A007954 -id:A007954 - cp's OEIS frontend

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).

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

A089898 Product of (digits of n each incremented by 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 8, 16, 24, 32, 40, 48, 56, 64

Offset: 0

Marc LeBrun, Nov 13 2003

Sum of products of all subsets of digits of n (with the empty subset contributing 1).
Number of nonnegative values k such that the lunar sum of k and n is n.
First 100 values are 10 X 10 multiplication table, read by rows/columns.

			a(12)=6 since (1+1)*(2+1)=2*3=6 and since (1*2)+(1)+(2)+(1)=2+1+2+1=6 and since the lunar sum of 12 with any of the six values {0,1,2,10,11,12} is 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Cf. A007954, A087061, A001316, A006047.

a089898 n = if n < 10 then n + 1 else (d + 1) * a089898 n'
            where (n', d) = divMod n 10
-- Reinhard Zumkeller, Jul 06 2014

seq(convert(map(`+`,convert(n,base,10),1),`*`), n = 0 .. 1000); # Robert Israel, Nov 17 2014

a089898[n_Integer] :=
Prepend[Array[Times @@ (IntegerDigits[#] + 1) &, n], 1]; a089898[77] (* Michael De Vlieger, Dec 22 2014 *)

a(n) = my(d=digits(n)); prod(i=1, #d, d[i]+1); \\ Michel Marcus, Apr 06 2014

a(n) = vecprod(apply(x->x+1, digits(n))); \\ Michel Marcus, Feb 01 2023

a(n) = a(floor(n/10))*(1+(n mod 10)). - Robert Israel, Nov 17 2014

G.f. g(x) satisfies g(x) = (10*x^11 - 11*x^10 + 1)*g(x^10)/(x-1)^2. - Robert Israel, Nov 17 2014

A246558 Product of the digits of the n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 3, 2, 12, 25, 72, 16, 18, 147, 0, 504, 315, 320, 32, 1260, 0, 49, 3360, 3456, 0, 162, 1728, 168, 720, 0, 7776, 0, 33600, 0, 30240, 0, 15680, 0, 311040, 0, 0, 326592, 435456, 0, 0, 0, 0, 0, 0, 0, 0, 0, 102060, 3951360, 24883200, 1411200

Offset: 0

Indrani Das, Nov 12 2014

a(n) > 0 iff n in A076564.

Probably, the last nonzero term is a(184). - Giovanni Resta, Jul 14 2015

			Fibonacci(7) = 13, thus a(7) = 1*3 = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000045, A004090, A076564.
Cf. A007954, A259773.
Cf. A261598.

a246558 = a007954 . a000045 -- Reinhard Zumkeller, Nov 17 2014

[0] cat [&*Intseq(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jan 04 2020

Array[Times@@IntegerDigits@Fibonacci[#]&, 100, 0] (* Vincenzo Librandi, Jan 04 2020 *)

a(n) = if (n, vecprod(digits(fibonacci(n))), 0); \\ Michel Marcus, Feb 11 2025

a(n) = A007954(A000045(n)). - Reinhard Zumkeller, Nov 17 2014

A249334 Numbers for which the digital sum contains the same distinct digits as the digital product.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 99, 123, 132, 213, 231, 312, 321, 1124, 1137, 1142, 1173, 1214, 1241, 1317, 1371, 1412, 1421, 1713, 1731, 2114, 2141, 2411, 3117, 3171, 3344, 3434, 3443, 3711, 4112, 4121, 4211, 4334, 4343, 4433, 7113, 7131, 7311, 11125, 11133

Offset: 1

Jaroslav Krizek, Oct 25 2014

Numbers k such that A007953(k) contains the same distinct digits as A007954(k). (But either of the two may contain some digit(s) more than once.)
Supersequence of A034710 (positive numbers for which the sum of digits is equal to the product of digits).
Union of A034710 and A249335.
The sequence is infinite since, e.g., A002275(n) = (10^n-1)/9 is in the sequence for all n = A002275(k), k>=0; and more generally N(k,d) = A002275(n)-1+d with n = (A002275(k)-1)*d+1, k>0 and 0M. F. Hasler, Oct 29 2014

			1137 is a term because 1+1+3+7 = 12 and 1*1*3*7 = 21.
3344 is a term because 3+3+4+4=14 has the same (distinct) digits as 3*3*4*4=144.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..201 from Jaroslav Krizek).

