A067077 -id:A067077 - cp's OEIS frontend

A057531 Numbers whose sum of digits and number of divisors are equal.

1, 2, 11, 22, 36, 84, 101, 152, 156, 170, 202, 208, 225, 228, 288, 301, 372, 396, 441, 444, 468, 516, 525, 530, 602, 684, 710, 732, 804, 828, 882, 952, 972, 1003, 1016, 1034, 1070, 1072, 1106, 1111, 1164, 1236, 1304, 1308, 1425, 1472, 1476, 1521, 1524

Offset: 1

Asher Auel, Sep 03 2000

[A007953(n)/A000005(n) = c] AND [A000005(n)/A007953(n) = c], c an integer. - Ctibor O. Zizka, Jun 26 2009

			36 is a term as the sum of the digits of 36 is 3+6 = 9 and the number of divisors is 9 too.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 210 terms from Daniel Arribas)
Code Golf StackExchange, Find the nth number where the digit sum equals the number of factors, coding challenge started Nov 28 2022.

Cf. A000005, A007953, A057532, A050689, A070274, A070275, A063737, A067077.

Select[ Range[ 1000 ], DivisorSigma[ 0, # ]==Plus@@IntegerDigits[ # ]& ] (* Harvey P. Dale, Feb 19 2004 *)

A050689 Composites whose sum of digits equals number of its prime factors, with multiplicity.

Original entry on oeis.org

12, 30, 32, 40, 102, 220, 240, 500, 600, 1002, 1012, 1104, 1152, 1210, 1320, 1500, 2001, 2002, 2020, 2040, 2120, 2240, 2300, 3010, 3040, 3300, 4032, 4100, 4320, 5100, 5200, 6400, 7000, 7200, 10001, 10002, 10011, 10030, 10040, 10080, 10140, 10220, 10304, 10800

Offset: 1

Patrick De Geest, Aug 15 1999

The sequence is infinite because there are infinitely many primes whose sum of digits is odd (see related comment in A119450). Let p be one of them, and let k be its digital sum. Then p*10^((k-1)/2) is a term. For example, 41*10^2 is a term. - Metin Sariyar, May 30 2020

			2002 is a term since 2+0+0+2 = 4, and 2002 = 2*7*11*13 has 4 prime factors.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 322 terms from Michael Turniansky)

Cf. A050690, A057531, A057532, A070274, A070275, A063737, A067077, A119450.

Select[Range[10300],!PrimeQ[#]&&PrimeOmega[#]==Total[IntegerDigits[#]]&] (* Jayanta Basu, May 30 2013 *)

isok(n) = sumdigits(n) == bigomega(n); \\ Michel Marcus, Feb 13 2017

from sympy import factorint
def ok(n): return 1 < sum(map(int, str(n))) == sum(factorint(n).values())
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, Dec 30 2021

A070275 Numbers k such that the sum of the digits of k equals the sum of the prime divisors of k.

Original entry on oeis.org

2, 3, 5, 7, 84, 160, 250, 336, 468, 735, 936, 975, 1344, 1375, 1408, 1600, 1694, 1872, 2352, 2401, 2500, 2625, 2808, 3744, 3920, 4116, 4913, 5145, 5616, 6084, 6318, 7296, 7497, 7695, 8424, 8624, 8664, 8704, 9126, 9639, 10240, 12168, 12636, 12675, 14896

Offset: 1

Benoit Cloitre, May 09 2002

If k=10^s*m is a term of the sequence where s > 0 and gcd(m,10)=1, then for each positive integer j, 10^j*m is in the sequence, because the sum of the digits of 10^j*k equals the sum of the digits of k and the sum of the distinct prime factors of 10^j*k equals the sum of the distinct prime factors of k. Also it is obvious that m isn't in the sequence. [Jahangeer Kholdi, Oct 07 2013]

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A008472, A057531, A057532, A050689, A070274, A063737, A067077, A285494.

