A034049 -id:A034049 - cp's OEIS frontend

A263471 Total number of positive integers < 10^n with multiplicative digital root value 2.

1, 9, 77, 543, 3213, 16673, 86093, 503815, 3529057, 25402097, 162303510, 884504882, 4156234265, 17270407962, 65375131342, 232901619970, 807191392546, 2795912956450, 9796747697594, 34556445906044, 120898966116007, 413105921852769, 1363586516014222

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263477.

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

Cf. A031347, A034049, A051822, A263477.

Length@ Select[Range[10^# - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 2 &] & /@ Range@ 6 (* Michael De Vlieger, Oct 19 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) == 2, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263470(n) + A000027(n) + a(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

a(8) from Michael De Vlieger, Oct 19 2015

a(9)-a(23) 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

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

A199986 Numbers with digital product = 2.

Original entry on oeis.org

2, 12, 21, 112, 121, 211, 1112, 1121, 1211, 2111, 11112, 11121, 11211, 12111, 21111, 111112, 111121, 111211, 112111, 121111, 211111, 1111112, 1111121, 1111211, 1112111, 1121111, 1211111, 2111111, 11111112, 11111121, 11111211, 11112111, 11121111, 11211111

Offset: 1

Jaroslav Krizek, Nov 13 2011

Subsequence of A034049.

Robert Israel, Table of n, a(n) for n = 1..10000
S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013-2015. See Sect. 3.2.1.

f:= proc(d) local b,i;
  b:= (10^d-1)/9;
  seq(b+10^i,i=0..d-1);
end proc:
seq(f(d),d=1..9); # Robert Israel, Jan 13 2021

one(n)=if(n,10^n\9,"")
for(n=0,9,for(m=0,n,print1(one(n-m)2one(m)", "))) \\ Charles R Greathouse IV, Nov 13 2011

def athrough(k1s):
  return [int("1"*(i-j)+"2"+"1"*j) for i in range(k1s+1) for j in range(i+1)]
print(athrough(8)) # Michael S. Branicky, Feb 12 2021

A199980 Primes whose multiplicative digital root is 2.

Original entry on oeis.org

2, 37, 43, 73, 137, 173, 211, 223, 317, 367, 389, 431, 673, 827, 839, 929, 983, 1223, 1279, 1297, 1367, 1447, 1621, 1637, 1693, 1999, 2111, 2161, 2179, 2213, 2269, 2339, 2393, 2663, 2699, 2719, 2791, 2917, 2969, 2971, 3167, 3169, 3221, 3329, 3463, 3499, 3617

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199981 with respect to A034049, numbers whose multiplicative digital root is 2.

			Prime 389 is in sequence because 3*8*9=216, 2*1*6 =12, 1*2=2.

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

Cf. A199981 (nonprime numbers whose multiplicative digital root is 2).
Includes A107612.
Cf. A031347, A034049.

mdr:= proc(n) option remember;
  local t;
  t:= convert(convert(n,base,10),`*`);
  if t < 10  then t else procname(t) fi
end proc:
select(t -> mdr(t) = 2 and isprime(t), [2, seq(i,i=3..10000,2)]); # Robert Israel, Nov 05 2020

t = {}; n = 0; While[Length[t] < 100, n = NextPrime[n]; s = n; While[s >= 10, s = Times @@ IntegerDigits[s]]; If[s == 2, AppendTo[t, n]]]; t (* T. D. Noe, Nov 15 2011 *)
Select[Prime[Range[600]],FixedPoint[Times@@IntegerDigits[#]&,#]==2&] (* Harvey P. Dale, Mar 28 2012 *)

A199981 Composite numbers whose multiplicative digital root is 2.

Original entry on oeis.org

12, 21, 26, 34, 62, 112, 121, 126, 134, 143, 162, 216, 232, 261, 278, 279, 287, 297, 299, 314, 322, 341, 369, 371, 376, 396, 398, 413, 447, 469, 474, 496, 612, 621, 637, 639, 649, 666, 693, 694, 713, 728, 729, 731, 736, 744, 763, 782, 792, 872, 893, 927, 936

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199980 with respect to A034049.

			Number 278 is in sequence because 2*7*8=112, 1*1*2 =2.

Cf. A199980 (primes whose multiplicative digital root is 2).

Cf. A034049 (numbers whose multiplicative digital root is 2).

t = {}; n = 0; While[Length[t] < 100, n = n + 1; If[! PrimeQ[n], s = n; While[s >= 10, s = Times @@ IntegerDigits[s]]; If[s == 2, AppendTo[t, n]]]]; t (* T. D. Noe, Nov 16 2011 *)

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A263471 Total number of positive integers < 10^n with multiplicative digital root value 2.

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

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A277061 Numbers with multiplicative digital root > 0.

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A199986 Numbers with digital product = 2.

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A199980 Primes whose multiplicative digital root is 2.

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A199981 Composite numbers whose multiplicative digital root is 2.

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