A003001 -id:A003001 - cp's OEIS frontend

A117884 Numbers at which the average multiplicative persistence of the interval (0,n) sets a new record.

0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96

Offset: 0

Sergio Pimentel, May 02 2006

nonn

This sequence seems to be finite. a(525)=9999 seems to be the last term, at a record of 2.282. No higher value found up to 100,000 Is the average multiplicative persistence of (0,Inf) = 1 ???

			E.g."39 belongs to the sequence because the average multiplicative persistence of the interval (0,39)= 1.05 which sets a new record. 40 is not because the average multiplicative persistence of the interval (0,40) = 1.048 which is not a record value"

Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A031346.

A121106 Trajectory of 6788 under "x -> product of digits of x" map.

Original entry on oeis.org

6788, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 6 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

NestList[Times@@IntegerDigits[#]&,6788,6] (* Harvey P. Dale, Nov 19 2021 *)

Edited by Charles R Greathouse IV, Aug 06 2010

A121107 Trajectory of 68889 under "x -> product of digits of x" map.

Original entry on oeis.org

68889, 27648, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 7 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

Edited by Charles R Greathouse IV, Aug 06 2010

A121108 Trajectory of 2677889 under "x -> product of digits of x" map.

Original entry on oeis.org

2677889, 338688, 27648, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 8 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

Edited by Charles R Greathouse IV, Aug 06 2010

A121109 Trajectory of 26888999 under "x -> product of digits of x" map.

Original entry on oeis.org

26888999, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 9 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

NestList[Times@@IntegerDigits[#]&,26888999,9] (* Harvey P. Dale, Aug 16 2015 *)

Edited by Charles R Greathouse IV, Aug 06 2010

A239477 Smallest number with additive and multiplicative persistence equal to n.

Original entry on oeis.org

0, 10, 28, 289, 2488888888888888999999999

Offset: 0

Giovanni Resta, Mar 20 2014

The corresponding smallest primes are 2, 11, 29, 487 and 2488888888888898999989999.

			a(3) = 289 because 289 is the smallest number with additive persistence 3, 289 -> 19 -> 10 -> 1 and multiplicative persistence 3, 289 -> 144 -> 16  -> 6.

Cf. A031346, A003001, A031286, A006050, A239427.

A239486 Smallest palindrome which has additive and multiplicative persistence n.

Original entry on oeis.org

0, 11, 99, 595, 467778888888979888888877764

Offset: 0

Giovanni Resta, Mar 20 2014

The corresponding sequence made of palindromic primes begins with 2, 11, 12421, 757 and 746788887898878898788887647.

			a(595) since 595 is the smallest palindrome with additive persistence 3 (595 -> 19 -> 10 -> 1) and multiplicative persistence 3 (595 -> 225 -> 20 -> 0).

Cf. A031346, A003001, A031286, A006050, A201991, A239427.

A245761 Numbers with a maximal multiplicative persistence of 1 in any base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 12

Offset: 1

Sergio Pimentel, Jul 31 2014

			12 is in the sequence since the persistence of 12 is at most 1 in any base. I.e. it takes at most one step to go from 12 to a single digit in any base, e.g., in base 2 we have 1100 -> 0. In base 5 we have 22 -> 4. In bases 12 and above the initial number is already a single digit.
This sequence is complete - there are no other terms. - _Alois P. Heinz_, Jul 31 2014

Cf. A003001, A031346, A064867, A064868, A046510, A245760.

A258584 Numbers n such that n = Sum_{j>=1} c(j) where c(0) = n, c(j) = floor(c(j-1)/10^k)*(c(j-1) mod 10^k) for j>0, and k is half the number of digits of n, rounded up if the number of digits of n is odd.

