A006753 -id:A006753 - cp's OEIS frontend

A213689 Smallest magic sum of an order-n magic square composed of consecutive Smith numbers.

252416863614, 10949070196804, 8719491322, 3873, 5551, 6496, 9179, 15681, 20978, 26062, 28301, 35936, 50560, 55330, 66877, 88738, 94465, 111840, 128860, 177659, 179929, 230580, 261107, 275220, 300070, 361843, 395338, 457169, 446645, 505470, 560168

Offset: 3

Max Alekseyev, Jun 18 2012

			3 X 3 magic square:
  84138954584 84138954498 84138954532
  84138954486 84138954538 84138954590
  84138954544 84138954578 84138954492
4 X 4 magic square:
  2737267549028 2737267549166 2737267549330 2737267549280
  2737267549305 2737267549265 2737267549134 2737267549100
  2737267549186 2737267549105 2737267549300 2737267549213
  2737267549285 2737267549268 2737267549040 2737267549211

Natalia Makarova, Smallest magic squares composed of consecutive Smith numbers (in Russian)

Cf. A006753, A170928, A189121.

A225133 Minimal index of order n Stanley's antimagic square composed of Smith numbers.

Original entry on oeis.org

4, 143, 669, 2088, 8318, 30885, 87643

Offset: 1

Natalia Makarova, Apr 29 2013

Stanley antimagic square of index d and order n is an n X n matrix where the sum of any n matrix elements in pairwise distinct rows and pairwise distinct columns equals d.

			Examples of order n Stanley's antimagic squares with minimal index S composed of Smith numbers:
.
n=2, S=143
  22  58
  85 121
.
For n=3, S=669 we have:
   22  58 202
   85 121 265
  346 382 526
Here 22+121+526 = 22+265+382 = 58+265+346 = 58+85+526 = 202+121+346 = 202+85+382 = 669.
.
n=4, S=2088
   85  94 121 517
  346 355 382 778
  526 535 562 958
  654 663 690 1086
.
n=5, S=8318 (author V. Pavlovsky)
    58  121  382  562 1111
   202  265  526  706 1255
   454  517  778  958 1507
  1858 1921 2182 2362 2911
  3802 3865 4126 4306 4855
.
n=6, S=30885
    319   346  1642  1678  1966  3226
    535   562  1858  1894  2182  3442
   1255  1282  2578  2614  2902  4162
   3595  3622  4918  4954  5242  6502
   4279  4306  5602  5638  5926  7186
  13639 13666 14962 14998 15286 16546
.
n=7, S=87643 (author J. K. Andersen)
    454   634  1858  2614  4054  4414 16474
   1642  1822  3046  3802  5242  5602 17662
   2038  2218  3442  4198  5638  5998 18058
   5674  5854  7078  7834  9274  9634 21694
   5935  6115  7339  8095  9535  9895 21955
  20362 20542 21766 22522 23962 24322 36382
  24214 24394 25618 26374 27814 28174 40234

Antimagic squares (in Russian), discussion at dxdy.ru
Carlos Rivera Puzzle 681

Cf. A006753, A210710.

A235812 a(n) is the start of the earliest run of n numbers such that the sum of their digits is equal to the sum of the digits of their prime factors.

Original entry on oeis.org

2, 2, 2, 2, 1458855, 1790478, 429990136, 4475873320, 1979414080360

Offset: 1

Giovanni Resta, Jan 16 2014

This sequence takes into account both primes and Smith numbers (A006753).

a(10) > 5*10^12.

			The four numbers 2, 3, 4, 5 are either prime (2, 3, 5) or Smith (4) numbers. In any case, the sum of their digits is equal to the sum of their prime factors (counted with multiplicity), hence a(1) = a(2) = a(3) = a(4) = 2.

C. Rivera, Puzzle 247. Consecutive Smith numbers

Cf. A006753, A059754.

A337294 Composite numbers k that are Smith numbers in a record number of bases 1 < b <= k.

