A034051 -id:A034051 - cp's OEIS frontend

A263472 Total number of positive integers < 10^n with multiplicative digital root value 4.

1, 10, 65, 279, 894, 2345, 6174, 66354, 1005399, 9737884, 66699415, 356586629, 1585685916, 6342292785, 30560724590, 264486626166, 2926013859615, 28611339267816, 232062904906327, 1588966488941437, 9421788721876388, 49444241961339027, 233673191793220620

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263478.

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

Cf. A031347, A034051, A263478.

lim = 6; t = Select[Range[1, 10^lim - 1], FixedPoint[Times @@ IntegerDigits@ # &, #] == 4 &]; 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) == 4, 1, 0)); \\ Altug Alkan, Oct 19 2015

A263470(n) + A000027(n) + A263471(n) + A000217(n) + a(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(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

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

A199987 Numbers with digital product = 4.

Original entry on oeis.org

4, 14, 22, 41, 114, 122, 141, 212, 221, 411, 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111, 11114, 11122, 11141, 11212, 11221, 11411, 12112, 12121, 12211, 14111, 21112, 21121, 21211, 22111, 41111, 111114, 111122, 111141, 111212, 111221, 111411

Offset: 1

Jaroslav Krizek, Nov 13 2011

Subsequence of A034051.

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

f:= proc(d) local b,i,t;
   b:= (10^d-1)/9;
   op(sort([seq(b+3*10^i,i=0..d-1),
    seq(b+10^t[1]+10^t[2],t=combinat:-choose([$0..d-1],2))]))
end proc:
seq(f(d),d=1..6); # Robert Israel, Jan 13 2021

Select[Range[200000], Times @@ IntegerDigits[#] == 4 &] (* T. D. Noe, Nov 16 2011 *)

A117678 Squares for which the multiplicative digital root is also a square.

Original entry on oeis.org

0, 1, 4, 9, 25, 100, 169, 196, 225, 256, 400, 529, 576, 625, 676, 900, 961, 1024, 1089, 1156, 1225, 1296, 1521, 1600, 2025, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3481, 3600, 3844, 3969, 4096, 4225, 4356, 4489, 4900, 5041, 5184, 5329

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 12 2006

From Robert Israel, Oct 22 2015: (Start)
1, 9, and squares in A034048 and A034051.
Are there infinitely many squares in A034051? (End)

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A000290, A031347, A034048, A034051, A116978.

A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A117678 := proc(n) option remember: local k, m: if(n=1)then return 0:fi: for k from procname(n-1)+1 do m:=k^2: while(length(m)>1)do m:=A007954(m): od: if(m in {0,1,4,9})then return k: fi: od: end: seq(A117678(n)^2, n=1..47); # Nathaniel Johnston, May 05 2011

Select[Range[0, 73]^2, IntegerQ@ Sqrt[FixedPoint[Times @@ IntegerDigits@ # &, #] &@ #] &] (* Michael De Vlieger, Oct 22 2015 *)

t(k) = {while(k>9, k=prod(i=1, #k=digits(k), k[i])); k}
for(n=0, 100, if(issquare(t(n^2)), print1(n^2, ", "))); \\ Altug Alkan, Oct 22 2015

Offset and some terms corrected by Nathaniel Johnston, May 05 2011

A199983 Primes whose multiplicative digital root is 4.

Original entry on oeis.org

41, 89, 127, 139, 193, 271, 277, 379, 383, 397, 463, 643, 677, 727, 739, 937, 1193, 1217, 1249, 1277, 1319, 1429, 1721, 1913, 1931, 1973, 2377, 2711, 3119, 3191, 3313, 3331, 3373, 3461, 3719, 3727, 3733, 3911, 3917, 4111, 4129, 4219, 6143, 7121, 7127, 7193

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199984 with respect to A034051.

Can this sequence be proved to be infinite? [Charles R Greathouse IV, Nov 13 2011]

			Prime 139 is in sequence because 1*3*9=27, 2*7=14, 1*4=4.

Cf. A199984 (composite numbers whose multiplicative digital root is 4).

Cf. A034051 (numbers whose multiplicative digital root is 4).

t = {}; n = 0; While[Length[t] < 100, n = NextPrime[n]; s = n; While[s >= 10, s = Times @@ IntegerDigits[s]]; If[s == 4, AppendTo[t, n]]]; t (* T. D. Noe, Nov 16 2011 *)

A199984 Composite numbers whose multiplicative digital root is 4.

Original entry on oeis.org

4, 14, 22, 27, 39, 72, 93, 98, 114, 122, 141, 172, 189, 198, 212, 217, 221, 249, 266, 294, 319, 333, 338, 346, 364, 391, 411, 429, 436, 492, 626, 634, 662, 712, 721, 767, 772, 776, 793, 819, 833, 891, 913, 918, 924, 931, 942, 973, 981, 1114, 1122, 1127, 1139

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199983 with respect to A034051.

			Number 172 is in sequence because 1*7*2=14, 1*4=4.

Cf. A199983 (primes whose multiplicative digital root is 4).

Cf. A034051 (numbers whose multiplicative digital root is 4).

cn4Q[n_]:=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]==4; Select[Select[ Range[ 1200],CompositeQ],cn4Q] (* Harvey P. Dale, Apr 28 2018 *)

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A263472 Total number of positive integers < 10^n with multiplicative digital root value 4.

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

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A277061 Numbers with multiplicative digital root > 0.

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A199987 Numbers with digital product = 4.

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A117678 Squares for which the multiplicative digital root is also a square.

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Maple

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A199983 Primes whose multiplicative digital root is 4.

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A199984 Composite numbers whose multiplicative digital root is 4.

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Mathematica