A034048 -id:A034048 - cp's OEIS frontend

A361337 Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).

0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 59, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 120, 125, 128, 129, 130, 134, 135, 136, 138, 140, 144, 145, 150, 152, 153, 154, 155, 156, 157, 158, 159

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on a posting to the Sequence Fans mailing list by Eric Angelini, Mar 20 2023

We always split the integer N into two integers, then multiply them (and iterate). For example, 2023 can be split into 20 and 23 (producing 20*23 = 460), or split into 202 and 3 (producing 202*3 = 606). The split 2 and 023 is forbidden, as 023 is not an integer (but 460 can be split into 46 and 0 as 0 is an integer).
The sequence lists numbers which reach 0 after a suitable sequence of splits and multiplications.
If we multiply ALL the digits at each step, we get A034048 (115 is the first term where they differ).
The complement (A361978) appears to be finite, containing only 219 members, the largest being 3111. - Michael S. Branicky, Apr 02 2023
More precisely, {811, 911, 913, 921, 1111, 1112, 1113, 1121, 1122, 1131, 1211, 1231, 1261, 1311, 1321, 1612, 2111, 2121, 2211, 3111} are the only numbers not in the sequence, between 792 and at least 10^7. - M. F. Hasler, Apr 05 2023

			We see that 115 reaches 0 when split into 11*5: 11*5 = 55 -> 5*5 = 25 -> 2*5 = 10 -> 1*0 = 0.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A031346, A034048, A361978.

Supersequence of A011540.

select( {is_A361337(n)=!vecmin(digits(n))|| for(p=1,logint(n,10), is_A361337(vecprod(divrem(n,10^p)))&& return(1))}, [1..160]) \\ M. F. Hasler, Apr 05 2023

def ok(n):
    if n < 10: return n == 0
    s = str(n)
    if "0" in s: return True
    return any(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s)))
print([k for k in range(116) if ok(k)]) # Michael S. Branicky, Apr 02 2023

ok = lambda n: '0' in (s:=str(n)) or any(ok(int(s[:i])*int(s[i:])) for i in range(1,len(s))) # M. F. Hasler, Apr 05 2023

a(2894 + k) = 3112 + k for all k >= 0 (conjectured). - M. F. Hasler, Apr 05 2023

a(38) and beyond from Michael S. Branicky, Apr 02 2023

A263470 Total number of positive integers < 10^n with multiplicative digital root value 0.

Original entry on oeis.org

0, 24, 476, 6739, 82401, 902608, 9394517, 96122290, 975700392, 9854082822, 99180099587, 995679223590, 9977627937023, 99879659224379, 999321444658475, 9996118748668338, 99978099721506172, 999879067589400315, 9999346524827012003, 99996542810942397874

Offset: 1

Martin Renner, Oct 19 2015

Partial sums of A263476. - Michel Marcus, Oct 22 2015

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

Cf. A031347, A034048, A263476.

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

a(n) + A000027(n) + A263471(n) + A000217(n) + A263472(n) + A263473(n) + A263474(n) + A000217(n) + A263475(n) + A000292(n) = A002283(n).

a(9)-a(20) 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

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

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

A199977 Primes whose multiplicative digital root is 0.

Original entry on oeis.org

59, 101, 103, 107, 109, 239, 251, 257, 269, 293, 307, 349, 353, 401, 409, 439, 457, 479, 503, 509, 521, 523, 541, 547, 563, 569, 577, 587, 599, 601, 607, 619, 653, 659, 691, 701, 709, 757, 787, 809, 853, 857, 859, 877, 907, 947, 997, 1009, 1013, 1019, 1021

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199978 with respect to A034048.

			Prime 239 is in sequence because 2*3*9 = 45, 4*5 = 20, 2*0 = 0.

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

Cf. A199978 (nonprime numbers whose multiplicative digital root is 0).

mdr:= proc(n) local L,r;
  r:= n;
  while r >= 10 do
    r:= convert(convert(r,base,10),`*`)
  od;
  r
end proc:
select(mdr=0, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 28 2021

digRoot[n_] := Module[{k = n}, While[k > 9, k = Times @@ IntegerDigits[k]]; k]; Select[Range[1200], PrimeQ[#] && digRoot[#] == 0 &] (* T. D. Noe, Nov 23 2011 *)

A199978 Nonprime numbers whose multiplicative digital root is 0.

Original entry on oeis.org

10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 102, 104, 105, 106, 108, 110, 120, 125, 130, 140, 145, 150, 152, 154, 155, 156, 158, 159, 160, 165, 169, 170, 178, 180, 185, 187, 190, 195, 196, 200, 201, 202

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A199977 with respect to A034048.

			Number 58 is in sequence because 5*8 = 40, 4*0 = 0.

Cf. A199977 (primes whose multiplicative digital root is 0).

digRoot[n_] := Module[{k = n}, While[k > 9, k = Times @@ IntegerDigits[k]]; k]; Select[Range[300], ! PrimeQ[#] && digRoot[#] == 0 &] (* T. D. Noe, Nov 23 2011 *)

A361978 Complement of A361337.

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, 116

Offset: 1

Michael S. Branicky, Apr 02 2023

More than the usual number of terms are shown to distinguish the sequence from A034048.

Appears to be finite with 219 members, the largest being 3111.

Michael S. Branicky, Table of n, a(n) for n = 1..219

Cf. A034048, A011540, A361337, A277061.

Subsequence of A052382.

A361978=select( {is_A361978(n)=vecmin(digits(n))&& !for(p=1, logint(n, 10), is_A361978(vecprod(divrem(n, 10^p)))|| return)}, [1..10^5]) \\ Conjecturedly the full list: no terms between 3112 and 10^5. - M. F. Hasler, Apr 05 2023

def ok(n):
    if n < 10: return n != 0
    s = str(n)
    if "0" in s: return False
    return all(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s)))
print([k for k in range(117) if ok(k)]) # Michael S. Branicky, Apr 02 2023

cp's OEIS Frontend

A361337 Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).

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A263470 Total number of positive integers < 10^n with multiplicative digital root value 0.

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

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Mathematica

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A277061 Numbers with multiplicative digital root > 0.

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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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A199977 Primes whose multiplicative digital root is 0.

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Maple

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A199978 Nonprime numbers whose multiplicative digital root is 0.

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