A006753 -id:A006753 - cp's OEIS frontend

A204341 Smith numbers with either no internal digits or all internal digits are 0.

4, 22, 27, 58, 85, 94, 202, 706, 8005, 80005, 700006, 800005, 7000000000000006, 80000000000000000005, 80000000000000000000005, 70000000000000000000000000006, 80000000000000000000000000000000000005, 700000000000000000000000000000000000000000000006

Offset: 1

Arkadiusz Wesolowski, Jan 14 2012

All semiprimes of the form 7*10^n + 6 and 8*10^n + 5 are in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..34
Eric Weisstein's World of Mathematics, Smith Number

Cf. A006753.

e = 47; Sort[Join[{27}, 2*Select[Flatten[Union[Table[(2*10^n + 2)/2, {n, 0, e}], Table[(a*10^n + 13 - a)/2, {a, {5, 7, 9}}, {n, e}]]], PrimeQ], 5*Select[Table[(8*10^n + 5)/5, {n, e}], PrimeQ]]]

A232541 Multiplicative Smith numbers: Composite numbers n such that the product of nonzero digits of n = product of nonzero digits of prime factors of n.

Original entry on oeis.org

4, 6, 8, 9, 95, 159, 195, 249, 326, 762, 973, 995, 998, 1057, 1086, 1111, 1189, 1236, 1255, 1337, 1338, 1383, 1389, 1395, 1419, 1509, 2139, 2248, 2623, 2679, 2737, 2928, 2949, 3029, 3065, 3202, 3344, 3345, 3419, 3432, 3437, 3464, 3706, 3945, 4344, 4502

Offset: 1

Derek Orr, Nov 25 2013

They follow the same formula for Smith numbers, however, instead of addition, we have multiplication (only nonzero digits are included).

Trivially, prime numbers satisfy this property but are not included in the sequence.

			1236 is a member of this sequence because 1236 = 2*2*3*103 and 1*2*3*6 = 2*2*3*1*3 (zeros are not included).
998 is a member of this sequence because 998 = 2*499 and 9*9*8 = 2*4*9*9.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006753, A051801.

f[n_] := Times @@ DeleteCases[IntegerDigits[n], 0]; pFactors[n_] := Module[{f = FactorInteger[n]}, Flatten[ConstantArray @@@ f]]; Select[Range[2, 10000], ! PrimeQ[#] && f[#] == Times @@ f /@ pFactors[#] &] (* T. D. Noe, Nov 28 2013 *)
msnQ[n_]:=Times@@(Flatten[IntegerDigits/@Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]/.(0->1))==Times@@(IntegerDigits[n]/.(0->1)); Select[ Range[ 5000],CompositeQ[#]&&msnQ[#]&] (* Harvey P. Dale, Jan 15 2022 *)

import sympy
from sympy import isprime
from sympy import factorint
def DigitProd(x):
    prod = 1
    for i in str(x):
        if i != '0':
            prod *= int(i)
    return prod
def f(x):
    lst = []
    for n in range(len(list(factorint(x)))):
        lst.append(str(list(factorint(x))[n])*list(factorint(x).values())[n])
    string = ''
    for i in lst:
        string += i
    prod = 1
    for a in string:
        if a != '0':
            prod *= int(a)
    if prod == DigitProd(x):
        return True
x = 4
while x < 10**3:
    if not isprime(x):
        if f(x):
            print(x)
    x += 1

def prodPrimeDig(x):
    F=factor(x)
    T=[item for sublist in [[y[0]]*y[1] for y in F] for item in sublist]
    return prod([prod(filter(lambda a: a!=0,h.digits(base=10))) for h in T])
n=3345 #Change n for more digits
[k for k in [1..n] if prod(filter(lambda a: a!=0,k.digits(base=10)))==prodPrimeDig(k) and not(is_prime(k))] # Tom Edgar, Nov 26 2013

Extended by T. D. Noe, Nov 28 2013

A274218 Numbers such that the sum of their digits is equal to the sum of digits of their aliquot parts.

Original entry on oeis.org

6, 33, 87, 249, 303, 519, 573, 681, 843, 951, 1059, 1329, 1383, 1923, 1977, 2463, 2733, 2787, 2949, 3057, 3273, 3327, 3543, 3651, 3867, 3921, 4083, 4353, 4677, 5163, 5433, 5703, 5919, 6081, 6243, 6297, 6621, 6891, 7053, 7323, 7377, 7647, 7971, 8079, 8133, 8187

Offset: 1

Paolo P. Lava, Jun 14 2016

			Sum of digits of 8884 is 8 + 8 + 8 + 4 = 28. Its aliquot parts are 1, 2, 4, 2221, 4442 and their sum is 1 + 2 + 4 + 2 + 2 + 2 + 1 + 4 + 4 + 4 + 2 = 28.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A007953, A006753.

with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do
y:=y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,k,n; for n from 1 to q do a:=sort([op(divisors(n))]);
if T(n)=add(T(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^6);

Select[Range[10^4], Total@ IntegerDigits@ # == Total[Total@ IntegerDigits@ # & /@ Most@ Divisors@ #] &] (* Michael De Vlieger, Jun 14 2016 *)

A279314 Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9).

