A087472 -id:A087472 - cp's OEIS frontend

A080464 Product of the two numbers formed by alternate digits of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0

Offset: 10

Amarnath Murthy, Mar 02 2003

			a(132546) = 124 * 356 = 44144.

M. F. Hasler, Table of n, a(n) for n = 10..10000

Cf. A007954, A080463, A080465, A087471, A087472, A087473, A087474.

nad[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Take[idn,{1,-1,2}]] FromDigits[ Take[idn,{2,-1,2}]]]; Array[nad,120,10] (* Harvey P. Dale, Aug 07 2019 *)

A080464(n,d=digits(n))={n=d*matrix(#d,2,z,s,if((z-s)%2,10^((#d-z)\2)));n[1]*n[2]}

a(n) < n for all n. - M. F. Hasler, Jan 10 2016

More terms from Ray Chandler, Oct 11 2003

A087471 Final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, 0, 8, 0, 8, 6, 8, 6, 0, 6, 0, 8, 4, 0, 9, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy and Paul D. Hanna, Sep 11 2003

A087472(n) gives the number of iterations required for Murthy's function, f(n), to reach a single digit. A087473(n) gives the smallest number that requires n iterations of Murthy's function to reach a single digit. The n-th row of triangle A087474 gives the n successive iterations of Murthy's function on A087473(n).

Apart from the undefined a(0), the sequence differs from A031347 first at n=121. [From R. J. Mathar, Sep 11 2008]

			a(1234) = a(13*24) = a(312) = a(32*1) = a(32) = a(3*2) = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A087472, A087473, A087474.

Table[NestWhile[With[{idn=IntegerDigits[#]},FromDigits[Take[idn,{1,-1,2}]] FromDigits[Take[idn,{2,-1,2}]]]&,n,#>9&],{n,110}] (* Harvey P. Dale, Dec 05 2014 *)

a(n) = a(f(n)), where f(n) is Murthy's function: f(1234)=13*24=312, f(12345)=135*24=3240, f(123456)=135*246=33210.

A087473 Smallest positive number that requires n iterations of f(k) to reach a single digit, where f(k) is the product of the two numbers formed from the alternating digits of k.

Original entry on oeis.org

1, 10, 25, 39, 77, 171, 199, 577, 887, 1592, 2682, 3988, 6913, 18747, 39577, 58439, 99428, 173442, 267343, 299137, 574182, 685812, 880543, 1635812, 1974447, 2771717, 18871813, 45797337, 49899368, 58935768, 158504329, 265956179, 566800111, 896125563

Offset: 0

Amarnath Murthy and Paul D. Hanna, Sep 11 2003

			a(4)= 77 since 77 is the smallest number that requires 4 iterations to reach a single digit: f(77)=7*7=49, f(49)=4*9=36, f(36)=3*6=18, f(18)=1*8=8.

Giovanni Resta, Table of n, a(n) for n = 0..40

Cf. A087471, A087472, A087474.

f[n_] := Block[{d = IntegerDigits@ n}, If[OddQ@ Length@ d, PrependTo[d, 0]]; Times @@ FromDigits /@ Transpose@ Partition[d, 2]]; a[n_] := Block[ {c=-1, m}, t=0; While[c != n, t++; m=t; c=0; While[m > 9, c++; m = f@ m]]; t]; a /@ Range[0, 12] (* Giovanni Resta, Aug 01 2018 *)

More terms from Ray Chandler, Sep 19 2003

a(30)-a(33) from Giovanni Resta, Aug 01 2018

A087474 Triangle T(n,k), read by rows, in which the n-th row gives the n successive iterations of f(m) on A087473(n), where f(m) is product of the two numbers formed from the alternating digits of m.

Original entry on oeis.org

1, 10, 0, 25, 10, 0, 39, 27, 14, 4, 77, 49, 36, 18, 8, 171, 77, 49, 36, 18, 8, 199, 171, 77, 49, 36, 18, 8, 577, 399, 351, 155, 75, 35, 15, 5, 887, 696, 594, 486, 368, 228, 56, 30, 0, 1592, 988, 784, 592, 468, 288, 224, 48, 32, 6, 2682, 1736, 988, 784, 592, 468, 288, 224

Offset: 0

Amarnath Murthy and Paul D. Hanna, Sep 11 2003

			The 4th row is {77,49,36,18,8} since f(77)=49, f(49)=36, f(36)=18, f(18)=8.
{1},
{10,0},
{25,10,0},
{39,27,14,4},
{77,49,36,18,8},
{171,77,49,36,18,8},
{199,171,77,49,36,18,8},
{577,399,351,155,75,35,15,5},...

Cf. A087471, A087472, A087473.

T(0, 0)=1, T(n, 0)=A087473(n), T(n, k+1) = f(T(n, k)), where f(m) is the product of the two numbers formed by the alternating digits of m.

A335808 Nonzero multiplicative persistence in base 10: number of iterations of "multiply nonzero digits in base 10" needed to reach a number < 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 1, 1, 2, 2, 3, 3, 2, 4, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 2, 3, 3, 3, 3, 3, 3, 2

Offset: 0

Lucas Colucci, Jun 24 2020

Coincides with A031346 up to n=204.

Differs from A087472 first at n=110. - R. J. Mathar, Aug 10 2020

R. K. Guy, Unsolved Problems in Number Theory, E16, pages 262-263.

Lucas Colucci, Table of n, a(n) for n = 0..9999
Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.

Cf. A031346, A051801.

A335808 := proc(n)
    option remember;
    if n < 10 then
        0 ;
    elif n < 20 then
        1 ;
    else
        A051801(n) ;
        1+procname(%) ;
    end if;
end proc: # R. J. Mathar, Aug 10 2020

Array[-1 + Length[NestWhileList[Times @@ DeleteCases[IntegerDigits[#], ?(# == 0 &)] &, #, # >= 10 &]] &, 105, 0] (* _Michael De Vlieger, Jun 24 2020 *)

a(n) = for (k=0, oo, if (n<10, return (k), n=vecprod(select(sign, digits(n))))) \\ Rémy Sigrist, Jul 18 2020

cp's OEIS Frontend

A080464 Product of the two numbers formed by alternate digits of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A087471 Final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A087473 Smallest positive number that requires n iterations of f(k) to reach a single digit, where f(k) is the product of the two numbers formed from the alternating digits of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A087474 Triangle T(n,k), read by rows, in which the n-th row gives the n successive iterations of f(m) on A087473(n), where f(m) is product of the two numbers formed from the alternating digits of m.

Views

Author

Keywords

Examples

Crossrefs

Formula

A335808 Nonzero multiplicative persistence in base 10: number of iterations of "multiply nonzero digits in base 10" needed to reach a number < 10.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI