A007954 -id:A007954 - cp's OEIS frontend

A046703 Multiplicative primes: product of digits is a prime.

2, 3, 5, 7, 13, 17, 31, 71, 113, 131, 151, 211, 311, 1117, 1151, 1171, 1511, 2111, 11113, 11117, 11131, 11171, 11311, 111121, 111211, 112111, 113111, 131111, 311111, 511111, 1111151, 1111211, 1111711, 1117111, 1171111, 11111117, 11111131, 11111171, 11111311, 11113111, 11131111

Offset: 1

Felice Russo

Primes with one prime digit and all other digits are 1. The linked table includes probable primes. - Jens Kruse Andersen, Jul 21 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000

Cf. A117835 ("noncomposite" variant), A007954 (product of digits), A028842 (product of digits is prime).

Select[Prime[Range[740000]],PrimeQ[Times@@IntegerDigits[#]]&] (* Harvey P. Dale, Oct 02 2011 *)
Select[FromDigits/@Flatten[Table[Permutations[PadRight[{p},n,1]],{n,8},{p,{2,3,5,7}}],2],PrimeQ]//Union (* Harvey P. Dale, Nov 21 2019 *)

f(n,b,d) = if(d, f(10*n+1, b, d-1); if(!b, forprime(q=2, 9, f(10*n+q, 1, d-1))), if(b && isprime(n), print1(n", ")))
for(d=1, 8, f(0,0,d)) \\ f(0,0,d) prints d-digit terms. Jens Kruse Andersen, Jul 21 2014

\\ From M. F. Hasler, Apr 23 2019: (Start)
select( is_A046703(n)=isprime(vecprod(digits(n)))&&ispseudoprime(n), [0..9999]) \\ This defines is_A046703(). In older PARI versions, vecprod=factorback.
next_A046703(n)={if( n>1, until( ispseudoprime(n), my(d=digits(n)); n=fromdigits( apply( t->if(t>1, nextprime(t+1), 1), d))+(d[1]>5)); n, 2)}
A046703_vec(N=99)=vector(N, i, t=next_A046703(if(i>1, t))) \\ (End)

Corrected by Harvey P. Dale, Oct 02 2011

A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 119, 1236, 19135, 19144, 261296, 3634992, 43139968

Offset: 1

Erich Friedman, Jun 30 2001

For a d-digit number with d >= 88, the sum and product of the digits together have fewer than d digits. So every element of this sequence has 87 or fewer digits, hence it is finite. - David W. Wilson, Apr 28 2005

Fixed points of the map A380873: concatenate sum and product of digits. - M. F. Hasler, Apr 01 2025

			1236 has sum of digits 12 and product of digits 36.

Cf. A007953 (sum of digs), A007954 (product of digs), A038364, A038369, A066282, A380873, A380872 (trajectories under map).

sdpdQ[n_]:=Module[{idn=IntegerDigits[n],s,p},s=Total[idn];p=Times@@idn;n==FromDigits[Join[IntegerDigits[s],IntegerDigits[p]]]]; Select[Range[44*10^6],sdpdQ] (* Harvey P. Dale, Nov 23 2024 *)

from math import prod
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def c(s):
    d = list(map(int, s))
    return sorted(s) == sorted(str(sum(d)) + str(prod(d)))
def ok(s):
    d = list(map(int, s))
    return s[0] != '0' and "".join(s) == str(sum(d)) + str(prod(d))
def nd(d): yield from ("".join(m) for m in mc("0123456789", d))
def b(): yield from (s for d in count(1) for s in nd(d) if c(s))
def a(): yield from (int("".join(p)) for s in b() for p in mp(s) if ok(p))
print(list(islice(a(), 16))) # Michael S. Branicky, Jun 30 2022

More terms from Harvey P. Dale, Jul 04 2001
More terms from David W. Wilson, Apr 28 2005; he reports on May 03 2005 that there are no further terms.
Offset corrected by Altug Alkan, Apr 10 2018

