A007954 -id:A007954 - cp's OEIS frontend

A343036 Positions of Zuckerman numbers within A342978, the ordered list of zeroless numbers according to k/A007954(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 154, 1629, 1630, 9014, 33504, 98062, 243287, 531870, 531871, 1063844, 1063845, 1063846, 3484080, 5810556, 5810557, 9305178, 9305179, 9305180, 14376561, 45251346, 63197812, 63197813, 63197814, 87055977, 87055978, 157169049, 206168352, 206168353

Offset: 1

Michel Marcus, Apr 03 2021

			The first 9 terms of A342978 are the first 9 Zuckerman numbers, so sequence begins with 1, 2, 3, 4, 5, 6, 7, 8, 9.
The next Zuckerman number in A342978 is 36 at position 154, so a(10) = 154.

David A. Corneth, Table of n, a(n) for n = 1..45
David A. Corneth, PARI program

Cf. A007602 (Zuckerman numbers), A007954 (product of decimal digits), A052382 (zeroless numbers), A342978.

See PARI link \\ David A. Corneth, Apr 07 2021

a(14)-a(22) from David A. Corneth, Apr 03 2021
a(23)-a(28) from Michel Marcus, Apr 03 2021
a(29)-a(37) from Michel Marcus, Apr 07 2021

A243218 Number of n-digit integers x such that x + A007954(x) has n digits, where A007954(x) is the product of decimal digits of x.

Original entry on oeis.org

5, 63, 756, 8268, 86225, 880519, 8898517, 89471520, 897248572, 8985712192, 89925825853, 899614672173, 8997997446679, 89989593213308, 899945924502919, 8999718992342921, 89998539650321017, 899992410699128981, 8999960560129165187, 89999795045731606967

Offset: 1

Michel Marcus, Jun 01 2014

			For n=1, the five 1-digit integers 0,1,2,3,4 satisfy the condition, with results being respectively 0,2,4,6 and 8, hence a(1)=5.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..35

Cf. A007954, A230099, A242945, A243219.

DP(n)= my(d = digits(n)); prod(i=1, #d, d[i]);
a(n) = {nb = 0; if (n==1, istart = 0, istart = 10^(n-1)); for (i=istart, 10^n-1, if (i + DP(i) < 10^n, nb++);); nb;}

a(9)-a(17) from Hiroaki Yamanouchi, Sep 29 2014

a(18)-a(20) from Hiroaki Yamanouchi, Jan 08 2016

A248794 a(n) = the smallest number k for which the sum of digits (A007953(k)) and the product of digits (A007954(k)) are both equal to A002473(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11125, 111126, 1111127, 111111135, 11111128, 111111129, 1111111111145, 1111111111137, 111111111111138, 11111111111111155, 11111111111111139, 1111111111111111147, 111111111111111111156, 1111111111111111111148, 1111111111111111111111157

Offset: 1

Jaroslav Krizek, Nov 02 2014

See comment in A034710 (positive numbers for which the sum of digits equals the product of digits).

			For n = 11; a(11) = 111126 because A002473(11) = 12, A007953(111126) = A007954(111126) = 12.

Cf. A002473, A034710, A007953, A007954.

A248794:=func; [A248794(n):n in[A002473(n)]]

Name clarified by Andrew Howroyd, Sep 20 2024

A249517 Numbers k for which the digital sum A007953(k) and the digital product A007954(k) both contain the same distinct digits as the number k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11111111111

Offset: 1

Jaroslav Krizek, Oct 31 2014

a(12) = (10^106-1)/9 + 122222222. - Max Alekseyev, Nov 15 2014

Other entries include (10^111-1)/9, (10^113-1)/9 + 177, (10^115-1)/9 + 122222222, (10^117-1)/9 + 11117, (10^125-1)/9 + 2224, (10^126-1)/9 + 333335, (10^135-1)/9 + 4666, (10^143-1)/9 + 446, (10^143-1)/9 + 2224, (10^144-1)/9 + 33335. All other entries with 150 or fewer digits are formed by permutations of the decimal digits of these entries (including a(12)). (10^((10^m-1)/9)-1)/9 are entries of the sequence for m >= 0. - Chai Wah Wu, Nov 15 2014

			11111111111 is a term since A007953(11111111111) = 11 and A007954(11111111111) = 1.

Intersection of A249515 and A249516. Subsequence of A249334.

