A006753 -id:A006753 - cp's OEIS frontend

A178204 Smith numbers of order 6; composite numbers n such that sum of digits^6 equal sum of digits^6 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

40844882, 113986781, 130852098, 141176320, 168137185, 170774472, 178180163, 181681157, 181693781, 183161897, 187117638, 215149451, 261666000, 284804842, 294557945, 307711074, 335524949, 337194240, 344552927, 347391040, 355318188, 358831104, 368657536

Offset: 1

Paul Weisenhorn, Dec 19 2010

			a(4) = 141176320 = 2^9*5*55147;
3*1^6+2^6+3^6+4^6+6^6+7^6 = 1^6+9*2^6+4^6+3*5^6+7^6 = 169197

Donovan Johnson, Table of n, a(n) for n = 1..1000
Eric W. Weisstein, Smith Number

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178193, A178203.

fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; id != fid && Plus @@ (id^6) == Plus @@ (fid^6)]; k = 2; lst = {}; While[k < 50000001, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst

Example corrected by Donovan Johnson, Jan 02 2013

A036920 Composite numbers n such that digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 27, 94, 105, 114, 121, 150, 166, 202, 204, 222, 224, 265, 274, 315, 342, 346, 355, 382, 438, 445, 450, 454, 517, 526, 540, 562, 612, 634, 640, 706, 841, 852, 913, 915, 922, 1068, 1086, 1111, 1120, 1122, 1138, 1165, 1185, 1200, 1219, 1221, 1230

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036921, A019506.

ds[n_]:=Total[IntegerDigits[n]]; t={}; Do[If[!PrimeQ[n]&&ds[n]==ds[Total[ Times@@@FactorInteger[n]]],AppendTo[t,n]],{n,4,1230}]; t (* Jayanta Basu, Jun 04 2013 *)

Title made more precise by Sean A. Irvine, Nov 30 2020

A050220 Larger of Smith brothers.

Original entry on oeis.org

729, 2965, 3865, 4960, 5936, 6188, 9387, 9634, 11696, 13765, 16537, 16592, 20785, 25429, 28809, 29624, 32697, 33633, 35806, 39586, 43737, 44734, 49028, 55345, 56337, 57664, 58306, 62635, 65913, 65975, 66651, 67068, 67729, 69280, 69836, 73616, 73617, 74169

Offset: 1

Eric W. Weisstein

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)
Eric Weisstein's World of Mathematics, Smith Brothers.

Cf. A006753, A050219.

issmith:= proc(n)
  if isprime(n) then return false fi;
  convert(convert(n,base,10),`+`) = add(t[2]*convert(convert(t[1],base,10),`+`),t=ifactors(n)[2])
end proc:
S:= select(issmith, {$4..10^5}):
sort(convert(S intersect map(`+`,S,1), list)); # Robert Israel, Jan 15 2018

smithQ[n_] := n > 1 && !PrimeQ[n] && Total[Flatten[IntegerDigits[Table[#[[1]], {#[[2]]}]& /@ FactorInteger[n]]]] == Total[IntegerDigits[n]];
Select[Range[10^5], smithQ[#] && smithQ[#-1]&] (* Jean-François Alcover, Jun 07 2020 *)

isone(n) = {if (!isprime(n), f = factor(n); sumdigits(n) == sum(k=1, #f~, f[k,2]*sumdigits(f[k,1])););}
isok(n) =  isone(n) && isone(n-1); \\ Michel Marcus, Jul 17 2015

from sympy import factorint
from itertools import count, islice
def sd(n): return sum(map(int, str(n)))
def smith():
    for k in count(1):
        f = factorint(k)
        if sum(f[p] for p in f) > 1 and sd(k) == sum(sd(p)*f[p] for p in f):
            yield k
def agen():
    prev = -1
    for s in smith():
        if s == prev + 1: yield s
        prev = s
print(list(islice(agen(), 38))) # Michael S. Branicky, Dec 23 2022

Offset corrected by Arkadiusz Wesolowski, May 08 2012

A050224 1/2-Smith numbers.