A098736 a(n) = product of n and all its digits.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 11, 24, 39, 56, 75, 96, 119, 144, 171, 0, 42, 88, 138, 192, 250, 312, 378, 448, 522, 0, 93, 192, 297, 408, 525, 648, 777, 912, 1053, 0, 164, 336, 516, 704, 900, 1104, 1316, 1536, 1764, 0, 255, 520, 795, 1080, 1375, 1680, 1995

Offset: 0

Alexandre Wajnberg, Sep 30 2004

			a(15) = 15*1*5=75

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A230101.

with transforms; [seq(n*digprod(n), n=0..200)]; # N. J. A. Sloane, Oct 12 2013

Array[# Times@@ IntegerDigits@#&, 60, 0] (* Vincenzo Librandi, Oct 13 2013 *)

a(n) = vecprod(digits(n))*n; \\ Michel Marcus, Aug 06 2020

If n=abcd (say) in decimal, then a(n) = abcd * a * b * c * d.

a(n) = n*A007954(n). - R. J. Mathar, Sep 27 2013

More terms from Sam Alexander, Jan 06 2005

Corrected by Vincenzo Librandi, Oct 13 2013

A038367 Numbers n with property that (product of digits of n) is divisible by (sum of digits of n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 22, 30, 36, 40, 44, 50, 60, 63, 66, 70, 80, 88, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 123, 130, 132, 138, 140, 145, 150, 154, 159, 160, 167, 170, 176, 180, 183, 189, 190, 195, 198, 200, 201, 202, 203

Offset: 1

Felice Russo

Equal to the disjoint union of A061013 and A011540 \ {0}. Contains in particular all positive single-digit integers, those with a digit 0, and 22*{1,...,18}. If x is in the sequence, any digit-permutation of x is also in the sequence. - M. F. Hasler, Feb 28 2018

Iain Fox, Table of n, a(n) for n = 1..10000

See A061013 for case where 0 digits are excluded. Cf. A055931.

[0] cat [n: n in [1..250] | IsIntegral(&*Intseq(n)/&+Intseq(n))]; // Bruno Berselli, Feb 09 2016

isA038367 := proc(n)
    if type( A007954(n)/A007953(n),'integer') then
        true;
     else
        false;
    end if;
end proc :
for n from 1 to 500 do
    if isA038367(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jun 30 2020

okQ[n_]:=Module[{idn=IntegerDigits[n]},Divisible[Times@@idn,Total[idn]]]
Select[Range[500],okQ] (* Harvey P. Dale, Nov 24 2010 *)

is(n)=n&&prod(i=1,#n=digits(n),n[i])%vecsum(n)==0 \\ M. F. Hasler, Feb 28 2018

Corrected by Vladeta Jovovic and Larry Reeves (larryr(AT)acm.org), Jun 08 2001

Erroneous 0 term removed by David A. Corneth, Jun 05 2016

A062998 Numbers whose sum of digits is less than or equal to its product of digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 123, 124, 125, 126, 127, 128, 129, 132, 133, 134, 135

Offset: 1

Henry Bottomley, Jun 29 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.

Not the same as A037344 (contains 124).

isA062998 := proc(n)
        local dgs,s,p ;
        dgs := convert(n,base,10) ;
        s := add(i,i=dgs) ;
        p := mul(i,i=dgs) ;
        if s <= p then
                true;
        else
                false;
        end if;
end proc:
for n from 2 to 150 do
        if isA062998(n) then
                printf("%d,",n) ;
        end if;
end do:   # R. J. Mathar, Aug 14 2025

Select[Range[100],Total[IntegerDigits[#]]<=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Feb 21 2017 *)

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

is_A062998(n)={normlp(n=digits(n),1)<=prod(i=1,#n,n[i])} \\ M. F. Hasler, Oct 29 2014

A171765 a(n) = 0 if n <= 10; for n >= 11, a(n) = product of digits of n.

