A035930 -id:A035930 - cp's OEIS frontend

A007954 Product of decimal digits of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

R. Muller

Moebius transform of A093811(n). a(n) = A093811(n) * A008683(n), where operation * denotes Dirichlet convolution, namely b(n) * c(n) = Sum_{d|n} b(d) * c(n/d). Simultaneously holds Dirichlet multiplication: a(n) * A000012(n) = A093811(n). - Jaroslav Krizek, Mar 22 2009
Apart from the 0's, all terms are in A002473. Further, for all m in A002473 there is some n such that a(n) = m, see A096867. - Charles R Greathouse IV, Sep 29 2013
a(n) = 0 asymptotically almost surely, namely for all n except for the set of numbers without digit '0'; this set is of density zero, since it is less and less probable to have no '0' as the number of digits of n grows. (See also A054054.) - M. F. Hasler, Oct 11 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Rigoberto Flórez, Robinson A. Higuita and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Florentin Smarandache, Only Problems, Not Solutions!.
Index entries for Colombian or self numbers and related sequences

Cf. A031347 (different from A035930), A007953, A007602, A010888, A093811, A008683, A000012, A061076 (partial sums), A230099.

Cf. A051802 (ignoring zeros).

a007954 n | n < 10 = n
          | otherwise = m * a007954 n' where (n', m) = divMod n 10
-- Reinhard Zumkeller, Oct 26 2012, Mar 14 2011

[0] cat [&*Intseq(n): n in [1..110]]; // Vincenzo Librandi, Jan 03 2020

A007954 := proc(n::integer)
    if n = 0 then
        0;
    else
        mul( d,d=convert(n,base,10)) ;
    end if;
end proc: # R. J. Mathar, Oct 02 2019

Array[Times @@ IntegerDigits@ # &, 108, 0] (* Robert G. Wilson v, Mar 15 2011 *)

A007954(n)= { local(resul = n % 10); n \= 10; while( n > 0, resul *= n %10; n \= 10; ); return(resul); } \\ R. J. Mathar, May 23 2006, edited by M. F. Hasler, Apr 23 2015

A007954(n)=prod(i=1,#n=Vecsmall(Str(n)),n[i]-48) \\ (...eval(Vec(...)),n[i]) is about 50% slower; (...digits(n)...) about 6% slower. \\ M. F. Hasler, Dec 06 2009

a(n)=if(n,factorback(digits(n)),0) \\ Charles R Greathouse IV, Apr 14 2020

from math import prod
def a(n): return prod(map(int, str(n)))
print([a(n) for n in range(108)]) # Michael S. Branicky, Jan 16 2022

(0 to 99).map(.toString.toCharArray.map( - 48).scanRight(1)( * ).head) // Alonso del Arte, Apr 14 2020

A000035(a(A014261(n))) = 1. - Reinhard Zumkeller, Nov 30 2007
a(n) = abs(A187844(n)). - Reinhard Zumkeller, Mar 14 2011
a(n) > 0 if and only if A054054(n) > 0. a(n) = d in {1, ..., 9} if n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. The statement holds with "if and only if" for d in {1, 2, 3, 5, 7}. For d = 4, 6, 8 or 9, one has a(n) = d if n = (10^k - 1)/9 + (a - 1)*10^m + (b - 1)*10^p with integers k > m > p >= 0 and a, b > 0 such that d = a*b. - M. F. Hasler, Oct 11 2015
From Robert Israel, May 17 2016: (Start)
G.f.: Sum_{n >= 0} Product_{j = 0..n} Sum_{k = 1..9} k*x^(k*10^j).
G.f. satisfies A(x) = (x + 2*x^2 + ... + 9*x^9)*(1 + A(x^10)). (End)
a(n) <= 9^(1 + log_10(n/9)). - Lucas A. Brown, Jun 22 2023

Error in term 25 corrected, Nov 15 1995

A257850 a(n) = floor(n/10) * (n mod 10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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

Offset: 0

M. F. Hasler, May 10 2015

Equivalently, write n in base 10, multiply the last digit by the number with its last digit removed.
See A142150(n-1) for the base 2 analog and A257843 - A257849 for the base 3 - base 9 variants.
The first 100 terms coincide with those of A035930 (maximal product of any two numbers whose concatenation is n), A171765 (product of digits of n, or 0 for n<10), A257297 ((initial digit of n)*(n with initial digit removed)), but the sequence is of course different from each of these three.
The terms a(10) - a(100) also coincide with those of A007954 (product of decimal digits of n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).

