A255732 -id:A255732 - cp's OEIS frontend

A099542 Rhonda numbers to base 10.

1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, 15625, 15698, 19435, 25284, 25662, 33475, 34935, 35581, 45951, 47265, 47594, 52374, 53176, 53742, 54479, 55272, 56356, 56718, 95232, 118465, 133857, 148653, 154462, 161785

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 21 2004

An integer m is a Rhonda number to base b if the product of its digits in base b equals b*(sum of prime factors of m (taken with multiplicity)).
Does every Rhonda number to base 10 contain at least one 5? - Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008
Yes, every Rhonda number m to base 10 contains at least one 5 and also one even digit, otherwise A007954(m) mod 10 > 0. - Reinhard Zumkeller, Dec 01 2012

			1568 has prime factorization 2^5 * 7^2. Sum of prime factors = 2*5 + 7*2 = 24. Product of digits of 1568 = 1*5*6*8 = 240 = 10*24, hence 1568 is a Rhonda number to base 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (first 180 terms from Harvey P. Dale)
Kevin Brown, Infinitely Many Rhondas.
Giovanni Resta, Rhonda numbers the 64507 base-10 Rhonda numbers up to 10^12.
Walter Schneider, Rhonda Numbers.
Eric Weisstein's World of Mathematics, Rhonda Number.

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A100973 (base 9), A100971 (base 12), A100972 (base 14), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60), see also A255880.
Cf. A001414, A027746, A007954.
Column k=5 of A291925.

import Data.List (unfoldr); import Data.Tuple (swap)
a099542 n = a099542_list !! (n-1)
a099542_list = filter (rhonda 10) [1..]
rhonda b x = a001414 x * b == product (unfoldr
       (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b) x)
-- Reinhard Zumkeller, Mar 05 2015, Dec 01 2012

Select[Range[200000],10Total[Times@@@FactorInteger[#]]==Times@@ IntegerDigits[ #]&] (* Harvey P. Dale, Oct 16 2011 *)

A007954(a(n)) = 10 * A001414(a(n)).

A100968 Integers that are Rhonda numbers to base 4.

Original entry on oeis.org

10206, 11935, 12150, 16031, 45030, 94185, 113022, 114415, 191149, 244713, 259753, 374782, 392121, 503773, 649902, 703326, 716250, 764526, 883630, 884446, 912766, 980694, 980837, 1005502, 1420250, 1474239, 1567335, 1685159, 1702822, 1824634, 1944190, 1948279

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 24 2004

See sequence of base 10 Rhonda numbers for more information and links.

			10206 is a Rhonda number to base 4 because the product of its base 4 digits is 2*1*3*3*1*3*2=108, the sum of the prime factors of 10206 is 2+6*3+7=27 and 27*4=108.
From _Reinhard Zumkeller_, Mar 05 2015: (Start)
a(18) = 764526 = 2*4^9 + 3*4^8 + 2*4^7 + 2*4^6 + 2*4^5 + 2*4^4 + 1*4^3 + 2*4^2 + 3*4^1 + 2*4^0 = 2*3*7*109*167
with 2*3*2*2*2*2*1*2*3*2 = 4 * (2+3+7+109+167) = 1152;
a(21) = 912766 = 3*4^9 + 1*4^8 + 3*4^7 + 2*4^6 + 3*4^5 + 1*4^4 + 1*4^3 + 3*4^2 + 3*4^1 + 2*4^0 = 2*53*79*109
with 3*1*3*2*3*1*1*3*3*2 = 4 * (2+53+79+109) = 972.  (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number

Rhonda numbers to other bases: 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), A255731 (base 60), see also A255872.
Cf. A001414, A027746, A007090, subsequence of A023705.
Column k=1 of A291925.

a100968 n = a100968_list !! (n-1)
a100968_list = filter (rhonda 4) a023705_list
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 08 2015

A100968Q[k_] := Times @@ IntegerDigits[k, 4] == 4*Total[Times @@@ FactorInteger[k]];
Select[Range[2000000], A100968Q] (* Paolo Xausa, Jul 01 2025 *)

a(18) and a(21) corrected, terms a(24) - a(32) by Reinhard Zumkeller, Mar 05 2015

A291925 A(n,k) is the n-th Rhonda number to base A002808(k), the k-th composite number; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

10206, 855, 11935, 1836, 1029, 12150, 15540, 6318, 3813, 16031, 1568, 21054, 6622, 5577, 45030, 560, 2835, 25331, 10530, 7040, 94185, 11475, 800, 4752, 44360, 14500, 7304, 113022, 2392, 18655, 3993, 5265, 44660, 14739, 15104, 114415, 1000, 2472, 20565, 4425, 5439, 44733, 17655, 19136, 191149

Offset: 1

Alois P. Heinz, Sep 05 2017

Integer m is Rhonda number to base b if the product of its base-b digits divided by b is equal to the integer log of m, A001414(m). This can only happen if b is a composite number, b in {A002808}.

