A019506 -id:A019506 - cp's OEIS frontend

A050223 Digit sums of hoax numbers A019506.

4, 13, 12, 13, 13, 10, 7, 13, 4, 9, 7, 13, 13, 11, 13, 12, 13, 13, 10, 13, 13, 13, 10, 15, 13, 15, 17, 15, 12, 13, 13, 13, 13, 15, 13, 14, 15, 11, 15, 12, 15, 15, 9, 13, 12, 15, 15, 22, 18, 14, 15, 22, 13, 15, 13, 21, 22, 22, 15, 4, 9, 14, 13, 13, 13, 13, 7, 12, 9, 17, 13

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hoax Number.

Cf. A007953, A019506.

Cf. A202393, A050218.

ishoax(m) = !isprime(m) && (sumdigits(m) == vecsum(apply(sumdigits, factor(m)[, 1]))); \\ A019506
lista(nn) = apply(sumdigits, select(ishoax, [1..nn])); \\ Michel Marcus, Feb 24 2023

a(n) = A007953(A019506(n)).

A202393 Digital root of Hoax Numbers A019506.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Dec 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hoax Number.

Cf. A050223, A202388.

a(n) = A010888(A019506(n)).

A329935 Numbers k such that k and k+1 are both hoax numbers (A019506).

Original entry on oeis.org

84, 516, 644, 860, 2325, 3344, 4188, 4980, 5268, 5484, 6259, 6603, 6692, 6980, 7051, 7195, 8076, 8420, 9716, 10704, 11774, 12795, 12955, 12956, 13747, 14475, 14715, 14724, 16473, 17148, 17149, 17225, 17661, 19175, 21828, 22143, 22347, 24259, 24272, 24980

Offset: 1

Amiram Eldar, Nov 24 2019

Analogous to A050219 (smaller of Smith brothers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).

			84 is in the sequence since 84 is a hoax number: 84 = 2^2 * 3 * 7 and 8 + 4 = 2 + 3 + 7 = 12, and 85 = 84 + 1 is also a hoax number: 85 = 5 * 17 and 8 + 5 = 5 + 1 + 7 = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 27, entry 85.

Cf. A006753, A019506, A050219, A235766.

digitSum[n_]  := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[ digitSum /@ FactorInteger[n][[;; , 1]] ] == digitSum[n]; seq = {}; isHoax1 = hoaxQ[1]; Do[isHoax2 = hoaxQ[n]; If[isHoax1 && isHoax2, AppendTo[seq, n-1]]; isHoax1 = isHoax2, {n, 2, 25000}]; seq

A357034 a(n) is the smallest number with exactly n divisors that are hoax numbers (A019506).

Original entry on oeis.org

1, 22, 308, 638, 3696, 4212, 18480, 26400, 55080, 52800, 73920, 108108, 220320, 216216, 275400, 324324, 432432, 550800, 734400, 1908000, 1144800, 1101600, 1377000, 1652400, 3027024, 2203200, 4039200, 2754000, 3304800, 5724000, 6528600, 9180000, 8586000, 5508000

Offset: 0

Marius A. Burtea, Sep 20 2022

			1 has no divisors in A019506, so a(0) = 1;
22 has divisors 1, 2, 11, 22, and 22 = A019506(1), so a(1) = 22.
308 has divisors 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308 and 22 = A019506(1), 308 = A019506(14), so a(2) = 308.

Cf. A019506.

hoax:=func; a:=[]; for n in [0..33] do k:=1; while #[d:d in Set(Divisors(k)) diff {1}|hoax(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

digitSum[n_] := Total @ IntegerDigits[n]; hoaxQ[n_] := CompositeQ[n] && Total[digitSum /@ FactorInteger[n][[;; , 1]]] == digitSum[n]; f[n_] := DivisorSum[n, 1 &, hoaxQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[10, 10^5] (* Amiram Eldar, Sep 26 2022 *)

A006753 Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity).