Original entry on oeis.org

86, 860, 1975, 2160, 3575, 19750, 21600, 35750, 43614, 51884, 65625, 479900, 868688, 967750, 1435575, 1548384, 1696875, 4799000, 8686880, 9677500, 28874200, 34095100, 38748800, 39214560, 47613625, 53415625, 148385715, 156293216, 288742000, 340951000, 387488000

Offset: 1

Pieter Post, Nov 06 2015

If n is an odd-digit decimal number, the first half is one digit smaller than the second half. For example, 43614 is in the sequence, because 43*614 = 26402, 26*402 = 10452, 10*452 = 4520, 4*520 = 2080, 2*80 = 160. Here the iteration stops because 160 has three digits, so the first half of the next multiplication is zero. 43614 = 26402 + 10452 + 4520 + 2080 + 160.
If n is an even-digit decimal number, the first half and the second half have the same length. For example, 868688 is in the sequence because 868*688 = 597184, 597*184 = 109848, 109*848 = 92432, 92*432 = 39744, 39*744 = 29016, 29*16 = 464, and here the iteration stops. 868688 = 597184 + 109848 + 92432 + 39744 + 29016 + 464.
If n is in the sequence and has an even number of digits, then 10*n is also in the sequence. - Jon E. Schoenfield, Nov 07 2015

			86 is in the sequence because 8*6 = 48, 4*8 = 32 and 3*2 = 6. And 86 = 48 + 32 + 6.

Cf. A003001, A014553, A135385.

fQ[n_] := Block[{i = Ceiling[IntegerLength[n]/2], g}, g[x_] := If[IntegerLength@ x <= i, x, Times @@ (FromDigits /@ {If[IntegerLength@ x - i == 0, Nothing, Take[IntegerDigits@ x, IntegerLength@ x - i]], Take[IntegerDigits@ x, -i]})]; Total@ Rest@ Most@ FixedPointList[g, n] == n]; Select[Range@ 500000, fQ] (* Michael De Vlieger, Nov 06 2015 *)

def pod(n, m):
    kk = 1
    while n > 0:
        kk= kk*(n%m)
        n =int(n//m)
    return kk
for b in range(0, 6):
    dd, bb=0, (b-1)//2+2
    j=10**bb
    for c in range (10*j, 100*j):
        d, a, ca=0, 0, pod(c, j)
        while ca>0:
            d, a=d+ca, a+1
            if ca

a(21)-a(31) from Jon E. Schoenfield, Nov 07 2015

Obsolete b-file deleted by N. J. A. Sloane, Jan 05 2019

A319507 Smallest number of multiplicative-additive divisors persistence n.

Original entry on oeis.org

1, 2, 36, 3489, 24778899, 566677899999, 47777778999999999999

Offset: 0

Pieter Post, Sep 21 2018

To compute the "multiplicative-additive divisors persistence" of a number, we proceed as follows. Form the product of the digits of the number (A007954) divided by the sum of the digits (A007953). Repeat this process until you reach 0 or 1. If we reach a non-integer, we write 0. The "multiplicative-additive divisors persistence" is the number of steps to reach 0 or 1.

For instance: the multiplicative-additive divisors persistence of 874 is 1, because 874 -> 8 * 7 * 4 / (8 + 7 + 4) = 224/19. This is not an integer, so the process stops after one step.

			The multiplicative additive divisors persistence of 24778899 is 4: 24778899 -> (2032128/54=) 37632 -> (756/21=) 36 -> (18/9=) 2 -> (2/2=) 1.

Cf. A038367, A126789, A003001, A006050, A007953, A007954, A031346.

Offset set to 0. - R. J. Mathar, Jun 30 2020

cp's OEIS Frontend

A117884 Numbers at which the average multiplicative persistence of the interval (0,n) sets a new record.

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A121106 Trajectory of 6788 under "x -> product of digits of x" map.

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A121107 Trajectory of 68889 under "x -> product of digits of x" map.

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A121108 Trajectory of 2677889 under "x -> product of digits of x" map.

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A121109 Trajectory of 26888999 under "x -> product of digits of x" map.

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A239477 Smallest number with additive and multiplicative persistence equal to n.

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A239486 Smallest palindrome which has additive and multiplicative persistence n.

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A245761 Numbers with a maximal multiplicative persistence of 1 in any base.

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A258584 Numbers n such that n = Sum_{j>=1} c(j) where c(0) = n, c(j) = floor(c(j-1)/10^k)*(c(j-1) mod 10^k) for j>0, and k is half the number of digits of n, rounded up if the number of digits of n is odd.

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A319507 Smallest number of multiplicative-additive divisors persistence n.

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