			A(1,2) = 855 is the first and smallest term of column k=2. The second composite number is A002808(2) = 6.  855 = (((3*6)+5)*6+4)*6+3 = 3543_6 = 3*3*5*19. And (3*5*4*3)/6 = A001414(855) = 3+3+5+19 = 30.
Square array A(n,k) begins:
:  10206,   855,  1836, 15540, 1568,  560, 11475, ...
:  11935,  1029,  6318, 21054, 2835,  800, 18655, ...
:  12150,  3813,  6622, 25331, 4752, 3993, 20565, ...
:  16031,  5577, 10530, 44360, 5265, 4425, 29631, ...
:  45030,  7040, 14500, 44660, 5439, 4602, 31725, ...
:  94185,  7304, 14739, 44733, 5664, 4888, 45387, ...
: 113022, 15104, 17655, 47652, 5824, 7315, 58404, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Rhonda Number

Columns k=1-11,19,42 give: A100968, A100969, A100970, A100973, A099542, A100971, A100972, A100974, A100975, A255735, A255732, A255736, A255731.
Row n=1 gives A255872.
Main diagonal gives A255880.
Cf. A001414, A002808, A100987, A100988, A217986.

A100969 Integers that are Rhonda numbers to base 6.

Original entry on oeis.org

855, 1029, 3813, 5577, 7040, 7304, 15104, 19136, 35350, 36992, 41031, 42009, 60368, 65536, 67821, 76880, 84525, 90601, 122831, 131175, 154570, 162565, 184009, 184585, 196504, 217021, 219830, 222200, 252161, 256041, 268677, 353115, 355737, 357568, 367517, 371229, 388367

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 24 2004

See sequence A099542 for definition of Rhonda numbers and links.

			The product of the base 6 digits of 507500 is 1*4*5*1*3*3*1*2=360 and the sum of the prime factors of 507500 is 2*2+4*5+7+29=60 and 360=6*60.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number
Wikipedia, Senary

Cf. Rhonda numbers to other bases: A100968 (base 4), 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), A255731 (base 60), see also A255872.
Cf. A001414, A027746, A007092, subsequence of A248910.
Column k=2 of A291925.

a100969 n = a100969_list !! (n-1)
a100969_list = filter (rhonda 6) a248910_list
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 08 2015

A100969Q[k_] := Times @@ IntegerDigits[k, 6] == 6*Total[Times @@@ FactorInteger[k]];
Select[Range[400000], A100969Q] (* Paolo Xausa, Jul 01 2025 *)

A100970 Integers that are Rhonda numbers to base 8.

Original entry on oeis.org

1836, 6318, 6622, 10530, 14500, 14739, 17655, 18550, 25398, 25956, 30562, 39215, 39325, 50875, 51429, 52887, 55611, 56420, 58548, 59731, 60604, 72358, 74620, 76581, 78780, 81370, 84180, 85949, 103350, 103788, 108750, 112914, 118233

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 24 2004

See A099542 for definition of Rhonda numbers and some links.

			Product of base 8 digits of 1836 is 3*4*5*4=240 and sum of prime factors of 1826 is 2*2+3*3+17=30 and 240=8*30.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number
Wikipedia, Octal

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), 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), A255731 (base 60), see also A255872.
Cf. A001414, A027746, A007094, subsequence of A255805.
Column k=3 of A291925.

a100970 n = a100970_list !! (n-1)
a100970_list = filter (rhonda 8) a255805_list
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 08 2015

rhda8Q[n_]:=Times@@IntegerDigits[n,8]==8*Total[Times@@@FactorInteger[n]]; Select[Range[120000],rhda8Q] (* Harvey P. Dale, Jul 10 2021 *)

A100973 Integers that are Rhonda numbers to base 9.

Original entry on oeis.org

15540, 21054, 25331, 44360, 44660, 44733, 47652, 50560, 54944, 76857, 77142, 83334, 83694, 96448, 97944, 106575, 108273, 117624, 125952, 138966, 141204, 144236, 153318, 158417, 159424, 188529, 188598, 189350, 192000, 192126, 196652, 202350, 203320, 205390, 246675, 247632

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004

See sequence A099542 for definition of Rhonda numbers and for some links.