Original entry on oeis.org

4, 10, 15, 27, 42, 60, 72, 78, 174, 204, 222, 378, 438, 663, 1352, 1446, 2022, 2526, 2598, 3462, 4038, 4542, 6054, 12102, 22182, 30336, 35432, 39318, 44358, 55446, 72582, 90726, 99798, 110886, 120966, 157254, 181446, 235878, 288294, 332646, 399174, 432438, 665286

Offset: 1

Amiram Eldar, Aug 21 2020

Values of A002808 at the indices of records of A060209.

The corresponding number of bases are 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 19, 20, 21, 22, 27, 29, 31, 33, 35, 40, 48, 59, 66, 67, 71, 76, 80, 81, 88, 97, 98, 101, 105, 118, 119, 130, 131, 152, 156, 167, 187, ...

			a(1) = 4 since it is the least composite number and it is not a Smith number in any base 1 < b <= 4.
a(2) = 10 since it is the least number that is a Smith number in any base 1 < b <= 10: 10 = 2 * 5 is, 22_4 = 2_4 * 11_4 in base 4, and 2 + 2 = 2 + (1 + 1) = 4.

Cf. A002808, A006753, A060209.

Similar sequences: A107129, A330813.

digSum[n_, b_] := Plus @@ IntegerDigits[n, b]; smithCount[n_] := If[! CompositeQ[n], 0, Module[{c = 0, f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Do[If[Total[e*(digSum[#, b] & /@ p)] == digSum[n, b], c++], {b, 2, n}]; c]]; seq = {}; cmax = -1; Do[If[CompositeQ[n] && (c = smithCount[n]) > cmax, cmax = c; AppendTo[seq, n]], {n, 1, 666}]; seq

A351618 Numbers that are both Zuckerman numbers and Smith numbers.

Original entry on oeis.org

4, 1111, 3168, 7119, 31488, 141184, 698112, 1169316, 1621248, 1687392, 1938816, 1967112, 12469248, 12822912, 14112672, 16616448, 41484288, 79817472, 116149248, 121911264, 128894976, 163319328, 166491936, 193916916, 218431488, 247984128, 798142464, 817883136

Offset: 1

Bernard Schott, Feb 15 2022

			3168 is a term since it is a Zuckerman number (3*1*6*8) = 144 is a divisor of 3168 and a Smith number (3168 = 2*2*2*2*2*3*3*11 and 2+2+2+2+2+3+3+1+1 = 3+1+6+8).

Giovanni Resta, Smith numbers, Numbers Aplenty.
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Intersection of A007602 and A006753.

Cf. A334527.

digSum[n_] := Plus @@ IntegerDigits[n]; smithQ[n_] := CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == digSum[n]; zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prodig]; Select[Range[10^6], zuckQ[#] && smithQ[#] &] (* Amiram Eldar, Feb 15 2022 *)

isok(m) = my(d=digits(m)); if (vecmin(d) && !(m % vecprod(d)) && !isprime(m) , my(f=factor(m)); sum(k=1, #f~, sumdigits(f[k,1])*f[k,2]) == vecsum(d)); \\ Michel Marcus, Feb 15 2022

More terms from Amiram Eldar, Feb 15 2022

A375724 a(n) is the first Smith number with at least n digits.

Original entry on oeis.org

4, 22, 121, 1086, 10086, 100066, 1000165, 10000426, 100000165, 1000000165, 10000000165, 100000000498, 1000000000066, 10000000000615, 100000000000786, 1000000000000426, 10000000000000246, 100000000000000642, 1000000000000000462, 10000000000000000246, 100000000000000000282, 1000000000000000000966

Offset: 1

Robert Israel, Aug 25 2024

a(n) is the least composite k >= 10^(n-1) such that the sum of the decimal digits of k is equal to the sum of the decimal digits of the prime factors of k, counted with multiplicity.

Almost certainly a(n) has exactly n digits, but "at least" is included in the Name since we have no proof of that.

			a(5) = 10086 because 10086 has digit-sum 15 and 10086 = 2 * 3 * 41^2 with 2 + 3 + (4 + 1) + (4 + 1) = 15, and no k from 10000 to 10085 works.