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

Offset: 0

Felice Russo, Oct 11 2010

Cf. A007954.

A171765 := proc(n)
    if n<= 10 then
        0;
    else
        A007954(n);
    end if:
end proc: # R. J. Mathar, Jan 11 2020

Table[If[n<11,0,Times@@IntegerDigits[n]],{n,0,100}] (* Harvey P. Dale, Aug 02 2016 *)

a(n)=if(n<=9,0,n=digits(n); prod(i=1,#n,n[i])) \\ Charles R Greathouse IV, Aug 25 2014

A061076 a(n) is the sum of the products of the digits of all the numbers from 1 to n.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 90, 92, 96, 102, 110, 120, 132, 146, 162, 180, 180, 183, 189, 198, 210, 225, 243, 264, 288, 315, 315, 319, 327, 339, 355, 375, 399, 427, 459, 495, 495, 500, 510, 525, 545, 570, 600, 635

Offset: 1

Amarnath Murthy, Apr 14 2001

What is the asymptotic behavior of this sequence? a(n) = a(n+1) for almost all n. A weak upper bound: a(n) << n^1.91. - Charles R Greathouse IV, Jan 13 2012
A check was done for k in {i^j | 1 <= i <= 10 AND 1 <= j <= 100}. For all these values, a(k) < k^1.733. Another check for k in {i^j | 101 <= i <= 110 AND 101 <= j <= 200} gave a(k) < k^1.65324. For k in {i | 10^6 <= i <= 10^7}, a(k) < k^1.6534. So I ask: is it true that a(n) < n^1.733 and a(n) -> n^(1.65323 + o(1)), or about n^(log(45)/log(10) + o(1))? - David A. Corneth, May 17 2016
For n = 10^(k-1), the closed-form formula from Mihai Teodor (see Formula section) gives a(n) = (45^k - 45)/44, so lim_{n->oo} log(a(n))/log_10(n) = log(45) = 3.80666248977.... - Jon E. Schoenfield, Apr 10 2022
For k >= 1, a(10^k-1) = a(10^k) = ... = a(10*R_k) where R = A002275; so there is a run of 10*R_{k-1} + 2 = A047855(k) consecutive terms equal to (45/44)*(45^k-1) when n runs from 10^k-1 up to 10*R_k, this is because those numbers have one or more 0's. Example: first runs with 2, 12, 112, 1112, ... consecutive terms equal to 45, 2070, 93195, 4193820, ... start at 9, 99, 999, 9999, ... and end at 10, 110, 1110, 11110, ... - Bernard Schott, Oct 18 2022

			a(9) = a(10) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1*0 = 1+2+3+4+5+6+7+8+9 = 45.

Amarnath Murthy, Smarandache friendly numbers and a few more sequences, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001.

Daria Micovic, Table of n, a(n) for n = 1..10000
Amarnath Murthy, Smarandache friendly numbers and a few more sequences, vixra, 2014.
Luca Onnis, On the general Smarandache's sigma product of digits, arXiv:2203.07227 [math.GM], 2022.

Cf. A002275, A007954, A037123, A047855.

A007954:= n -> convert(convert(n,base,10),`*`):
ListTools:-PartialSums(map(A007954,[$1..100])); # Robert Israel, May 17 2016

Accumulate[Times@@IntegerDigits[#]& /@ Range[100]]

pd(n) = my(d = digits(n)); prod(i=1, #d, d[i]);
a(n) = sum(k=1, n, pd(k)); \\ Michel Marcus, Feb 01 2015

a(n) = {n=digits(n); p=1; d=#n; for(i=1,#n,if(n[i]==0,d=i-1;break));
(45/44) * (45^(#n-1)-1) + sum(i=1,d,p*=n[i]; p * (n[i]-1) * (45/44) * (45^(#n -i) - 45^(#n-i-1)) / 2)+p*(d==#n)} \\ David A. Corneth, May 17 2016