			The product of the base 9 digits of 15540 is 2*3*2*7*6=504, the sum of the prime factors of 15540 is 2*2+3+5+7+37=56 and 504=9*56. So 15540 is a Rhonda number to base 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A099542 (base 10), A100971 (base 12), A100972 (base 14), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60), see also A255872.
Cf. A001414, A027746, A007095, subsequence of A255808.
Column k=4 of A291925.

a100973 n = a100973_list !! (n-1)
a100973_list = filter (rhonda 9) a255808_list
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 08 2015

A100973Q[k_] := Times @@ IntegerDigits[k, 9] == 9*Total[Times @@@ FactorInteger[k]];
Select[Range[250000], A100973Q] (* Paolo Xausa, Jul 01 2025 *)

A100975 Integers that are Rhonda numbers to base 16.

Original entry on oeis.org

1000, 1134, 6776, 15912, 19624, 20043, 20355, 23946, 26296, 29070, 31906, 32292, 34236, 34521, 36465, 39066, 50055, 50986, 52341, 54340, 58088, 59541, 75867, 85870, 87150, 88182, 93058, 95944, 99674, 101460, 109650, 109810, 119480, 124500

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004

See A099542 for definition of Rhonda numbers and for some links.

			The product of the base 16 digits of 1000 is 3*14*8=336. The sum of the prime factors of 1000 is 3*2+3*5=21. 336=16*21 and so 1000 is a Rhonda number to base 16.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number
Wikipedia, Hexadecimal

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), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60).
Cf. A001414, A027746, A255872.
Column k=9 of A291925.

a100975 n = a100975_list !! (n-1)
a100975_list = filter (rhonda 16) $ iterate z 1 where
   z x = 1 + if r < 15 then x else 16 * z x' where (x', r) = divMod x 16
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 07 2015

Offset fixed by Reinhard Zumkeller, Mar 05 2015

A100971 Integers that are Rhonda numbers to base 12.

Original entry on oeis.org

560, 800, 3993, 4425, 4602, 4888, 7315, 8296, 9315, 11849, 12028, 13034, 14828, 15052, 16264, 18511, 18906, 25619, 25875, 27176, 32742, 37264, 37523, 46035, 50765, 52338, 58261, 58504, 59166, 62002, 66176, 66752, 66928, 67195, 68502

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004

See sequence A099542 for the definition of Rhonda numbers and some links. 560 is the smallest integer that is a Rhonda number to some base.

			The product of the base 12 digits of 560 is 3*10*8=240; the sum of the prime factors of 560 is 4*2+5+7=20; and 240=12*20 so 560 is a Rhonda number to base 12.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number
Wikipedia, Duodecimal

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A100973 (base 9), A099542 (base 10), A100972 (base 14), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60).
Cf. A001414, A027746, A255872.
Column k=6 of A291925.

a100971 n = a100971_list !! (n-1)
a100971_list = filter (rhonda 12) $ iterate z 1 where
   z x = 1 + if r < 11 then x else 12 * z x' where (x', r) = divMod x 12
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 07 2015

A100972 Integers that are Rhonda numbers to base 14.

Original entry on oeis.org

11475, 18655, 20565, 29631, 31725, 45387, 58404, 58667, 59950, 63945, 67525, 68904, 91245, 99603, 125543, 135196, 141141, 148645, 149575, 168270, 175577, 204611, 207230, 237540, 240006, 240787, 275429, 287950, 321867, 338709, 358984

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 25 2004

See sequence A099542 for definition of Rhonda numbers and for some links.

			The product of the base 14 digits of 11475 is 4*2*7*9=504, the sum of the prime factors of 11475 is 3*3+2*5+17=46 and 504=14*36, so 11475 is a Rhonda number to base 14.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number

Cf. Rhonda numbers to other bases: A100968 (base 4), A100969 (base 6), A100970 (base 8), A100973 (base 9), A099542 (base 10), A100971 (base 12), A100974 (base 15), A100975 (base 16), A255735 (base 18), A255732 (base 20), A255736 (base 30), A255731 (base 60).
Cf. A001414, A027746, A255872.
Column k=7 of A291925.

a100972 n = a100972_list !! (n-1)
a100972_list = filter (rhonda 14) $ iterate z 1 where
   z x = 1 + if r < 13 then x else 14 * z x' where (x', r) = divMod x 14
-- Function rhonda as in A099542.
-- Reinhard Zumkeller, Mar 07 2015

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

See A099542 for definition of Rhonda numbers and for some links.

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rhonda Number

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).
Cf. A001414, A027746, A255872.
Column k=8 of A291925.

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

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A099542 Rhonda numbers to base 10.

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A100968 Integers that are Rhonda numbers to base 4.

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A291925 A(n,k) is the n-th Rhonda number to base A002808(k), the k-th composite number; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A100969 Integers that are Rhonda numbers to base 6.

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A100970 Integers that are Rhonda numbers to base 8.

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A100973 Integers that are Rhonda numbers to base 9.

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A100975 Integers that are Rhonda numbers to base 16.

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A100971 Integers that are Rhonda numbers to base 12.

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