Cf. A142150 (the base 2 analog), A115273, A257844 - A257849.

Cf. also A007954, A035930, A171765, A257297.

[Floor(n/10)*(n mod 10): n in [0..100]]; // Vincenzo Librandi, May 11 2015

Table[Floor[n/10] Mod[n, 10], {n, 100}] (* Vincenzo Librandi, May 11 2015 *)

PARI
```
a(n,b=10)=(n=divrem(n,b))[1]*n[2]
```

def A257850(n): return n//10*(n%10) # M. F. Hasler, Sep 01 2021

a(n) = 2*a(n-10)-a(n-20). - Colin Barker, May 11 2015

G.f.: x^11*(9*x^8+8*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+2*x+1) / ((x-1)^2*(x+1)^2*(x^4-x^3+x^2-x+1)^2*(x^4+x^3+x^2+x+1)^2). - Colin Barker, May 11 2015

A088117 Let the decimal expansion of n be abcd...; then a(n) = (abcd... + bacd... + cabd... + dabc... + ...) + (abcd... + bcad... + cdab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) (deleted digit string).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 11

Offset: 0

Amarnath Murthy, Sep 25 2003

Each substring is used in only product.

a(n) = 0 for 0 <= n <= 10.

			a(1234) = (1*234 + 2*134 + 3*124 + 4*123) + (12*34 + 23*14) = 2096.
a(12345) = (1*2345 + 2*1345 + 3*1245 + 4*1235 + 5*1234) + (12*345 + 15*234 + 23*145 + 34*125 + 45*123) = 40650.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A007954, A088116.

Different from A035930, A171765, A257850.

a:= n-> (s-> add(add(parse(s[i..j])*parse(cat(s[1..i-1],
    s[j+1..length(s)])), i=1..j), j=1..length(s)-1))(""||n):
seq(a(n), n=0..120);  # Alois P. Heinz, May 22 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

a(0)=0 inserted and examples corrected by Alois P. Heinz, May 22 2021

A257297 a(n) = (initial digit of n) * (n with initial digit removed).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 1, 2, 3

Offset: 0

M. F. Hasler, May 10 2015

The initial 100 terms match those of A035930 (maximal product of any two numbers whose concatenation is n) and also those of A171765 (product of digits of n, or 0 for n<10), and except for initial terms, also A007954 (product of decimal digits of n) and A115300 (greatest digit of n * least digit of n).
Iterations of this map always end in 0, since a(n) < n. Sequence A257299 lists numbers for which these iterations reach 0 in exactly 9 steps, with the additional constraint of having each time a different initial digit.
If "initial" is replaced by "last" in the definition (A257850), then we get the same values up to a(100), but (10, 20, 30, ...) for n = 101, 102, 103, ..., again different from each of the 4 other sequences mentioned in the first comment. - M. F. Hasler, Sep 01 2021

			For n<10, a(n) = n*0 = 0, since removing the initial and only digit leaves nothing, i.e., zero (by convention).
a(10) = 1*0 = 0, a(12) = 1*2 = 2, ..., a(20) = 2*0 = 0, a(21) = 2*1 = 2, a(22) = 2*2 = 4, ...
a(99) = 9*9 = 81, a(100) = 1*00 = 0, a(101) = 1*01 = 1, ..., a(123) = 1*23, ...

Cf. A257850, A007954, A035930, A080464, A115300, A169669, A111707, A088117, A187844.

a:= n-> `if`(n<10, 0, (s-> parse(s[1])*parse(s[2..-1]))(""||n)):
seq(a(n), n=0..120);  # Alois P. Heinz, Feb 12 2024

Table[Times@@FromDigits/@TakeDrop[IntegerDigits@n,1],{n,0,103}] (* Giorgos Kalogeropoulos, Sep 03 2021 *)

apply( {A257297(n)=vecprod(divrem(n,10^logint(n+!n,10)))}, [0..111]) \\ Edited by M. F. Hasler, Sep 01 2021

def a(n): s = str(n); return 0 if len(s) < 2 else int(s[0])*int(s[1:])
print([a(n) for n in range(104)]) # Michael S. Branicky, Sep 01 2021

For 1 <= m <= 9 and n < 10^k, a(m*10^k + n) = m*n.

a(101..103) corrected by M. F. Hasler, Sep 01 2021

A035935 Smallest number that always takes at least n steps to reach 0 under "k -> any product of 2 numbers whose concatenation is k".