Cf. A006753.

f:= proc(n) local t,x;
    for x from 10^(n-1) do
      if isprime(x) then next fi;
      if convert(convert(x,base,10),`+`) = add(t[2]*convert(convert(t[1],base,10),`+`), t = ifactors(x)[2]) then return x fi;
    od
end proc:
map(f, [$1..30]);

from sympy import factorint
from itertools import count
def sd(n): return sum(map(int, str(n)))
def is_smith(n):
    f = factorint(n)
    return sum(f[p] for p in f) > 1 and sd(n) == sum(sd(p)*f[p] for p in f)
def a(n): return next(k for k in count(10**(n-1)) if is_smith(k))
print([a(n) for n in range(1, 23)]) # Michael S. Branicky, Aug 25 2024

A375727 a(n) is the least number that is a Smith number in all bases 2 to n but not in base n+1.

Original entry on oeis.org

15, 475, 1023, 222, 1924475, 26910204, 191999912, 240365505, 10127638898, 15357419170

Offset: 2

Robert Israel, Aug 25 2024

			a(3) = 475 = 5^2 * 19.
In base 2, 475 = 111011011_2 with digit sum 7, 5 = 101_2 with digit sum 2, 19 = 10011_2 with digit sum 3, and 7 = 2 * 2 + 3.
In base 3, 475 = 122121_3 with digit sum 9, 5 = 12_3 with digit sum 3, 19 = 201_3 with digit sum 3, and 9 = 2 * 3 + 3.
In base 4, 475 = 13123_4 with digit sum 10, 5 = 11_4 with digit sum 2, 19 = 103_4 with digit sum 4, and 10 <> 2 * 2 + 4.

Cf. A006753, A278909.

f:= proc(n) local F, t,b;
     if isprime(n) then return 0 fi;
     F:= ifactors(n)[2];
     for b from 2 while convert(convert(n,base,b),`+`) = add(t[2]*convert(convert(t[1],base,b),`+`), t = F) do od:
     if b = 2 then 0 else b-1 fi
end proc:
N:= 7: # for a(2) .. a(N)
V:= Vector(N): count:= 0:
for n from 4 while count < N-1 do
  v:= f(n);
  if v > 0 and V[v] = 0 then V[v]:= n; count:= count+1 fi;
od:
convert(V[2..N],list);

smith(k) = {my(f = factor(k), p = f[,1], e = f[,2], b = 2); while(sumdigits(k, b) == sum(i = 1, #p, e[i] * sumdigits(p[i], b)), b++); b-1;}
lista(len) = {my(m = len + 1, v = vector(m), c = 0, i); forcomposite(k = 1, , i = smith(k); if(i <= m && v[i] == 0, c++; v[i] = k; if(c == m, break))); vecextract(v, "^1");} \\ Amiram Eldar, Aug 29 2024

from sympy.ntheory import digits
from sympy import factorint, isprime
from itertools import count, islice
def sd(n, base=10): return sum(digits(n, base)[1:])
def f(n, factors):
    for b in count(2):
        if sd(n, base=b) != sum(sd(p, base=b)*factors[p] for p in factors):
            break
    return b-1
def agen(): # generator of terms
    adict, n = dict(), 2
    for k in count(1):
        if isprime(k): continue
        v = f(k, factorint(k))
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 4))) # Michael S. Branicky, Aug 25 2024

a(8)-a(9) from Michael S. Branicky, Aug 26 2024

a(10)-a(11) from Amiram Eldar, Aug 29 2024

A376157 Numbers k such that the sum of the digits of k equals the sum of its prime factors plus the sum of the multiplicities of each prime factor.

Original entry on oeis.org

4, 25, 36, 54, 125, 192, 289, 297, 343, 392, 448, 676, 756, 1089, 1536, 1764, 1936, 2646, 2888, 3872, 4802, 4860, 6174, 6250, 6776, 6860, 7290, 7488, 7680, 8750, 8775, 9408, 9747, 10648, 14739, 15309, 16848, 18432, 18865, 21296, 22869, 25725, 29988, 33750, 33957

Offset: 1

Jordan Brooks, Sep 12 2024

			For k = 54, its prime factorization is 2^1*3^3: 5+4 = 2+1+3+3 = 9.
For k = 756, its prime factorization is 2^2*3^3*7^1: 7+5+6 = 2+2+3+3+7+1 = 18.

