A034053 -id:A034053 - 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

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

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

A199988 Numbers whose product of digits is 6.

Original entry on oeis.org

6, 16, 23, 32, 61, 116, 123, 132, 161, 213, 231, 312, 321, 611, 1116, 1123, 1132, 1161, 1213, 1231, 1312, 1321, 1611, 2113, 2131, 2311, 3112, 3121, 3211, 6111, 11116, 11123, 11132, 11161, 11213, 11231, 11312, 11321, 11611, 12113, 12131, 12311, 13112, 13121

Offset: 1

Jaroslav Krizek, Nov 13 2011

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

Subsequence of A034053.

Cf. A007954.

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

Select[Range[20000], Times @@ IntegerDigits[#] == 6 &] (* T. D. Noe, Nov 16 2011 *)

from sympy import prod
from sympy.utilities.iterables import multiset_permutations
def agen(maxdigits):
    for digs in range(1, maxdigits+1):
        for mp in multiset_permutations("1"*(digs-1) + "236", digs):
            if prod(map(int, mp)) == 6: yield int("".join(mp))
print(list(agen(5))) # Michael S. Branicky, Jun 16 2021

A201020 Composite numbers whose multiplicative digital root is 6.

Original entry on oeis.org

6, 16, 28, 32, 44, 48, 68, 74, 82, 84, 86, 116, 123, 128, 132, 144, 147, 148, 161, 168, 174, 182, 184, 186, 213, 218, 224, 228, 231, 238, 242, 244, 246, 264, 267, 272, 276, 282, 288, 289, 298, 312, 321, 328, 344, 368, 374, 377, 378, 382, 386, 387, 414, 417, 418, 422

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201019 with respect to A034053.

			Number 128 is in sequence because 1*2*8=16, 1*6=6.

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

Cf. A201019 (primes whose multiplicative digital root is 6), A034053 (numbers whose multiplicative digital root is 6).

mdr6Q[n_]:=CompositeQ[n]&&NestWhile[Times@@IntegerDigits[#]&,n,#>9&] ==6; Select[Range[500],mdr6Q] (* Harvey P. Dale, May 30 2015 *)

A201019 Primes whose multiplicative digital root is 6.

Original entry on oeis.org

23, 47, 61, 227, 281, 283, 347, 443, 449, 467, 487, 647, 683, 743, 769, 773, 797, 821, 823, 829, 863, 887, 967, 977, 1123, 1213, 1231, 1283, 1289, 1321, 1471, 1481, 1487, 1489, 1627, 1697, 1741, 1783, 1823, 1847, 1861, 1873, 2113, 2131, 2237, 2243, 2267, 2273, 2281, 2287, 2311

Offset: 1

Jaroslav Krizek, Nov 25 2011

Complement of A201020 with respect to A034053.

			Prime 227 is in sequence because 2*2*7=28, 2*8=16, 1*6=6.

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

Cf. A201020 (composite numbers whose multiplicative digital root is 6), A034053 (numbers whose multiplicative digital root is 6).

dr6Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==6; Select[ Prime[ Range[ 400]],dr6Q] (* Harvey P. Dale, Jun 16 2016 *)

cp's OEIS Frontend

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

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

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A277061 Numbers with multiplicative digital root > 0.

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A199988 Numbers whose product of digits is 6.

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A201020 Composite numbers whose multiplicative digital root is 6.

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Mathematica

A201019 Primes whose multiplicative digital root is 6.

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Mathematica