Original entry on oeis.org

4, 22, 27, 58, 85, 94, 105, 114, 121, 150, 166, 202, 204, 222, 224, 265, 274, 315, 319, 342, 346, 355, 378, 382, 391, 438, 445, 450, 454, 483, 517, 526, 535, 540, 560, 562, 576, 588, 612, 627, 634, 636, 640, 645, 648, 654, 663, 666, 690, 697, 706, 728, 729, 762, 778, 825, 840, 841, 852

Offset: 1

Ely Golden, Dec 09 2016

nonn

Supersequence of A006753 (Smith numbers).
Sequence is proven infinite due to the infinitude of the Smith numbers.
Can be generalized for other moduli. Setting the modulus to 1 yields the composite numbers. Setting the modulus to m (m>=2) yields the supersequence which includes the Smith numbers in base (m+1). Of course, m=1 includes all Smith numbers for any base.

			105 is a member as 105 = 3*5*7 with 105 mod 9 = 6 and (3+5+7) mod 9 = 15 mod 9 = 6.

Ely Golden, Table of n, a(n) for n = 1..10000

Cf. A006753.

Select[Range[4, 860], Function[n, And[CompositeQ@ n, Mod[#, 9] == Mod[n, 9] &@ Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)
cnnQ[n_]:=CompositeQ[n]&&Mod[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]],9]==Mod[n,9]; Select[Range[900],cnnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 17 2017 *)

isok(n) = !isprime(n) && (f=factor(n)) && ((n % 9) == (sum(k=1, #f~, f[k,1]*f[k,2]) % 9)); \\ Michel Marcus, Dec 10 2016

def factorSum(f):
    s=0
    for c in range(len(f)):
        s+=(f[c][0]*f[c][1])
    return s
#this variable affects the modulus
modulus=9
c=2
index=1
while(index<=10000):
    f=list(factor(c))
    if(((len(f)>1)|(f[0][1]>1))&(factorSum(f)%modulus==c%modulus)):
        print(str(index)+" "+str(c))
        index+=1
    c+=1
print("complete")

A308821 Semiprimes where the sum of the digits equals the difference between the prime factors.

Original entry on oeis.org

14, 95, 527, 851, 1247, 3551, 4307, 8051, 14351, 26969, 30227, 37769, 64769, 87953, 152051, 163769, 199553, 202451, 256793, 275369, 341969, 455369, 1070969, 1095953, 1159673, 1232051, 1625369, 1702769, 2005007, 2081993

Offset: 1

James Beyer, Jun 26 2019

14 is the only even number in the sequence, since 2 is the only even prime and p-2 grows much faster than the digit sum of 2p.

			14=2*7 and 1+4=7-2.
95=5*19 and 9+5=19-5.
527=17*31 and 5+2+7=31-17.

James Beyer, Table of n, a(n) for n = 1..1000
James Beyer, Numbers Like Fourteen
Wikipedia, Digit sum
Wikipedia, Semiprime

Cf. A001358, A006753, A006881.

[n:n in [2..2100000]|IsSquarefree(n) and #PrimeDivisors(n) eq 2 and PrimeDivisors(n)[2]-PrimeDivisors(n)[1] eq &+Intseq(n)]; // Marius A. Burtea, Jul 27 2019

Take[Sort@ Reap[ Do[ If[PrimeQ[q + g] && g == Total@ IntegerDigits[n = q (q + g)], Sow@n], {g, 9*9}, {q, Prime@ Range@ 2000}]][[2, 1]], 100] (* Giovanni Resta, Jul 25 2019 *)
spdpfQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},PrimeOmega[n]== 2 && Total[ IntegerDigits[n]]==f[[2]]-f[[1]]]; Select[Range[ 21*10^5],spdpfQ]// Quiet (* or *) Times@@@Select[Subsets[Prime[ Range[ 300]],{2}],#[[2]]-#[[1]]==Total[IntegerDigits[#[[1]]#[[2]]]]&] (* Harvey P. Dale, Oct 14 2021 *)

isok(n) = (bigomega(n) == 2) && (f=factor(n)) && (#f~ == 2) && (sumdigits(n) == f[2,1] - f[1,1]); \\ Michel Marcus, Jun 29 2019

A033663 Least Smith number having digital sum A033662(n).