Cf. A007954, A249334, A249515, A249516.

[0] cat [n: n in [0..10^7] | Set(Intseq(n)) eq Set(Intseq(&*Intseq(n))) and Set(Intseq(n)) eq Set(Intseq(&+Intseq(n)))];

is(n)=if(n<=9,return(1)); my(d=digits(n),s=Set(d)); s==Set(digits(sum(i=1,#d,d[i]))) && s==Set(digits(prod(i=1,#d,d[i]))) \\ Charles R Greathouse IV, Nov 13 2014

from itertools import product
from operator import mul
from functools import reduce
A249517_list = [0]
for g in range(1,15):
    xp, ylist = [], []
    for i in range(9*g,-1,-1):
        x = set(str(i))
        if not (('0' in x) or (x in xp)):
            xv = [int(d) for d in x]
            imin = int(''.join(sorted(str(i))))
            if max(xv)*(g-len(x)) >= imin-sum(xv) and i-sum(xv) >=  min(xv)*(g-len(x)):
                xp.append(x)
                for y in product(x,repeat=g):
                    if set(y) == x:
                        yd = [int(d) for d in y]
                        if set(str(sum(yd))) == x == set(str(reduce(mul, yd, 1))):
                            ylist.append(int(''.join(y)))
    A249517_list.extend(sorted(ylist)) # Chai Wah Wu, Nov 15 2014

a(11) = 11111111111 confirmed by Sean A. Irvine, Nov 13 2014, by direct search.

A334803 Numbers k such that k*p is divisible by k+p and k-p, where k > p > 0 and p = A007954(k) = the product of digits of k.

Original entry on oeis.org

24, 36, 3276, 1886976

Offset: 1

Scott R. Shannon, May 12 2020

If a(5) exists it is at least 3*10^12.

a(5) > 1.5*10^14, if it exists. - Giovanni Resta, May 12 2020

			24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by both 24-8 = 16 and 24+8 = 32.
36 is a term as p = 3*6 = 18 and 38*18 = 648 is divisible by both 36-18 = 18 and 36+18 = 54.
3276 is a term as p = 3*2*7*6 = 252 and 3276*252 = 825552 is divisible by both 3276-252 = 3024 and 3276+252 = 3528.
1886976 is a term as p = 1*8*8*6*9*7*6 = 145152 and 1886976*145152 = 273898340352 is divisible by both 1886976-145152 = 1741824 and 1886976+145152 = 2032128.

Cf. A007954, A049102, A085124.

Subsequence of A052382. Intersection of A334679 and A330880.

isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)) && !((m*p) % (m+p)); \\ Michel Marcus, May 12 2020

A340908 Primitive numbers m without zero digits such that pod(m + pod(m)) = pod(m) where pod is the product of digits, A007954.

Original entry on oeis.org

28, 214, 239, 266, 318, 326, 364, 494, 497, 563, 598, 613, 637, 695, 819, 2114, 2139, 2168, 2285, 2313, 2356, 2369, 2419, 2594, 2639, 2791, 3118, 3126, 3148, 3213, 3235, 3238, 3259, 3354, 3365, 3561, 3698, 3786, 4138, 4145, 4188, 4219, 4338, 4346, 4353, 4368, 4395

Offset: 1

Bernard Schott, Jan 31 2021

When a number k belongs to A327750, the integer 111..11//k obtained by concatenation of 111..11 and k is another term; hence, there exist primitive terms as 28, 214, 239, ... that are listed in this sequence.

Equivalently, terms of A327750 that do not begin with 1.

			pod(28 + pod(28)) = pod(28 + 2*8) = pod(28 + 16) = pod(44) = 4*4 = 16 = pod(28), hence 28 that does not begin with 1 is a term.

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, and Ivan Yashchenko, Moscow-Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, pp. 15 and 97.

Index to sequences related to Olympiads.

Cf. A007954, A247888, A340907.

