A034055 -id:A034055 - cp's OEIS frontend

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

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

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

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115

Offset: 1

J. Lowell, Sep 26 2016

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.

To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Index entries for 10-automatic sequences.

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)

is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

More terms from Michael De Vlieger, Sep 26 2016

A199989 Numbers with digital product = 8.

Original entry on oeis.org

8, 18, 24, 42, 81, 118, 124, 142, 181, 214, 222, 241, 412, 421, 811, 1118, 1124, 1142, 1181, 1214, 1222, 1241, 1412, 1421, 1811, 2114, 2122, 2141, 2212, 2221, 2411, 4112, 4121, 4211, 8111, 11118, 11124, 11142, 11181, 11214, 11222, 11241, 11412, 11421, 11811

Offset: 1

Jaroslav Krizek, Nov 13 2011

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A034055.

Cf. A007954.

f:= proc(d) local b,i,t;
   b:= (10^d-1)/9;
   op(sort([seq(b+7*10^i,i=0..d-1),seq(b+10^t[1]+3*10^t[2],t = combinat:-permute([$0..d-1],2)),seq(b+10^t[1]+10^t[2]+10^t[3],t=combinat:-choose([$0..d-1],3))]))
end proc:
seq(f(d),d=1..5); # Robert Israel, Jan 13 2021

Select[Range[20000], Times @@ IntegerDigits[#] == 8 &] (* T. D. Noe, Nov 16 2011 *)

A201023 Composite numbers whose multiplicative digital root is 8.

Original entry on oeis.org

8, 18, 24, 36, 38, 42, 46, 49, 63, 64, 66, 76, 77, 81, 88, 92, 94, 99, 118, 124, 129, 136, 138, 142, 146, 164, 166, 176, 177, 183, 188, 192, 194, 214, 219, 222, 226, 234, 236, 237, 243, 248, 262, 273, 284, 291, 292, 316, 318, 323, 324, 326, 327, 332, 334, 339

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201022 with respect to A034055.

Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Cf. A201022 (primes whose multiplicative digital root is 8), A034055 (numbers whose multiplicative digital root is 8).

Composite number 124 is in the sequence because 1*2*4=8.

A201022 Primes whose multiplicative digital root is 8.

Original entry on oeis.org

29, 67, 79, 83, 97, 149, 163, 167, 179, 181, 197, 199, 229, 233, 241, 263, 337, 373, 419, 421, 433, 461, 491, 499, 613, 617, 631, 641, 661, 719, 733, 761, 811, 881, 883, 919, 941, 971, 991, 1129, 1163, 1181, 1229, 1237, 1291, 1327, 1361, 1373, 1381, 1399, 1423

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201023 with respect to A034055.

			Prime 149 is in the sequence because 1*4*9=36, 3*6=18, 1*8=8.

Eric Weisstein's World of Mathematics, Multiplicative Digital Root

Cf. A201023 (composite numbers whose multiplicative digital root is 8), A034055 (numbers whose multiplicative digital root is 8).

cp's OEIS Frontend

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

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

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A277061 Numbers with multiplicative digital root > 0.

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A199989 Numbers with digital product = 8.

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A201023 Composite numbers whose multiplicative digital root is 8.

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A201022 Primes whose multiplicative digital root is 8.

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