A031347 -id:A031347 - cp's OEIS frontend

A263474 Total number of positive integers < 10^n with multiplicative digital root value 6.

1, 14, 155, 1172, 6843, 43538, 318457, 2223803, 14185700, 84670477, 477808607, 2577052118, 13759255632, 75251167843, 418157757456, 2267313716636, 11616142299625, 55909713312571, 257522103127082, 1182251998919171, 5791219719115580, 32715779086392723

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263480.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034053, A263480.

lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 6 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 6, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + a(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

a(9)-a(22) from Hiroaki Yamanouchi, Oct 25 2015

A263475 Total number of positive integers < 10^n with multiplicative digital root value 8.

Original entry on oeis.org

1, 23, 161, 1050, 5971, 32658, 187197, 1057467, 5495088, 25862850, 112452321, 501114082, 2867532188, 21469965415, 164448147485, 1116524049413, 6550885669936, 33615367021792, 154093286995596, 651413912544125, 2703190211181211, 12293485890559055

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263481.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034055, A263481.

lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 8 &]; Count[t, n_ /; n <= 10^#] & /@ Range@ lim (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=1, 10^n - 1, if(t(i) == 8, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263470(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + a(n) + A000292(n) = A002283(n).

a(9)-a(22) from Hiroaki Yamanouchi, Oct 25 2015

A263476 Total number of n-digit positive integers with multiplicative digital root value 0.

Original entry on oeis.org

0, 24, 452, 6263, 75662, 820207, 8491909, 86727773, 879578102, 8878382430, 89326016765, 896499124003, 8981948713433, 89902031287356, 899441785434096, 8996797304009863, 89981980972837834, 899900967867894143, 8999467457237611688, 89997196286115385871

Offset: 1

Martin Renner, Oct 19 2015

First differences of A263470.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034048, A263470.

Last /@ Tally@ IntegerLength@ Select[Range[0, 10^6 - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 0 &] (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 0, 1, 0));  \\ Altug Alkan, Oct 19 2015

a(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

a(9)-a(20) from Hiroaki Yamanouchi, Oct 25 2015

A263477 Total number of n-digit positive integers with multiplicative digital root value 2.

Original entry on oeis.org

1, 8, 68, 466, 2670, 13460, 69420, 417722, 3025242, 21873040, 136901413, 722201372, 3271729383, 13114173697, 48104723380, 167526488628, 574289772576, 1988721563904, 7000834741144, 24759698208450, 86342520209963, 292206955736762, 950480594161453

Offset: 1

Martin Renner, Oct 19 2015

First differences of A263471.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Cf. A031347, A034049, A051813, A263471.

Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 2 &] (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 2, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263476(n) + A000012(n) + a(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015

A263478 Total number of n-digit positive integers with multiplicative digital root value 4.

Original entry on oeis.org

1, 9, 55, 214, 615, 1451, 3829, 60180, 939045, 8732485, 56961531, 289887214, 1229099287, 4756606869, 24218431805, 233925901576, 2661527233449, 25685325408201, 203451565638511, 1356903584035110, 7832822232934951, 40022453239462639, 184228949831881593

Offset: 1

Martin Renner, Oct 19 2015

First differences of A263472.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034051, A263472.

Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 4 &] (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 4, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263476(n) + A000012(n) + A263477(n) + A000027(n) + a(n) + A263479(n) + A263480(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015

A263480 Total number of n-digit positive integers with multiplicative digital root value 6.

Original entry on oeis.org

1, 13, 141, 1017, 5671, 36695, 274919, 1905346, 11961897, 70484777, 393138130, 2099243511, 11182203514, 61491912211, 342906589613, 1849155959180, 9348828582989, 44293571012946, 201612389814511, 924729895792089, 4608967720196409, 26924559367277143

Offset: 1

Martin Renner, Oct 19 2015

First differences of A263474.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034053, A263474.

Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 6 &] (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 6, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + a(n) + A000027(n) + A263481(n) + A000217(n) = A052268(n).

a(9)-a(22) from Hiroaki Yamanouchi, Oct 25 2015

A263481 Total number of n-digit positive integers with multiplicative digital root value 8.

Original entry on oeis.org

1, 22, 138, 889, 4921, 26687, 154539, 870270, 4437621, 20367762, 86589471, 388661761, 2366418106, 18602433227, 142978182070, 952075901928, 5434361620523, 27064481351856, 120477919973804, 497320625548529, 2051776298637086, 9590295679377844, 54933121828772931

Offset: 1

Martin Renner, Oct 19 2015

First differences of A263475.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..50

Cf. A031347, A034055, A263475.