Subsequence of A327750.

pod[n_] := Times @@ IntegerDigits[n]; q[n_] := First[IntegerDigits[n]] > 1 && (p = pod[n]) > 0 && pod[n + p] == p; Select[Range[5000], q] (* Amiram Eldar, Jan 31 2021 *)

isok(n) = my(d = digits(n), p); vecmin(d) && ((d[1]!=1) && p=vecprod(d)) && (vecprod(digits(n+p)) == p); \\ Michel Marcus, Feb 01 2021

A342978 Zeroless numbers k ordered according to k/A007954(k), and in case of ties by k, where A007954(k) is the product of digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 99, 89, 79, 69, 59, 49, 98, 999, 88, 899, 78, 799, 68, 699, 39, 58, 599, 48, 9999, 989, 97, 499, 998, 8999, 889, 87, 898, 789, 7999, 77, 38, 798, 689, 67, 6999, 29, 698, 57, 589, 399, 5999, 598, 47, 99999, 9899, 489, 9989, 4999, 9998

Offset: 1

Michel Marcus, Apr 02 2021

			For k=1 to 9, k/A007954(k) = 1, the least possible value, so a(k)=k for k=1 to 9.
Then we have 99, 89, 79, 69, 59, 49, 98 with 1.22, 1.23, 1.25, 1.27, 1.31, 1.36, 1.36.

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A007954 (product of decimal digits), A052382 (zeroless numbers).

\\ up to d digits
nonzero(n) = vecmin(digits(n));
pd(n) = n/vecprod(digits(n));
lista(d) = {my(lim = 10^d-1, vk = select(nonzero, [1..lim]), vpk = vector(#vk, k, pd(vk[k])), vsk = vecsort(vpk, ,1)); my(vall = vector(#vsk, k, vk[vsk[k]])); my(out = List(), k = 1); while(vall[k] != lim, listput(out, vall[k]); k++); listput(out, vall[k]); Vec(out);}

\\ See Corneth link. David A. Corneth, Apr 02 2021

A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 36, 41, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 61, 62, 63, 64, 66, 68, 71, 74, 76, 78, 82, 83, 84, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113, 114, 115, 116, 118, 121, 122, 123

Offset: 1

Bernard Schott, Feb 19 2022

Inspired by A351650 where pod is replaced by sod.
All terms are zeroless (A052382).
Repunits form a subsequence (A002275).
Integers m without 0 and such that m^2 has a 0 form a subsequence (A134844).
The smallest term k such that the corresponding quotient = n is A351809(n).

			Product of digits of 27 = 2*7 = 14; then 27^2 = 729, product of digits of 729 = 7*2*9 = 81; as 81 divides 729, 27 is a term.

Cf. A007954, A002473, A351808 (corresponding quotients), A351809.

Subsequences: A002275, A134844.

pod[n_] := Times @@ IntegerDigits[n]; Select[Range[120], FreeQ[IntegerDigits[#], 0] && Divisible[pod[#^2], pod[#]] &] (* Amiram Eldar, Feb 19 2022 *)

isok(m) = my(d=digits(m)); vecmin(d) && denominator(vecprod(digits(m^2))/vecprod(d)) == 1; \\ Michel Marcus, Feb 19 2022

from math import prod
def pod(n): return prod(map(int, str(n)))
def ok(m): pdm = pod(m); return pdm > 0 and pod(m*m)%pdm == 0
print([m for m in range(124) if ok(m)]) # Michael S. Branicky, Feb 19 2022

More terms from Amiram Eldar, Feb 19 2022

A351808 a(n) is the quotient obtained when pod(m) divides pod(m^2), with pod = product of digits = A007954 and m = A351807(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 2, 8, 18, 4, 10, 3, 2, 8, 32, 15, 6, 21, 9, 14, 18, 0, 0, 6, 12, 21, 24, 0, 0, 0, 0, 0, 0, 0, 0, 7, 32, 81, 0, 10, 4, 0, 30, 35, 0, 21, 144, 0, 64, 32, 0, 2, 0, 0, 0, 12, 80, 252, 243, 12, 60, 27, 48, 256, 15, 30, 140, 36, 8, 14, 336, 96, 144

Offset: 1

Bernard Schott, Feb 20 2022

a(n) = 0 iff m = A351807(n) is a term of A134844.

As pod(m) is 7-smooth number and pod(m^2) can be 0 (see example), all terms of the sequence are in {0} union A002473. The smallest term k such that the corresponding quotient = 0 or A002473(n) is A351809(n).

			A351807(9) = 13, then pod(13) = 1*3 = 3 while pod(13^2) = pod(169) = 1*6*9 = 54; hence, a(9) = 54/3 = 18.
A351807(23) = 33, then pod(33) = 3*3 = 9 while pod(33^2) = pod(1089) = 1*0*8*9 = 0; hence, a(23) = 0.