Similar to A006753, A019506, A050689, A063737, A070275, A285494.

Cf. A007953, A008474.

Select[Range[34000], DigitSum[#]==Total[Flatten[FactorInteger[#]]] &] (* Stefano Spezia, Sep 14 2024 *)

isok(k)={my(f=factor(k)); vecsum(f[,1]) + vecsum(f[,2]) == sumdigits(k)} \\ Andrew Howroyd, Sep 26 2024

from sympy.ntheory import factorint
c = 2
while c < 10000:
    charsum = 0
    for char in str(c):
        charsum += int(char)
    pf = factorint(c)
    cand = 0
    for p in pf.keys():
        cand += p
        cand += pf[p]
    if charsum == cand:
        print(c)
        print(pf)
    c += 1

{ k : A007953(k) = A008474(k) }.

A382922 Numbers k such that Fibonacci(k) is a Smith number.

Original entry on oeis.org

31, 77, 231, 354, 523, 535, 631, 819, 827, 830, 991, 1234

Offset: 1

Shyam Sunder Gupta, Apr 08 2025

			31 is a term since Fibonacci(31) = 1346269 = 557 * 2417, and 1346269 is a Smith number (1 + 3 + 4 + 6 + 2 + 6 + 9 = 5 + 5 + 7 + 2 + 4 + 1 + 7).

Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.

Cf. A000045, A006753.

a(5)-a(12) from Amiram Eldar, Apr 09 2025

A385932 Composite numbers m such that the sum of digits of m divides the sum of digits of prime factors of m (counted with multiplicity).

Original entry on oeis.org

4, 10, 22, 27, 32, 42, 58, 60, 70, 85, 94, 100, 104, 121, 152, 166, 200, 202, 231, 265, 274, 315, 316, 319, 322, 330, 342, 346, 355, 361, 378, 382, 391, 402, 406, 430, 438, 450, 454, 483, 510, 517, 526, 535, 540, 562, 576, 588, 602, 610, 612, 627, 632, 634, 636, 645, 648

Offset: 1

Stefano Spezia, Jul 12 2025

Equivalently, numbers m such that A007953(m) | A118503(m).

Union of the k-Smith numbers for all the positive integers k.

			10 = 2*5 is a term since it is a 7-Smith number: 1 + 0 = 1 | 7 = 2 + 5;
60 = 2^2*3*5 is term since it is a 2-Smith number: 6 + 0 = 6 | 12 = 2 + 2 + 3 + 5;
382 = 2*191 is a term since it is a Smith number (k=1): 3 + 8 + 2 = 13 | 13 = 2 + 1 + 9 + 1;
635 = 5*127 is not a term since 6 + 3 + 5 = 14 does not divide 15 = 5 + 1 + 2 + 7.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.1.14 and 3.1.16 on pages 84-85.

Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).

Supersequence of A006753, A103125, A103126, A104390, A104391.

Cf. A002808, A007953, A118503.

fQ[n_]:=!PrimeQ[n] && n>1 && Divisible[Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]] & /@ FactorInteger[n]]], Total[IntegerDigits[n]]]; Select[ Range@ 650, fQ]

cp's OEIS Frontend

A213689 Smallest magic sum of an order-n magic square composed of consecutive Smith numbers.

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A225133 Minimal index of order n Stanley's antimagic square composed of Smith numbers.

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A235812 a(n) is the start of the earliest run of n numbers such that the sum of their digits is equal to the sum of the digits of their prime factors.

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A337294 Composite numbers k that are Smith numbers in a record number of bases 1 < b <= k.

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Mathematica

A351618 Numbers that are both Zuckerman numbers and Smith numbers.

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Mathematica

PARI

Extensions

A375724 a(n) is the first Smith number with at least n digits.

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Maple

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A375727 a(n) is the least number that is a Smith number in all bases 2 to n but not in base n+1.

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Maple

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A376157 Numbers k such that the sum of the digits of k equals the sum of its prime factors plus the sum of the multiplicities of each prime factor.

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Mathematica

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Python

Formula

A382922 Numbers k such that Fibonacci(k) is a Smith number.

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