Original entry on oeis.org

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219

Offset: 1

N. J. A. Sloane

Of course primes also have this property, trivially.
a(133809) = 4937775 is the first Smith number historically: 4937775 = 3*5*5*65837 and 4+9+3+7+7+7+5 = 3+5+5+(6+5+8+3+7) = 42, Albert Wilansky coined the term Smith number when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.
There are 248483 7-digit Smith numbers, corresponding to US phone numbers without area codes (like 4937775). - Charles R Greathouse IV, May 19 2013
A007953(a(n)) = Sum_{k=1..A001222(a(n))} A007953(A027746(a(n),k)), and A066247(a(n))=1. - Reinhard Zumkeller, Dec 19 2011
3^3, 3^6, 3^9, 3^27 are in the sequence. - Sergey Pavlov, Apr 01 2017
As mentioned by Giovanni Resta, there are no other terms of the form 3^t for 0 < t < 300000 and, probably, no other terms of such form for t >= 300000. It seems that, if there exists any other term of form 3^t with integer t, then t == 0 (mod 3) or, perhaps, t = {3^k; 2*3^k} where k is an integer, k > 10. - Sergey Pavlov, Apr 03 2017

			58 = 2*29; sum of digits of 58 is 13, sum of digits of 2 + sum of digits of 29 = 2+11 is also 13.

M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 300.
R. K. Guy, Unsolved Problems in the Theory of Numbers, Section B49.
C. A. Pickover, "A Brief History of Smith Numbers" in "Wonders of Numbers: Adventures in Mathematics, Mind and Meaning", pp. 247-248, Oxford University Press, 2000.
J. E. Roberts, Lure of the Integers, pp. 269-270 MAA 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. D. Spencer, Key Dates in Number Theory History, Camelot Pub. Co. FL, 1995, pp. 94.
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.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 180.

T. D. Noe, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Smith Numbers and Rhonda Numbers
C. K. Caldwell, The Prime Glossary, Smith number
P. J. Costello, Smith Numbers
M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
Ely Golden, General program for generating Smith number sequences
S. S. Gupta, Smith Numbers
T. Jason, Smith number
Madras Math's Amazing Number Facts, Smith Numbers.
Wayne L. McDaniel, The Existence of Infinitely Many k-Smith Numbers, The Fibonacci Quarterly, 25(1), 76-80, (1987).
Sham Oltikar, and Keith Wayland, Construction of Smith Numbers, Mathematics Magazine, vol. 56(1), 1983, pp. 36-37.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Carlos Rivera, Problem 107: Consecutive Smith numbers, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 108: Methods for generating Smith numbers, The Prime Puzzles and Problems Connection.
W. Schneider, Smith Numbers
Eric Weisstein's World of Mathematics, Smith Number
Wikipedia, Smith number
A. Wilansky, Smith numbers, Two-Year Coll. Math. J., 13 (1982), p. 21.
A. Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6

Cf. A002808, A007953, A019506, A050218, A050224, A050255, A098834-A098840, A103123-A103126, A104166-A104171, A104390, A104391, A202387, A202388.

a006753 n = a006753_list !! (n-1)
a006753_list = [x | x <- a002808_list,
                    a007953 x == sum (map a007953 (a027746_row x))]
-- Reinhard Zumkeller, Dec 19 2011

q:= n-> not isprime(n) and (s-> s(n)=add(s(i[1])*i[2], i=
     ifactors(n)[2]))(h-> add(i, i=convert(h, base, 10))):
select(q, [$1..2000])[];  # Alois P. Heinz, Apr 22 2021

fQ[n_] := !PrimeQ@ n && n>1 && Plus @@ Flatten[ IntegerDigits@ Table[ #[[1]], {#[[2]] }] & /@ FactorInteger@ n] == Plus @@ IntegerDigits@ n; Select[ Range@ 1200, fQ]

isA006753(n) = if(isprime(n), 0, my(f=factor(n)); sum(i=1,#f[,1], sumdigits(f[i,1])*f[i,2]) == sumdigits(n)); \\ Charles R Greathouse IV, Jan 03 2012; updated by Max Alekseyev, Oct 21 2016

from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def ok(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)
print(list(filter(ok, range(1220)))) # Michael S. Branicky, Apr 22 2021

is_A006753 = lambda n: n > 1 and not is_prime(n) and sum(n.digits()) == sum(sum(p.digits())*m for p,m in factor(n)) # D. S. McNeil, Dec 28 2010

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

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 *)

A202387 Squarefree Smith numbers, cf. A006753.