Cf. A002473, A007954, A134844, A351807, A351809.

pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, 1/2]; Select[r /@ Range[200], IntegerQ] (* Amiram Eldar, Feb 21 2022 *)

lista(nn) = {my(list=List()); for (m=1, nn, my(d=digits(m), q); if (vecmin(d) && denominator(q = vecprod(digits(m^2))/vecprod(d)) == 1, listput(list, q);); ); Vec(list);} \\ Michel Marcus, Feb 21 2022

from math import prod
from itertools import count, islice
def A351808_gen(): # generator of terms
    return (q for q, r in (divmod(prod(int(d) for d in str(m**2)),prod(int(d) for d in str(m))) for m in count(1) if '0' not in str(m)) if r == 0)
A351808_list = list(islice(A351808_gen(),20)) # Chai Wah Wu, Feb 25 2022

More terms from Amiram Eldar, Feb 21 2022

A351809 a(0) = 32; then, for n >= 1, a(n) is the smallest positive integer k such that pod(k^2)/pod(k) = A002473(n) where pod = product of digits = A007954.

Original entry on oeis.org

32, 1, 2, 3, 15, 381, 25, 61, 12, 27, 16, 41, 28, 23, 336, 13, 1766, 26, 43, 2675, 118, 278, 74, 22, 76, 128, 392, 343, 228, 121, 418, 976, 258, 193, 116, 194, 93, 218, 441, 1231, 112, 63, 219, 984, 136, 4165, 2271, 1894, 183, 615, 434, 22831, 523, 1592, 2435

Offset: 0

Bernard Schott, Feb 24 2022

As pod(m) is a 7-smooth number and pod(m^2) can be 0, all terms of A351808 are in {0} union A002473. See example section for why a(0) = 32.

			pod(32) = 3*2 = 6, pod(32^2) = pod(1024) = 1*0*2*4 = 0, and k = 32 is the smallest positive integer k such that pod(k^2) = 0 while pod(k) <> 0, so a(0) = 32.
A002473(5) = 5; pod(381) = 3*8*1 = 24, pod(381^2) = pod(145161) = 1*4*5*1*6*1 = 120; as 120/24 = 5, and 381 is the smallest positive integer k such that pod(k^2)/pod(k) = 5 then a(5) = 381.
A002473(11) = 12; pod(41)= 4*1 = 4, pod(41^2) = pod(1681) = 1*6*8*1 = 48; as 48/4 = 12 and 41 is the smallest positive integer k such that pod(k^2)/pod(k) = 12, then a(11) = 41.

Cf. A002473, A007954, A280012, A351807, A351808.

sevenSmooths = Select[Range[150], Max[FactorInteger[#][[;; , 1]]] <= 7 &]; pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, -1]; s = Array[r, 3*10^4]; TakeWhile[FirstPosition[s, #] & /@ Join[{0}, sevenSmooths] // Flatten, NumberQ] (* Amiram Eldar, Feb 24 2022 *)

pod(k) = vecprod(digits(k)); \\ A007954
smp(m) = my(k=1); while (!pod(k) || (pod(k^2)/pod(k) != m), k++); k;
isss(n) = (n<11) || (vecmax(factor(n, 7)[, 1])<8); \\ A002473
lista(nn) = apply(smp, select(isss, [0..nn]));
lista(200) \\ Michel Marcus, Feb 24 2022

More terms from Amiram Eldar, Feb 24 2022

cp's OEIS Frontend

A343036 Positions of Zuckerman numbers within A342978, the ordered list of zeroless numbers according to k/A007954(k).

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A243218 Number of n-digit integers x such that x + A007954(x) has n digits, where A007954(x) is the product of decimal digits of x.

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A248794 a(n) = the smallest number k for which the sum of digits (A007953(k)) and the product of digits (A007954(k)) are both equal to A002473(n).

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A249517 Numbers k for which the digital sum A007953(k) and the digital product A007954(k) both contain the same distinct digits as the number k.

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A334803 Numbers k such that k*p is divisible by k+p and k-p, where k > p > 0 and p = A007954(k) = the product of digits of k.

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A340908 Primitive numbers m without zero digits such that pod(m + pod(m)) = pod(m) where pod is the product of digits, A007954.

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A342978 Zeroless numbers k ordered according to k/A007954(k), and in case of ties by k, where A007954(k) is the product of digits of k.

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A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

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