Rest[Select[Range[20000],Total[Transpose[FactorInteger[#]][[1]]] == Total[ IntegerDigits[#]] &]] (* Harvey P. Dale, Dec 15 2010 *)

isok(n) = sumdigits(n) == vecsum(factor(n)[,1]); \\ Michel Marcus, May 27 2018

A070274 Numbers n such that sum of digits of n equals the squarefree part of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 12, 24, 100, 150, 200, 300, 320, 375, 500, 600, 640, 700, 704, 735, 832, 960, 1014, 1088, 1200, 1815, 2023, 2400, 2535, 2940, 3549, 3610, 3840, 4046, 4335, 4913, 5054, 5376, 5415, 5491, 6069, 6137, 6358, 6647, 7260, 7581, 7942, 8959, 9386

Offset: 1

Benoit Cloitre, May 09 2002

The squarefree part of x, core(x), is the smallest integer such that x*core(x) is a square.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007913, A057531, A057532, A050689, A070275, A063737, A067077, A285494.

core[n_]:=Block[{f=FactorInteger[n], p, e}, {p, e} = Transpose[f]; Times@@ (p^Mod[e, 2])]; Select[Range[10^4], core[#] == Plus @@ IntegerDigits[#] &] (* Giovanni Resta, Apr 21 2017 *)

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(core(n)==sumdigits(n[1]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

Data corrected by Giovanni Resta, Apr 21 2017

A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

Original entry on oeis.org

20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010, 10001, 10002, 10011, 10030, 10120, 11001, 11020, 11110, 12000, 20000, 20001, 21010, 30000, 30030, 30100, 100001, 100030, 100101, 100130, 100210, 100300, 101001, 101101, 101200, 102102, 110001, 110200

Offset: 1

Jonathan Frech, Apr 19 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001221, A007953, A057531, A057532, A050689, A070274, A070275, A063737, A067077.

Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&] (* or *)
nd = 10; s = 1; Sort@ Flatten@ Reap[ While[ Times @@ Prime[ Range@ s] < 10^nd, pa = IntegerPartitions[s, {nd}, Range[0, 9]]; Do[Sow@ Select[ FromDigits /@ Permutations[p], PrimeNu[#] == s &], {p, pa}]; s++]][[2, 1]] (* terms < 10^10, Giovanni Resta, Apr 21 2017 *)

isok(n) = sumdigits(n) == omega(n); \\ Michel Marcus, Apr 20 2017

More terms from Michel Marcus, Apr 20 2017

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.

A050690 Sum of digits of zero-absent composite a(n) equals number of prime factors.

Original entry on oeis.org

12, 32, 1152, 11232, 13122, 14112, 21312, 111132, 112112, 3121152, 11231232, 11354112, 812122112, 1251213312, 2211121152, 2211213312, 5121114112, 26122125312, 56321114112, 62214111232, 431711322112, 3421411213312, 11111212122112, 11112113242112

Offset: 1

Patrick De Geest, Aug 15 1999

10^11 < a(21) <= 431711322112. a(22) <= 3421411213312. - Donovan Johnson, May 30 2010

Do all terms end in 2, i.e., is each term = 2 mod 10? - Harvey P. Dale, May 26 2024

			E.g., 21312 (no zero in the string) gives 2+1+3+1+2 = 9 prime factors, namely, 2*2*2*2*2*2*3*3*37.

Cf. A050689, A057531, A057532, A070274, A070275, A063737, A067077.

t={}; Do[If[FreeQ[x=IntegerDigits[n],0]&&PrimeOmega[n]==Total[x],AppendTo[t,n]],{n,2,3220000,2}]; t (* Jayanta Basu, May 30 2013 *)

a(15)-a(20) from Donovan Johnson, May 30 2010
a(21)-a(22) confirmed by Giovanni Resta, Jun 02 2013
a(23)-a(24) from Giovanni Resta, Apr 23 2017

cp's OEIS Frontend

A057531 Numbers whose sum of digits and number of divisors are equal.

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A050689 Composites whose sum of digits equals number of its prime factors, with multiplicity.

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A070275 Numbers k such that the sum of the digits of k equals the sum of the prime divisors of k.

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A070274 Numbers n such that sum of digits of n equals the squarefree part of n.

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A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

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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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Maple

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.

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