Original entry on oeis.org

4, 27, 58, 438, 728, 378, 2944, 2576, 588, 778, 5936, 2679, 9880, 2888, 4788, 15778, 16688, 9849, 5998, 9968, 17889, 69856, 26999, 76896, 177688, 167888, 159996, 79978, 487697, 188889, 895696, 777896, 399699, 869896, 1879976, 957999, 779998

Offset: 1

David W. Wilson

Cf. A006753.

A036922 Even composite numbers whose digit sum equals the digit sum of (sum of prime factors, counted with multiplicity).

Original entry on oeis.org

4, 22, 94, 114, 150, 166, 202, 204, 222, 224, 274, 342, 346, 382, 438, 450, 454, 526, 540, 562, 612, 634, 640, 706, 852, 922, 1068, 1086, 1120, 1122, 1138, 1200, 1230, 1232, 1282, 1314, 1318, 1400, 1626, 1642, 1770, 1820, 1822, 1894, 1966, 2070, 2080

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036920, A019506.

isok(n)=my(f=factor(n)); n%2==0 && n>2 && sumdigits(sum(i=1, #f~, f[i,1]*f[i,2])) == sumdigits(n) \\ Andrew Howroyd, Jun 19 2021

Title reworded by Sean A. Irvine, Nov 30 2020

A036923 Odd composite numbers n such that the digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 105, 121, 265, 315, 355, 445, 517, 841, 913, 915, 1111, 1165, 1185, 1219, 1221, 1239, 1255, 1345, 1363, 1507, 1633, 1903, 2067, 2101, 2155, 2173, 2209, 2227, 2245, 2265, 2335, 2409, 2515, 2533, 2605, 2965, 3091, 3129, 3219, 3235, 3417, 3505, 3507

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036920, A019506.

ds[n_]:=Total[IntegerDigits[n]]; t={}; Do[If[!PrimeQ[n]&&ds[n]==ds[Total[ Times@@@FactorInteger[n]]],AppendTo[t,n]],{n,9,3508,2}]; t (* Jayanta Basu, Jun 04 2013 *)

Title reworded by Sean A. Irvine, Nov 30 2020

A036926 Digit sum of 'even' number equals digit sum of 'sum' and 'juxtaposition' of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 94, 166, 202, 274, 346, 382, 438, 454, 526, 562, 634, 706, 852, 922, 1086, 1282, 1626, 1642, 1822, 1894, 1966, 2182, 2326, 2362, 2434, 2614, 2722, 2902, 2974, 3046, 3138, 3226, 3246, 3258, 3442, 3622, 3694, 3802, 3946, 4054, 4126, 4162, 4306, 4414

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036921, A019506.

A036927 Digit sum of 'odd' number equals digit sum of 'sum' and 'juxtaposition' of its prime factors (counted with multiplicity).

Original entry on oeis.org

27, 121, 265, 355, 517, 913, 915, 1111, 1165, 1219, 1255, 1507, 1633, 1903, 2067, 2155, 2173, 2227, 2265, 2409, 2515, 2605, 2965, 3091, 3505, 3615, 3865, 4209, 4765, 4855, 5071, 5305, 6115, 6315, 6457, 7051, 7447, 7465, 7915, 8005, 8023, 9015, 9031

Offset: 1

Patrick De Geest, Jan 04 1999

Cf. A006753, A036921, A019506.

cp's OEIS Frontend

A204341 Smith numbers with either no internal digits or all internal digits are 0.

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A232541 Multiplicative Smith numbers: Composite numbers n such that the product of nonzero digits of n = product of nonzero digits of prime factors of n.

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A274218 Numbers such that the sum of their digits is equal to the sum of digits of their aliquot parts.

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A279314 Composite numbers n such that the sum of the prime factors of n, with multiplicity, is congruent to n (mod 9).

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A308821 Semiprimes where the sum of the digits equals the difference between the prime factors.

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Magma

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A033663 Least Smith number having digital sum A033662(n).

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A036922 Even composite numbers whose digit sum equals the digit sum of (sum of prime factors, counted with multiplicity).

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Author

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PARI

Extensions

A036923 Odd composite numbers n such that the digit sum of n equals digit sum of sum of its prime factors (counted with multiplicity).

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Author

Keywords

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Programs

Mathematica

Extensions

A036926 Digit sum of 'even' number equals digit sum of 'sum' and 'juxtaposition' of its prime factors (counted with multiplicity).

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