Original entry on oeis.org

88, 169, 286, 484, 598, 682, 808, 844, 897, 961, 1339, 1573, 1599, 1878, 1986, 2266, 2488, 2626, 2662, 2743, 2938, 3193, 3289, 3751, 3887, 4084, 4444, 4642, 4738, 4804, 4972, 4976, 4983, 5566, 5665, 5764, 5797, 5863

Offset: 1

Eric W. Weisstein

			88 is a 2^(-1) Smith number because the digit sum of 88, i.e., S(88) = 8 + 8 = 16, which is equal to twice the sum of the digits of its prime factors, i.e., 2 * Sp (88) = 2 * Sp (11 * 2 * 2 * 2) = 2 * (1 + 1 + 2 + 2 + 2) = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Shyam Sunder Gupta, Smith Numbers.
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Wayne L. McDaniel, The Existence of infinitely Many k-Smith numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 76-80.
Eric Weisstein's World of Mathematics, Smith Numbers

Cf. A006753, A050225.

snoQ[n_]:=Total[IntegerDigits[n]]==2Total[Flatten[IntegerDigits/@ Flatten[ Table[First[#],{Last[#]}]&/@FactorInteger[n]]]]; Select[Range[ 6000], snoQ] (* Harvey P. Dale, Oct 15 2011 *)

More terms from Shyam Sunder Gupta, Mar 11 2005

A177927 3-Monica numbers.

Original entry on oeis.org

4, 9, 10, 22, 24, 25, 27, 34, 42, 46, 55, 58, 60, 72, 78, 81, 82, 85, 94, 105, 106, 114, 115, 118, 121, 126, 128, 132, 142, 145, 150, 166, 178, 180, 186, 187, 192, 195, 202, 204, 205, 214, 216, 222, 224, 226, 231, 234, 235, 243, 253, 256, 258, 262, 265, 274, 276, 285, 289, 295

Offset: 1

Chris Fry, Dec 26 2010

3-Monica numbers are composite positive integers k for which 3 divides S(k)-Sp(k), where S(k) denotes the sum of the digits of k and Sp(k) denotes the sum of the digits in an extended prime factorization of k.

			S(10)=1+0=1, 10=2*5, Sp(10)=2+5=7, S(10)-Sp(10)=-6 which is divisible by 3.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 93.
E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pages 1192-1193.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Monica Set.

Cf. A006753 (Smith numbers are a subset of every n-Monica sequence).
Cf. A102217 (n-Suzanne numbers are a subset of n-Monica numbers).
Cf. A102219 (This list of '3-Monica' numbers is incorrect. It does not contain all the Smith numbers and appears to be based on S(n)+Sp(n) ==0 (mod 3), instead of S(n)-Sp(n) == 0 (mod 3)).
Cf. A007953, A118503.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon3Q[n_] := CompositeQ[n] && Divisible[s[n] - sp[n], 3]; Select[Range[300], mon3Q] (* Amiram Eldar, Apr 23 2021 *)

A334527 Numbers that are both Niven numbers and Smith numbers.

Original entry on oeis.org

4, 27, 378, 576, 588, 645, 648, 666, 690, 825, 915, 1872, 1908, 1962, 2265, 2286, 2484, 2556, 2688, 2934, 2970, 3168, 3258, 3294, 3345, 3366, 3390, 3564, 3615, 3690, 3852, 3864, 3930, 4428, 4464, 4557, 4880, 5526, 6084, 6315, 7695, 8154, 8736, 8883, 9015, 9036

Offset: 1

Amiram Eldar, May 05 2020

McDaniel (1990) proved that there exist infinitely many numbers which are both base-b Niven numbers and base-b Smith numbers, for all bases b >= 8.

			27 is a term since it is a Niven number (2 + 7 = 9 is a divisor of 27) and a Smith number (27 = 3 * 3 * 3 and 2 + 7 = 3 + 3 + 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Wayne L. McDaniel, On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers, Missouri Journal of Mathematical Sciences, Vol. 2, No. 3 (1990), pp. 132-136.