Cf. A034710, A007953, A007954, A249335.

Cf. A061672.

[0] cat [n: n in [1..10^6] | Set(Intseq(&*Intseq(n))) eq Set(Intseq(&+Intseq(n)))];

Select[Range[0,12000],Union[IntegerDigits[Total[IntegerDigits[#]]]]==Union[IntegerDigits[Times@@IntegerDigits[#]]]&] (* Harvey P. Dale, Aug 17 2025 *)

is_A249334(n)=Set(digits(sumdigits(n)))==Set(digits(prod(i=1,#n=digits(n),n[i]))) \\ M. F. Hasler, Oct 29 2014

A336876 a(n) is the least m such that A336826(n) = m*p(m) (where p(m) is the product of decimal digits of m).

Original entry on oeis.org

0, 1, 2, 3, 11, 4, 12, 5, 6, 13, 21, 7, 14, 8, 15, 9, 22, 31, 16, 111, 17, 23, 18, 41, 19, 24, 112, 121, 25, 51, 33, 26, 42, 113, 61, 27, 131, 34, 211, 28, 114, 122, 71, 43, 52, 29, 35, 141, 115, 36, 116, 44, 123, 62, 151, 37, 132, 53, 91, 212, 221, 45, 38

Offset: 1

Rémy Sigrist, Aug 06 2020

Some terms of A336826 have several representations as the product of a number and of its decimal digits; for example 549504 has four such representations: 1696 * 1 * 6 * 9 * 6, 2862 * 2 * 8 * 6 * 2, 3392 * 3 * 3 * 9 * 2 and 3816 * 3 * 8 * 1 * 6.

			For n = 26:
- A336826(26) = 192,
- the divisors d of 192, alongside d*p(d), are:
  d    d*p(d)
  ---  ------
    1       1
    2       4
    3       9
    4      16
    6      36
    8      64
   12      24
   16      96
   24     192
   32     192
   48    1536
   64    1536
   96    5184
  192    3456
- so a(26) = min(24, 32) = 24.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A336876

Cf. A007954, A098736, A336826, A336879.

C
```
See Links section.
```

A098736(a(n)) = A336826(n).

A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

Original entry on oeis.org

338688, 826686, 2239488, 3188646, 6613488, 14224896, 3416267673274176, 6499837226778624

Offset: 1

Daniel Mondot, Jan 15 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 8.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
These are the numbers k in A002473 such that A031346(k) = 7,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 7 steps.
Postulated to be finite and complete.
a(9), if it exists, is > 10^20000, and likely > 10^119000.

			338688 is in this sequence because:
- 338688 goes to a single digit in 7 steps: p(338688) = 27648, p(27648) = 2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(4478976) = p(13477889) = 338688, etc.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).

Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350187 (numbers with mp 1 to 6 and 8 to 10 that are themselves 7-smooth numbers).

mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&]  (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

A050626 Product of digits of n is a nonzero square.

Original entry on oeis.org

1, 4, 9, 11, 14, 19, 22, 28, 33, 41, 44, 49, 55, 66, 77, 82, 88, 91, 94, 99, 111, 114, 119, 122, 128, 133, 141, 144, 149, 155, 166, 177, 182, 188, 191, 194, 199, 212, 218, 221, 224, 229, 236, 242, 248, 263, 281, 284, 289, 292, 298, 313, 326, 331, 334, 339, 343

Offset: 1

Patrick De Geest, Jun 15 1999

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

Cf. A007954, A028845, A028839.

[ n: n in [1..350] | s gt 0 and IsSquare(s) where s is &*Intseq(n,10) ];  // Bruno Berselli, May 26 2011

Select[Range[500],DigitCount[#,10,0]==0&&IntegerQ[Sqrt[Times@@ IntegerDigits[ #]]]&] (* Harvey P. Dale, Jun 09 2020 *)

is(n)=my(v=digits(n),pr=prod(i=1,#v,v[i]));pr&&issquare(pr) \\ Charles R Greathouse IV, May 19 2013

A053668 Product of digits of n^3.

