55 * 48 = 60 * (2+2+3+3+3+31) = 2640;
a(10) = 21312 = 5*60^2 + 55*60^1 + 12*60^0 = 2*2*2*2*2*2*3*3*37,
with 5 * 55 * 12 = 60 * (2+2+2+2+2+2+3+3+37) = 3300.
Cf. Rhonda numbers to other bases:
A100968 (base 4),
A100969 (base 6),
A100970 (base 8),
A100973 (base 9),
A099542 (base 10),
A100971 (base 12),
A100972 (base 14),
A100974 (base 15),
A100975 (base 16),
A255735 (base 18),
A255732 (base 20),
A255736 (base 30).
A255872
Smallest Rhonda number to base b = n-th composite number, A002808(n).
Original entry on oeis.org
10206, 855, 1836, 15540, 1568, 560, 11475, 2392, 1000, 1470, 1815, 1632, 2695, 2080, 6764, 7788, 4797, 3094, 3024, 1944, 756, 5661, 8232, 1000, 12296, 5824, 4624, 4851, 8262, 6561, 16583, 14616, 6545, 7225, 11310, 18382, 1995, 16896, 2940, 23465, 8464, 3348
Offset: 1
. n | b | a(n) | a(n) in base b | factorization
. ----+----+--------------------+-----------------+--------------
. 1 | 4 | 10206 = A100968(1) | [2,1,3,3,1,3,2] | 2*3^6*7
. 2 | 6 | 855 = A100969(1) | [3,5,4,3] | 3^2*5*19
. 3 | 8 | 1836 = A100970(1) | [3,4,5,4] | 2^2*3^3*17
. 4 | 9 | 15540 = A100973(1) | [2,3,2,7,6] | 2^2*3*5*7*37
. 5 | 10 | 1568 = A099542(1) | [1,5,6,8] | 2^5*7^2
. 6 | 12 | 560 = A100971(1) | [3,10,8] | 2^4*5*7
. 7 | 14 | 11475 = A100972(1) | [4,2,7,9] | 3^3*5^2*17
. 8 | 15 | 2392 = A100974(1) | [10,9,7] | 2^3*13*23
. 9 | 16 | 1000 = A100975(1) | [3,14,8] | 2^3*5^3
. 10 | 18 | 1470 = A255735(1) | [4,9,12] | 2*3*5*7^2
. 11 | 20 | 1815 = A255732(1) | [4,10,15] | 3*5*11^2
. 12 | 21 | 1632 | [3,14,15] | 2^5*3*17
. 13 | 22 | 2695 | [5,12,11] | 5*7^2*11
. 14 | 24 | 2080 | [3,14,16] | 2^5*5*13
. 15 | 25 | 6764 | [10,20,14] | 2^2*19*89
. 16 | 26 | 7788 | [11,13,14] | 2^2*3*11*59
. 17 | 27 | 4797 | [6,15,18] | 3^2*13*41
. 18 | 28 | 3094 | [3,26,14] | 2*7*13*17
. 19 | 30 | 3024 = A255736(1) | [3,10,24] | 2^4*3^3*7
. 20 | 32 | 1944 | [1,28,24] | 2^3*3^5
Cf.
A002808,
A100968,
A100969,
A100970,
A100973,
A099542,
A100971,
A100972,
A100974,
A100975,
A255735,
A255732,
A255736.
A255880
a(n) = n-th Rhonda number to base b = n-th composite number, cf. A002808.