Intersection of A005349 and A006753.

digSum[n_] := Plus @@ IntegerDigits[n]; nivenSmithQ[n_] := Divisible[n, (ds = digSum[n])] && CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == ds; Select[Range[10^4], nivenSmithQ]

from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def ok(n):
  sdn = sd(n)
  if sdn == 0 or n%sdn != 0: return False # not Niven
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and sdn == sum(sd(p)*f[p] for p in f)
print(list(filter(ok, range(9999)))) # Michael S. Branicky, Apr 27 2021

A384444 Positive integers k for which the sum of their digits equals the product of their prime digits.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 20, 22, 30, 50, 70, 100, 123, 132, 200, 202, 213, 220, 231, 300, 312, 321, 500, 700, 1000, 1023, 1032, 1203, 1230, 1247, 1274, 1302, 1320, 1356, 1365, 1427, 1472, 1536, 1563, 1635, 1653, 1724, 1742, 2000, 2002, 2013, 2020, 2031, 2103, 2130, 2147

Offset: 1

Felix Huber, Jun 03 2025

Numbers k for which A007953(k) = A384443(k).

If t is a term then t*10^m is also a term for any positive integer m.

			1302 is a term, because 1 + 3 + 0 + 2 = 3*2 = 6.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384445, A384505.

A384444:=proc(n)
    option remember;
    local k,c;
    if n=1 then
        1
    else
        for k from procname(n-1)+1 do
            c:=convert(k,'base',10);
            if mul(select(isprime,c))=add(c) then
                return k
            fi
        od
    fi;
end proc;
seq(A384444(n),n=1..51);

Select[Range[2147],Total[IntegerDigits[#]]==Times@@Select[IntegerDigits[#],PrimeQ]&] (* James C. McMahon, Jun 20 2025 *)

isok(k) = my(d=digits(k)); vecprod(select(isprime, d)) == vecsum(d); \\ Michel Marcus, Jun 04 2025

A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.

Original entry on oeis.org

5, 6, 7, 10, 23, 43, 74, 125, 199, 305, 449, 637, 885, 1216, 1649, 2184, 2852, 3664, 4657, 5863, 7298, 9002, 10993, 13312, 16000, 19084, 22613, 26606, 31120, 36192, 41867, 48220, 55317, 63232, 72022, 81746, 92479, 104282, 117229, 131393, 146843, 163652, 181892

Offset: 1

Felix Huber, Jun 03 2025

			a(3) = 7 because exactly for the 7 multisets with 3 digits {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 0, 5}, {0, 0, 7}, {0, 2, 2} and {1, 2, 3} their sum equals the product of the prime digits.
a(4) = 10 because exactly for the 10 multisets with 4 digits {0, 0, 0, 1}, {0, 1, 2, 3}, {1, 2, 4, 7}, {1, 3, 5, 6}, {0, 0, 0, 2}, {0, 0, 2, 2}, {0, 0, 0, 3}, {0, 0, 0, 5}, {5, 5, 6, 9} and {0, 0, 0, 7} their sum equals the product of the prime digits.

Felix Huber, Table of n, a(n) for n = 1..200

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444, A384505.

f:=proc(p,n)
    local c,d,i,l,m,r,s,t,u,w,x,y,z;
    m:={0,1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    y:={l='Union'(t),w='Set'(l),t=~'Atom'};
    d:=(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))};
    Order:=p+1;
    r:=combstruct:-agfseries(y,d,'unlabeled',z,[[u,s]])[w(z,u)];
    r:=collect(convert(r,'polynom'),[z,u],'recursive');
    c:=coeff(r,z,p);
    coeff(c,u,n)
end proc:
A384445:=proc(n)
    local a,k,m,s,p,j,L;
    a:=1;
        for k from 9*n to 1 by -1 do
            L:=ifactors(k)[2];
            m:=nops(L);
            if m>0 and L[m,1]<=7 then
                p:=n-add(L[j,2],j=1..m);
                s:=k-add(L[j,1]*L[j,2],j=1..m);
                if s=0 and p>=0 then
                    a:=a+1
                elif p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
end proc;
seq(A384445(n),n=1..43);

A384505 a(n) is the number of multisets of n positive decimal digits where the sum of the digits equals the product of the prime digits.