Original entry on oeis.org

0, 1, 8, 14, 24, 10, 12, 36, 10, 126, 0, 9, 112, 126, 224, 315, 0, 108, 240, 2160, 0, 108, 0, 84, 192, 300, 1470, 1296, 180, 1728, 0, 1134, 2016, 2835, 0, 2240, 4320, 0, 2240, 1215, 0, 864, 0, 0, 1280, 90, 3402, 0, 0, 1512, 0, 180, 0, 12544, 3360, 3780, 1260, 1080, 90, 0

Offset: 0

Enoch Haga, Feb 17 2000

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A007954.

seq(convert(convert(n^3,base,10),`*`), n=0..100); # Robert Israel, Feb 03 2019

Times@@IntegerDigits[#]&/@(Range[0,60]^3) (* Harvey P. Dale, Jul 15 2019 *)

a(n) = my(d = digits(n^3)); prod(i=1, #d, d[i]); \\ Michel Marcus, Jun 06 2014

a(n) = A007954(n^3). - Michel Marcus, Jun 06 2014

A056770 Smallest number that is n times the product of its digits or 0 if impossible.

Original entry on oeis.org

1, 36, 15, 384, 175, 12, 735, 128, 135, 0, 11, 1296, 624, 224, 0, 0, 816, 216, 1197, 0, 315, 132, 115, 0, 0, 0, 2916, 1176, 3915, 0, 93744, 0, 51975, 78962688, 0, 82944, 1184, 0, 0, 0, 31488, 0, 0, 77616, 77175, 4416, 0, 12288, 1715, 0, 612

Offset: 1

Robert G. Wilson v, Aug 16 2000

			a(4) = 384 because 4*(product of digits of 384) = 4*96 = 384, and no number smaller than 384 has this property.

David W. Wilson, Table of n, a(n) for n = 1..1000

Cf. A007954, A007602, A003634.

Do[k = n; If[Mod[n, 10] == 0, Print[0]; Continue[]]; While[Apply[Times, RealDigits[k][[1]]]*n != k, k += n]; Print[k], {n, 1, 14}]

from itertools import count, combinations_with_replacement
from math import prod
def A056770(n):
    if not n%10: return 0
    for l in count(1):
        if 9**l*n < 10**(l-1): return 0
        c = 10**l
        for d in combinations_with_replacement(range(1,10),l):
            if sorted(str(a:=prod(d)*n)) == list(str(e) for e in d):
                c = min(c,a)
        if c < 10**l:
            return c # Chai Wah Wu, May 09 2023

a(15) onwards from David W. Wilson, Jan 20 2016

A062997 Numbers whose sum of digits is strictly greater than its product of digits.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 30, 31, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 130, 131, 140, 141, 150, 151, 160, 161, 170

Offset: 1

Amarnath Murthy, Jun 27 2001

Every multiple of 10 is a term.

			118 is a term as 1 + 1 + 8 = 10, 10 > 8 and 8 = 1 * 1 * 8.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.

Cf. A007953, A007954, A034710, A062329, A062996, A062998, A062999.

Select[Range[170], (Plus @@ IntegerDigits[ # ]) > (Times @@ IntegerDigits[ # ]) &] (* Alonso del Arte, May 16 2005 *)

isok(k)={my(d=digits(k)); vecsum(d) > vecprod(d)} \\ Harry J. Smith, Aug 15 2009

Extended by Larry Reeves (larryr(AT)acm.org) and Henry Bottomley, Jun 29 2001

cp's OEIS Frontend

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).

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A089898 Product of (digits of n each incremented by 1).

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A246558 Product of the digits of the n-th Fibonacci number.

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Magma

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A249334 Numbers for which the digital sum contains the same distinct digits as the digital product.

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Magma

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A336876 a(n) is the least m such that A336826(n) = m*p(m) (where p(m) is the product of decimal digits of m).

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C

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A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

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Mathematica

A050626 Product of digits of n is a nonzero square.

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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).