A277061 -id:A277061 - cp's OEIS frontend

A299690 Numbers without digit 1 whose multiplicative digital root is not 0.

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 53, 57, 62, 63, 64, 66, 67, 68, 72, 73, 74, 75, 76, 77, 79, 82, 83, 84, 86, 88, 89, 92, 93, 94, 97, 98, 99, 222, 223, 224, 226, 227, 228, 229, 232

Offset: 1

J. Lowell, Feb 19 2018

Is this sequence infinite?
There are no members of this sequence with 45 to 2000 decimal digits. Perhaps the last term is a(614640917006263790) = 77333222222222222222222222222222222222222222. - Charles R Greathouse IV, Feb 26 2018
This sequence is finite. Proof: Let k be the smallest term with more than 2000 decimal digits. Then the product of decimal digits pk of k has fewer than 2001 decimal digits (otherwise k isn't the smallest term with more than 2000 decimal digits). This number pk has at least as many decimal digits as 2^2001 has, which are 603. But then it doesn't have a nonzero multiplicative digital root per the computations of Charles R Greathouse IV. QED. - David A. Corneth, Aug 23 2018

			5 times 4 = 20 and 2 times 0 = 0, so 54 is not in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A031347, A052383, A277061.

multDigRoot[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Select[Range[500], DigitCount[#, 10, 1] == 0 && multDigRoot[#] != 0 &] (* Alonso del Arte, Feb 19 2018, based on Robert G. Wilson v's program for A031347 *)

mdr(n)=while(n>9,n=factorback(digits(n)));n
do(n)=my(v=List());forvec(u=vector(n,i,[2,9]), if(mdr(factorback(u)), listput(v, fromdigits(u)))); Vec(v) \\ Gives n-digit elements
\\ Charles R Greathouse IV, Feb 19 2018

{ A052383 } intersect { A277061 }.

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

A318275 Numbers with digits in nondecreasing order and 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, 22, 23, 24, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 46, 47, 48, 49, 57, 66, 67, 68, 77, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126, 127, 128, 129, 133, 134

Offset: 1

David A. Corneth, Aug 23 2018

This sequence is a primitive sequence of A277061, it has digits in nondecreasing order. Terms in A277061 can be found by permuting digits of terms of this sequence.

Cf. A009994, A277061, A299690.

Select[Range@ 134, And[FixedPoint[Times @@ IntegerDigits@ # &, #] != 0, AllTrue[Differences@ IntegerDigits@ #, # >= 0 &]] &] (* Michael De Vlieger, Aug 25 2018 *)

A318273 Numbers with digits in nondecreasing order such that additive and multiplicative digital roots coincide.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 123, 137, 139, 168, 179, 188, 233, 267, 299, 346, 389, 899, 1124, 1157, 1347, 1355, 1469, 1779, 1788, 2236, 2346, 2348, 2778, 3335, 3779, 11126, 11133, 11148, 11177, 11222, 11238, 11279, 11339, 11369, 11579, 11666, 11677, 11679, 11699

Offset: 1

David A. Corneth, Aug 23 2018

A299690 can be used to find terms for this sequence below some bound by prepending ones to terms while staying below that bound so the additive and multiplicative root that term matches.

For example, 27 is in A299690 and has multiplicative root 4. 27 has the additive root 9. Prepending 4 ones gives the number 111127 which has multiplicative root 4, the same as 27 has, but it also has an additive root of 4. Furthermore, the digits are in nondecreasing order hence is in this sequence.

Cf. A009994, A010888, A031347, A064702, A277061, A299690.

is(n) = my(cn=n); d=digits(n); if(d!=vecsort(d), return(0)); while(cn>9, d=digits(cn); cn=prod(i=1, #d, d[i])); cn-1 == (n-1)%9 || n == 0

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A299690 Numbers without digit 1 whose multiplicative digital root is not 0.

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A361978 Complement of A361337.

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A318275 Numbers with digits in nondecreasing order and with multiplicative digital root > 0.

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A318273 Numbers with digits in nondecreasing order such that additive and multiplicative digital roots coincide.

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