from math import prod
def A061076(n): return sum(prod(int(d) for d in str(i)) for i in range(1,n+1)) # Chai Wah Wu, Mar 21 2022

def A061076(n):
    p = 0
    i = 0
    while i < n + 1:
        p += prod(int(digit) for digit in str(i))
        i += 1
    return p # Daria Micovic, Apr 13 2016

a(n) = Sum_{k = 1..n} (product of the digits of k).
a(10^k-1) = (45/44)*(45^k-1). - Giovanni Resta, Oct 18 2012
From Robert Israel, May 17 2016: (Start)
Partial sums of A007954.
G.f.: (1-x)^(-1) * 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+(1-x^10)*A(x^10))/(1-x).
(End)
Let b(1), b(2), ..., b(k) be the digits of the base-10 expansion of n: n = b(1)*10^(k-1) + b(2)*10^(k-2) + ... + b(k). Then a(n) = b(1)*b(2)*...*b(k) + (45^k-45)/44 + (1/2)*Sum_{i=1..k} b(1)*b(2)*...*b(i)*(b(i)-1)*45^(k-i). - Mihai Teodor, Apr 09 2022

Corrected and extended by Matthew Conroy, Apr 16 2001

A062996 Numbers whose sum of digits is greater than or equal to its product of digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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, 123, 130, 131, 132

Offset: 1

Henry Bottomley, Jun 29 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954, A034710, A062329, A062996, A062997, A062998, A062999.

Select[Range[150],Total[IntegerDigits[#]]>=Times@@IntegerDigits[#]&] (* Harvey P. Dale, Sep 27 2023 *)

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

A070565 n - product of digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, 40, 37, 34, 31, 28, 25, 22, 19, 16, 13, 50, 46, 42, 38, 34, 30, 26, 22, 18, 14, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 70, 64, 58

Offset: 0

N. J. A. Sloane, May 07 2002

			a(20) = 20 - 2*0 = 20, a(22) = 22 - 2*2 = 18.

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

Cf. A007954, A062329.

Table[n - Times @@ IntegerDigits[n], {n, 0, 75}]

for(n=1,100,s=ceil(log(n)/log(10)); print1(n-prod(i=0,s-1,floor(n/10^i*1.)-10*floor(n/10^(i+1)*1.)),", "))

a(n) = if (n==0, 0, n - vecprod(digits(n))); \\ Michel Marcus, Jul 22 2025

More terms from Benoit Cloitre and Robert G. Wilson v, May 09 2002

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 40, 42, 50, 60, 70, 80, 81, 90, 100, 105, 108, 112, 120, 140, 150, 160, 180, 200, 210, 240, 250, 270, 280, 300, 320, 350, 360, 400, 405, 420, 450, 480, 490, 500, 504, 540, 560, 600, 630, 640, 700, 720, 750, 800

Offset: 1

Daniel Mondot, Dec 18 2021

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 2.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.

			10 is in this sequence because:
- 10 goes to a single digit in 1 step: p(10) = 0.
- 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
12 is in this sequence because:
- 12 goes to a single digit in 1 step: p(12) = 2.
- 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
(12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
14 is in this sequence because:
- 14 goes to a single digit in 1 step: p(14) = 4.
- 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
(27 and 72 are in the next layer of classes, A350181, the rest are not).

Daniel Mondot, Table of n, a(n) for n = 1..20000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046510
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).

mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
isok(n)={mp(n)==1 && isparent(n)} \\ Andrew Howroyd, Dec 20 2021

cp's OEIS Frontend

A046703 Multiplicative primes: product of digits is a prime.

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A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

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A098736 a(n) = product of n and all its digits.

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A038367 Numbers n with property that (product of digits of n) is divisible by (sum of digits of n).

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A062998 Numbers whose sum of digits is less than or equal to its product of digits.

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A171765 a(n) = 0 if n <= 10; for n >= 11, a(n) = product of digits of n.

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A061076 a(n) is the sum of the products of the digits of all the numbers from 1 to n.

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