Original entry on oeis.org

0, 1, 11, 26, 39, 77, 666

Offset: 0

Erich Friedman

Next term is at least 3*10^7. 666 may be the last term. - Naohiro Nomoto, Apr 07 2001

Checked up to 1.55*10^10. No other numbers apart 666 with minimal path of length 6. The largest number found with minimal path equal to 5 is 711111. - Giovanni Resta, Oct 03 2012

Cf. A035930-A035934.

A035931 Number of steps to reach 0 under "k->max product of two numbers whose concatenation is k".

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 1, 2, 2, 3, 3, 2, 3, 4, 3, 4, 1, 2, 2, 3, 2, 3, 2, 4, 2, 3, 1, 2, 3, 3, 3, 2, 4, 3, 4, 3, 1, 2, 3, 3, 4, 4, 3, 5, 3, 4, 1, 2, 3, 3, 3, 2, 4, 3, 4, 4, 1, 2, 3, 4, 4, 3, 3, 4, 4, 3, 1

Offset: 0

Erich Friedman

			a(341)=5 since 341->123->36->18->8->0.

Scott R. Shannon, Table of n, a(n) for n = 0..10000.

Cf. A035930, A035932, A035933, A035934, A035935.

f[n_] := If[n<10, 0, With[{d = IntegerDigits[n]}, Table[FromDigits[Take[d, k]]*FromDigits[Drop[d, k]], {k, 1, Length[d]-1}] // Max]];
a[n_] := If[n == 0, 0, Length[FixedPointList[f, n]]-2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 03 2017 *)

A069816 a(n) = (sum of digits of n)^2 - (sum of digits^2 of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 0, 12, 24, 36, 48, 60, 72, 84, 96, 108, 0, 14, 28, 42, 56, 70, 84

Offset: 0

N. J. A. Sloane, May 03 2002

a(n) = 0 iff n is a single-digit number or is a power of ten times a single-digit number. - Robert G. Wilson v, May 04 2002

			n=14: sum of digits = 5, squaring gives 25; sum of squared digits of n = 1^2 + 4^2 = 17, so a(14) = 25 - 17 = 8.

Cf. A067552.

A007953 := proc(n) add(d, d=convert(n,base,10)) ; end proc: A003132:= proc(n) add(d^2, d=convert(n,base,10)) ; end proc: A118881 := proc(n) (A007953(n))^2 ; end proc: A069816 := proc(n) A118881(n)-A003132(n) ; end proc: seq(A069816(n),n=0..80) ; # R. J. Mathar, Jun 23 2009

f[n_] := Plus @@ IntegerDigits[n]^2 - Plus @@ (IntegerDigits[n]^2); Table[ f[n], {n, 0, 100}]

From R. J. Mathar, Jun 23 2009: (Start)
a(n) = 2*A035930(n), n <= 100.
a(n) = A118881(n) - A003132(n). (End)

Extended by Robert G. Wilson v, May 04 2002

A035934 Smallest number that can be made to take n steps to reach 0 under "k -> any product of 2 numbers whose concatenation is k".

Original entry on oeis.org

0, 1, 11, 26, 39, 77, 117, 139, 429, 529, 777, 1117, 1669, 2238, 2993, 3697, 4779, 5319, 5919, 10998, 11794, 14989, 21179, 26869, 27797, 36177, 38993, 62958, 74297, 85797, 95339, 113319, 125919, 139919, 199683, 201799, 247817, 333329, 360497, 419926

Offset: 0

Erich Friedman

			a(6) = 117 since 117 -> 77 -> 49 -> 36 -> 18 -> 8 -> 0.