Last /@ Tally@ IntegerLength@ Select[Range@ 1000000, FixedPoint[Times @@ IntegerDigits@ # &, #] == 8 &] (* Michael De Vlieger, Oct 21 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
a(n) = sum(i=10^(n-1), 10^n - 1, if(t(i) == 8, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263476(n) + A000012(n) + A263477(n) + A000027(n) + A263478(n) + A263479(n) + A263480(n) + A000027(n) + a(n) + A000217(n) = A052268(n).

a(9)-a(23) from Hiroaki Yamanouchi, Oct 25 2015

A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

Original entry on oeis.org

338688, 826686, 2239488, 3188646, 6613488, 14224896, 3416267673274176, 6499837226778624

Offset: 1

Daniel Mondot, Jan 15 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 8.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
These are the numbers k in A002473 such that A031346(k) = 7,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 7 steps.
Postulated to be finite and complete.
a(9), if it exists, is > 10^20000, and likely > 10^119000.

			338688 is in this sequence because:
- 338688 goes to a single digit in 7 steps: p(338688) = 27648, p(27648) = 2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(4478976) = p(13477889) = 338688, etc.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).

Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350187 (numbers with mp 1 to 6 and 8 to 10 that are themselves 7-smooth numbers).

mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&]  (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

A064702 Nonnegative numbers such that additive and multiplicative digital roots coincide.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 132, 137, 139, 168, 173, 179, 186, 188, 193, 197, 213, 231, 233, 267, 276, 299, 312, 317, 319, 321, 323, 332, 346, 364, 371, 389, 391, 398, 436, 463, 618, 627, 634, 643, 672, 681, 713, 719, 726, 731, 762, 791, 816, 818, 839

Offset: 1

Santi Spadaro, Oct 12 2001

If k is in this sequence then all permutations of (the digits of) k are in this sequence.

A010888(a(n)) = A031347(a(n)). - Reinhard Zumkeller, Jul 10 2013

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digital Root
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Wikipedia, Digital root
Wikipedia, Multiplicative digital root
Index entries for 10-automatic sequences.

Cf. A010888, A031347, A239427.

a064702 n = a064702_list !! (n-1)
a064702_list = filter (\x -> a010888 x == a031347 x) [1..]
-- Reinhard Zumkeller, Jul 10 2013

A007954 := proc(n) return mul(d, d=convert(n,base,10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: A064702 := proc(n) option remember: local k: if(n=1)then return 1:fi: for k from procname(n-1)+1 do if(A031347(k)-1 = (k-1) mod 9)then return k: fi: od: end: seq(A064702(n),n=1..56); # Nathaniel Johnston, May 04 2011

okQ[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]== NestWhile[ Total[ IntegerDigits[ #]]&, n,#>9&]; Select[Range[1000],okQ]  (* Harvey P. Dale, Apr 20 2011 *)

is(n) = my(cn = n); while(cn > 9, d = digits(cn); cn = prod(i = 1, #d, d[i])); cn - 1 == (n-1)%9 \\ David A. Corneth, Aug 23 2018

from math import prod
def A010888(n):
    while n > 9: n = sum(map(int, str(n)))
    return n
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
def ok(n): return A010888(n) == A031347(n)
print([k for k in range(840) if ok(k)]) # Michael S. Branicky, Sep 17 2022

Definition rephrased by Reinhard Zumkeller, Jul 10 2013

Initial 0 added by Halfdan Skjerning, Aug 21 2018

A087471 Final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, 0, 8, 0, 8, 6, 8, 6, 0, 6, 0, 8, 4, 0, 9, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy and Paul D. Hanna, Sep 11 2003

A087472(n) gives the number of iterations required for Murthy's function, f(n), to reach a single digit. A087473(n) gives the smallest number that requires n iterations of Murthy's function to reach a single digit. The n-th row of triangle A087474 gives the n successive iterations of Murthy's function on A087473(n).

Apart from the undefined a(0), the sequence differs from A031347 first at n=121. [From R. J. Mathar, Sep 11 2008]

			a(1234) = a(13*24) = a(312) = a(32*1) = a(32) = a(3*2) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A087472, A087473, A087474.

Table[NestWhile[With[{idn=IntegerDigits[#]},FromDigits[Take[idn,{1,-1,2}]] FromDigits[Take[idn,{2,-1,2}]]]&,n,#>9&],{n,110}] (* Harvey P. Dale, Dec 05 2014 *)

a(n) = a(f(n)), where f(n) is Murthy's function: f(1234)=13*24=312, f(12345)=135*24=3240, f(123456)=135*246=33210.

cp's OEIS Frontend

A263474 Total number of positive integers < 10^n with multiplicative digital root value 6.

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A263475 Total number of positive integers < 10^n with multiplicative digital root value 8.

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A263476 Total number of n-digit positive integers with multiplicative digital root value 0.

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A263477 Total number of n-digit positive integers with multiplicative digital root value 2.

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A263478 Total number of n-digit positive integers with multiplicative digital root value 4.

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A263480 Total number of n-digit positive integers with multiplicative digital root value 6.

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A263481 Total number of n-digit positive integers with multiplicative digital root value 8.

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