Original entry on oeis.org

22, 58, 85, 94, 166, 202, 265, 274, 319, 346, 355, 382, 391, 438, 454, 483, 517, 526, 535, 562, 627, 634, 645, 654, 663, 690, 706, 762, 778, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1507, 1581, 1626, 1633, 1642, 1678, 1795, 1822

Offset: 1

Reinhard Zumkeller, Dec 19 2011

Intersection of A006753 and A005117;
also squarefree hoax numbers: intersection of A019506 and A005117;
squarefree composite numbers m such that sum of digits of m = sum of digits of all prime factors of m.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smith Number
Eric Weisstein's World of Mathematics, Hoax Number
Wikipedia, Smith number

Cf. A007953, A027746, A120944.

a202387 n = a202387_list !! (n-1)
a202387_list = [x | x <- a120944_list,
                    a007953 x == sum (map a007953 (a027746_row x))]

A329936 Binary hoax numbers: composite numbers k such that sum of bits of k equals the sum of bits of the distinct prime divisors of k.

Original entry on oeis.org

4, 8, 9, 15, 16, 32, 45, 49, 50, 51, 55, 64, 75, 85, 100, 117, 126, 128, 135, 153, 159, 162, 171, 185, 190, 200, 205, 207, 215, 222, 225, 238, 246, 249, 252, 253, 256, 287, 303, 319, 324, 333, 338, 350, 369, 374, 378, 380, 400, 407, 438, 442, 444, 469, 471

Offset: 1

Amiram Eldar, Nov 24 2019

Analogous to A278909 (binary Smith numbers) as A019506 (hoax numbers) is analogous to A006753 (Smith numbers).

Includes all the powers of 2 except for 1 and 2.

			4 = 2^2 is in the sequence since the binary representation of 4 is 100 and 1 + 0 + 0 = 1, and the binary representation of 2 is 10 and 1 + 0 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006753, A019506, A278909.

filter:= proc(n)
if isprime(n) then return false fi;
convert(convert(n,base,2),`+`) = add(convert(convert(t,base,2),`+`),t=numtheory:-factorset(n))
end proc:
select(filter, [$2..1000]); # Robert Israel, Nov 28 2019

binWt[n_] := Total @ IntegerDigits[n, 2]; binHoaxQ[n_] := CompositeQ[n] && Total[binWt /@ FactorInteger[n][[;; , 1]]] == binWt[n]; Select[Range[500], binHoaxQ]

is(n)= my(f=factor(n)[,1]); sum(i=1,#f, hammingweight(f[i]))==hammingweight(n) && !isprime(n) \\ Charles R Greathouse IV, Nov 28 2019

A036924 Digit sum of composite even number equals digit sum of juxtaposition of its prime factors (counted with multiplicity).

Original entry on oeis.org

4, 22, 58, 94, 166, 202, 274, 346, 378, 382, 438, 454, 526, 562, 576, 588, 634, 636, 648, 654, 666, 690, 706, 728, 762, 778, 852, 922, 958, 1086, 1282, 1284, 1376, 1626, 1642, 1678, 1736, 1776, 1822, 1842, 1858, 1872, 1894, 1908, 1952, 1962, 1966, 2038

Offset: 1

Patrick De Geest, Jan 04 1999

Even Smith numbers. - Robert Israel, Aug 24 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006753, A019506, A036925.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  convert(convert(n,base,10),`+`) = convert(map(t -> t[2]*convert(convert(t[1],base,10),`+`), F),`+`)
end proc:
select(filter, [seq(i,i=4..10000,2)]); # Robert Israel, Aug 24 2024

d[n_] := IntegerDigits[n]; co[n_,k_] := Nest[Flatten[d[{#,n}]]&, n, k-1]; t={}; Do[If[!PrimeQ[n] && Total[d[n]] == Total[Flatten[d[co@@@FactorInteger[n]]]], AppendTo[t,n]], {n,4,2040,2}]; t (* Jayanta Basu, Jun 04 2013 *)

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

cp's OEIS Frontend

A050223 Digit sums of hoax numbers A019506.

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A202393 Digital root of Hoax Numbers A019506.

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A329935 Numbers k such that k and k+1 are both hoax numbers (A019506).

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A357034 a(n) is the smallest number with exactly n divisors that are hoax numbers (A019506).

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A006753 Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity).

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

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A202387 Squarefree Smith numbers, cf. A006753.

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A329936 Binary hoax numbers: composite numbers k such that sum of bits of k equals the sum of bits of the distinct prime divisors of k.

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