Cf. A035930-A035935.

tbl=Table[1, {10}]; Do[tbl=Append[tbl, b=IntegerDigits[k]; If[(First[b]==0||Last[b]==0), 1, Max[Part[tbl, Table[FromDigits[Take[b, i]]*FromDigits[Take[b, i-Length[b]]], {i, 1, Length[b]-1}]]]+1]], {k, 11, 170000}]; tbl; m=Max[tbl]; Prepend[Flatten[Table[Position[tbl, i, 1, 1], {i, 1, m}]], 0]

More terms from Naohiro Nomoto, Apr 11 2001
More terms from Vit Planocka (planocka(AT)mistral.cz), Feb 01 2003
More terms from Martin Fuller, Jun 02 2006

A330633 The concatenation of the products of every pair of consecutive digits of n (with a(n) = 0 for 0 <= n <= 9).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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

Offset: 0

Scott R. Shannon, Dec 21 2019

If the decimal expansion of n is d_1 d_2 ... d_k then a(n) is the number formed by concatenating the decimal numbers d_1*d_2, d_2*d_3, ..., d_{k-1}*d_k.

Due to the fact that for two digit numbers the sequence is simply the multiplication of those two numbers, this sequence matches numerous others for the first 100 terms. See the sequences in the cross references. The terms begin to differ beyond n = 100.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A007954, A088117, A040115, A053392, A035930, A257850, A080464, A171765, A169669, A257297.

read("transforms") :
A330633 := proc(n)
    local dgs,L,i ;
    if n <=9 then
        0;
    else
        dgs := ListTools[Reverse](convert(n,base,10)) ;
        L := [] ;
        for i from 2 to nops(dgs) do
            L := [op(L), op(i-1,dgs)*op(i,dgs)] ;
        end do:
        digcatL(L) ;
    end if;
end proc: # R. J. Mathar, Jan 11 2020

Array[If[Or[# == 0, IntegerLength@ # == 1], 0, FromDigits[Join @@ IntegerDigits[Times @@ # & /@ Partition[IntegerDigits@ #, 2, 1]]]] &, 81, 0] (* Michael De Vlieger, Dec 23 2019 *)

a(n) = my(d=digits(n), s="0"); for (k=1, #d-1, s=concat(s, d[k]*d[k+1])); eval(s); \\ Michel Marcus, Apr 28 2020

a(10) = 0 as 1 * 0 = 0.
a(29) = 18 as 2 * 9 = 18.
a(100) = 0 as 1 * 0 = 0 and 0 = 0 = 0, and '00' is reduced to 0.
a(110) = 10 as 1 * 1 = 1 and 1 * 0 = 0. This is the first term that differs from A007954 and A171765, the multiplication of all digits of n.

A035932 Smallest number that takes n steps to reach 0 under "k->max product of 2 numbers whose concatenation is k".

Original entry on oeis.org

0, 1, 11, 26, 39, 77, 117, 139, 449, 529, 777, 1117, 2229, 2982, 4267, 4779, 5319, 5919, 8693, 12699, 14119, 17907, 27779, 47877, 80299, 103199, 135199, 274834, 293938, 312794, 606963, 653993, 773989, 1160892, 1296741, 1616696, 1986576

Offset: 0

Erich Friedman

Giovanni Resta, Table of n, a(n) for n = 0..100 (terms 0..50 from Scott R. Shannon)
Scott R. Shannon, The 50 steps to 0 for start value n = 74438327.

Cf. A035930, A035931, A035933, A035934, A035935.

More terms from Naohiro Nomoto, Apr 03 2001

cp's OEIS Frontend

A007954 Product of decimal digits of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Scala

Formula

Extensions

A257850 a(n) = floor(n/10) * (n mod 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A088117 Let the decimal expansion of n be abcd...; then a(n) = (a*bcd... + b*acd... + c*abd... + d*abc... + ...) + (ab*cd... + bc*ad... + cd*ab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) * (deleted digit string).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A257297 a(n) = (initial digit of n) * (n with initial digit removed).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A035935 Smallest number that always takes at least n steps to reach 0 under "k -> any product of 2 numbers whose concatenation is k".

Views

Author

Keywords

Comments

Crossrefs

A035931 Number of steps to reach 0 under "k->max product of two numbers whose concatenation is k".

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A069816 a(n) = (sum of digits of n)^2 - (sum of digits^2 of n).

Views

Author

Keywords

Comments

Examples

Crossrefs

A088117 Let the decimal expansion of n be abcd...; then a(n) = (abcd... + bacd... + cabd... + dabc... + ...) + (abcd... + bcad... + cdab... + ...) + ... . That is, a(n) = sum over all the digit strings of the product (number obtained by deleting a digit string) (deleted digit string).