Original entry on oeis.org
10206, 1029, 6622, 44360, 5439, 4888, 58404, 20079, 26296, 36549, 52059, 61376, 131427, 29106, 165504, 137007, 63525, 61115, 22784, 135705, 658896, 563159, 208369, 115506, 1078784, 228436, 152308, 185571, 539213, 152532, 2289001, 193963, 2499742, 298768
Offset: 1
Diagonalization of Rhonda numbers to base b = A002808(n), n = 1 .. 8:
. b | n\n 1 2 3 4 5 6 7 8
. ----+---+---------------------------------------------------------------
. 4 | 1 | A100968 [10206] 11935 12150 16031 45030 94185 113022 114415
. 6 | 2 | A100969 855 [1029] 3813 5577 7040 7304 15104 19136
. 8 | 3 | A100970 1836 6318 [6622] 10530 14500 14739 17655 18550
. 9 | 4 | A100973 15540 21054 25331 [44360] 44660 44733 47652 50560
. 10 | 5 | A099542 1568 2835 4752 5265 [5439] 5664 5824 5832
. 12 | 6 | A100971 560 800 3993 4425 4602 [4888] 7315 8296
. 14 | 7 | A100972 11475 18655 20565 29631 31725 45387 [58404] 58667
. 15 | 8 | A100974 2392 2472 11468 15873 17424 18126 19152 [20079]
-
a255880 n = (filter (rhonda b) $ iterate zeroless 1) !! (n - 1) where
-- function rhonda as defined in A099542
zeroless x = 1 + if r < b - 1 then x else b * zeroless x'
where (x', r) = divMod x b
b = a002808 n
-
nc = 34; (* number of composite bases *)
compos = Select[Range[FindRoot[n == nc + PrimePi[n] + 1, {n, nc, 2nc}][[1, 2]] // Floor], CompositeQ];
RhondaQ[n_, b_] := Times @@ IntegerDigits[n, b] == b Total[Times @@@ FactorInteger[n]];
a[n_] := a[n] = Module[{b = compos[[n]], cnt = 0, k}, For[k = 1, True, k++, If[RhondaQ[k, b], cnt++; If[cnt == n, Return[k]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, nc}] (* Jean-François Alcover, Nov 15 2021 *)
A100987
Integers that are Rhonda numbers to some base.
Original entry on oeis.org
560, 756, 800, 855, 1000, 1029, 1134, 1470, 1568, 1632, 1750, 1815, 1836, 1944, 1995, 2080, 2100, 2392, 2472, 2662, 2695, 2709, 2835, 2940, 3000, 3024, 3060, 3087, 3094, 3168, 3240, 3264, 3348, 3456, 3510, 3600, 3672, 3675, 3744, 3750, 3813, 3888, 3952, 3976
Offset: 1
Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004
560 is a Rhonda number to base 12. 756 is a Rhonda number to base 33. 800 is a Rhonda number to base 12 etc. No integer smaller than 560 is a Rhonda number and there exists no Rhonda number between 560 and 756.
A100988
Integers that are Rhonda numbers to more than one base.
Original entry on oeis.org
1000, 2940, 4200, 4212, 4725, 5670, 5824, 5832, 6776, 6864, 7040, 7140, 8296, 9476, 9633, 10200, 11016, 11050, 11160, 11495, 11935, 12393, 12474, 13068, 13260, 13671, 14014, 14322, 14406, 15680, 15750, 15912, 16240, 16821, 17056, 17820, 18270, 18655, 18700
Offset: 1
Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004
1000 is a Rhonda number to bases 16 and 36. 2940 is a Rhonda number to bases 56 and 76. 5670 is a Rhonda number to bases 36, 106, 108 and 196.
Cf.
A099542 for definition of Rhonda numbers.
A100968 to
A100975 for Rhonda numbers to specific bases.
A100987 for integers that are Rhonda numbers to some base.
A217986
The smallest k that is a Rhonda number in exactly n bases.
Original entry on oeis.org
560, 1000, 10200, 5670, 63945, 158400, 322920, 140800, 1200420, 889200, 792064, 4706352, 6331446, 12042800, 1000350, 8429960, 101892288, 129600900, 365575680, 340692480, 264269250, 304646400, 511999488, 118857024
Offset: 1
a(3)=10200 since 10200=2^3*3*5^2*17, so sopf=36. Its representations in bases 130,174, and 238 are (78,60)_130, (58,108)_174 and (42, 204)_238.
Finally, we have 78*60=130*36, 58*108=174*36, 42*204=238*36 and no smaller number is a Rhonda number w.r.t. exactly 3 bases.
a(24)=118857024 is a Rhonda number in bases {65412, 73593, 97020, 111176, 138996, 190125, 215644, 239057, 250120, 312576, 329004, 354497, 451308, 465426, 544128, 562104, 640692, 778752, 888888, 930402, 1026168, 1101672, 1165944, 1213082}.
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