Original entry on oeis.org

5, 1, 1, 3, 13, 20, 31, 51, 74, 106, 144, 188, 248, 331, 433, 535, 668, 812, 993, 1206, 1435, 1704, 1991, 2319, 2688, 3084, 3529, 3993, 4514, 5072, 5675, 6353, 7097, 7915, 8790, 9724, 10733, 11803, 12947, 14164, 15450, 16809, 18240, 19757, 21374, 23073, 24876, 26759

Offset: 1

Felix Huber, Jun 11 2025

			a(1) = 5 because exactly for the 5 multisets with 1 digits {1}, {2}, {3}, {5}, and {7} their sum equals the product of the prime digits.
a(2) = 1 because only for 1 multiset with 2 positive digits {2, 2} their sum equals the product of the prime digits: 2 + 2 = 2*2 = 4.
a(3) = 1 because only for 1 multiset with 3 positive digits {1, 2, 3} their sum equals the product of the prime digits: 1 + 2 + 3 = 2*3 = 6.
a(4) = 3 because exactly for the 3 multisets with 4 digits {1, 2, 4, 7}, {1, 3, 5, 6}, and {5, 5, 6, 9} their sum equals the product of the prime digits: 1 + 2 + 4 + 7 = 2 * 7 = 14, 1 + 3 + 5 + 6 = 3*5 = 15, 5 + 5 + 6 + 9 = 5*5 = 25.

Felix Huber, Table of n, a(n) for n = 1..200

Partial sums give A384445.

Cf. A002110, A006753, A007947, A007954, A066306, A067077, A384443, A384444.

f:=proc(p,n)
    local i,l,m,s,t,u,w,x,z;
    m:={1,4,6,8,9};
    t:=seq(cat(x,i),i in m);
    Order:=p+1;
    coeff(coeff(collect(convert(combstruct:-agfseries({l='Union'(t),w='Set'(l),t=~'Atom'},(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))},'unlabeled',z,[[u,s]])[w(z,u)],'polynom'),[z,u],'recursive'),z,p),u,n)
end proc:
A384505:=proc(n)
    local a,k,m,s,p,j,L;
    if n=1 then
        5
    elif n=2 then
        1
    else
        a:=0;
        for k from 9*n to 1 by -1 do
            L:=ifactors(k)[2];
            m:=nops(L);
            if m>0 and L[m,1]<=7 then
                p:=n-add(L[j,2],j=1..m);
                s:=k-add(L[j,1]*L[j,2],j=1..m);
                if p>0 and s>0 then
                    a:=a+f(p,s)
                fi
            fi
	od;
	return a
	fi;
end proc;
seq(A384505(n),n=1..48);

a(n) = A384445(n) - A384445(n-1) for n > 1.

A036921 Numbers n such that digit sum of n equals digit sum of 'juxtaposition' and 'sum' of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 27, 94, 121, 166, 202, 265, 274, 346, 355, 382, 438, 454, 517, 526, 562, 634, 706, 852, 913, 915, 922, 1086, 1111, 1165, 1219, 1255, 1282, 1507, 1626, 1633, 1642, 1822, 1894, 1903, 1966, 2067, 2155, 2173, 2182, 2227, 2265, 2326, 2362, 2409, 2434

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036920, A019506.

d[n_]:=IntegerDigits[n]; co[n_,k_]:=Nest[Flatten[d[{#,n}]]&,n,k-1]; t={}; Do[If[!PrimeQ[n]&&Total[d[n]]==Total[d[Total[Times@@@(x=FactorInteger[n])]]]==Total[Flatten[d[co@@@x]]],AppendTo[t,n]],{n,4,2435}]; t (* Jayanta Basu, Jun 04 2013 *)

cp's OEIS Frontend

A178204 Smith numbers of order 6; composite numbers n such that sum of digits^6 equal sum of digits^6 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

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A036920 Composite numbers n such that digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).

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A050220 Larger of Smith brothers.

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A050224 1/2-Smith numbers.

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A177927 3-Monica numbers.

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A334527 Numbers that are both Niven numbers and Smith numbers.

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A384444 Positive integers k for which the sum of their digits equals the product of their prime digits.

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A384445 a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.

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