A174460 -id:A174460 - cp's OEIS frontend

A046022 Primes together with 1 and 4.

1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Eric W. Weisstein

Also the numbers which are incrementally largest values of A002034. - validated by Franklin T. Adams-Watters, Jul 13 2012
Solutions to A000005(x) + A000010(x) - x - 1 = 0. - Labos Elemer, Aug 23 2001
Also numbers m such that m, phi(m) and tau(m) form an integer triangle, where phi=A000010 is the totient and tau=A000005 the number of divisors (see also A084820). - Reinhard Zumkeller, Jun 04 2003
Terms > 1 are n such that n does not divide (n-1)!. - Benoit Cloitre, Nov 12 2003
Terms > 1 are the sum of their prime factors; 4 (= 2+2) is the only such composite number. - Stuart Orford (sjorford(AT)yahoo.co.uk), Aug 04 2005
From Jonathan Vos Post, Aug 23 2010, Robert G. Wilson v, Aug 25 2010, proof by D. S. McNeil, Aug 29 2010: (Start)
Also the numbers n which divide A001414(n), or equivalently divide A075254(n). Proof:
Theorem: for a multiset of m >= 2 integers a_i, each a_i >= 2, Product_{i=1..m} a_i >= Sum_{i=1..m} a_i, with equality only at (a_1,a_2) = (2,2).
Lemma: For integers x,y >= 2, if x > 2 or y > 2, x*y > x + y. This follows from distributing (x-1)*(y-1) > 1.
[Proof of the theorem by induction on m:
first consider m=2. We have equality at (2,2) and for any product(a_i) > 4 there is some a_i > 2, so the lemma gives a_1*a_2 > a_1+a_2.
Then the induction m->m+1: Product_{i=1..m+1} a_i = a_(m+1)*Product_{i=1..m} a_i >= a_(m+1) * Sum_{i=1..m} a_i.
Since a_(m+1) >= 2 and the sum >= 4, the lemma applies, and we find a_(m+1) * Sum+{i=1..m} a_i > a_(m+1) + Sum_{i=1..m} a_i = Sum_{i=1..m+1} a_i and thus Product_{i=1..m+1} a_i > Sum_{i=1..m+1} a_i, QED.]
For composite n > 4, applying the theorem to the multiset of prime factors with multiplicity yields n > sopfr(n), so there are no composite numbers greater than 4 such that they divide sopfr(n).
(End)
Numbers k such that the k-th Fibonacci number is relatively prime to all smaller Fibonacci numbers. - Charles R Greathouse IV, Jul 13 2012
Numbers k such that (-1)^k*floor(d(k)*(-1)^k/2) = 1, where d(k) is the number of divisors of k. - Wesley Ivan Hurt, Oct 11 2013
Also, union of odd primes (A065091) and the divisors of 4. Also, union of A008578 and 4. - Omar E. Pol, Nov 04 2013
Numbers k such that sigma(k!) is divisible by sigma((k-1)!). - Altug Alkan, Jul 18 2016

J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A002034, A046021, A001751, A178156, A174460, A000040.

a046022 n = a046022_list !! (n-1)
a046022_list = [1..4] ++ drop 2 a000040_list
-- Reinhard Zumkeller, Apr 06 2014

A046022:=n-> `if`((-1)^n*floor(numtheory[tau](n)*(-1)^n/2) = 1, n, NULL); seq(A046022(j), j=1..260); # Wesley Ivan Hurt, Oct 11 2013

max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]*m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, w]; max = w], {n, 1, 1000}]; a (* Artur Jasinski, Apr 06 2008 *)

a(n)=if(n<6,n,prime(n-2)) \\ Charles R Greathouse IV, Apr 28 2015

from sympy import prime
def A046022(n): return prime(n-2) if n>4 else n # Chai Wah Wu, Oct 17 2024

A141295(a(n)) = a(n). - Reinhard Zumkeller, Jun 23 2008
A018194(a(n)) = 1. - Reinhard Zumkeller, Mar 09 2012
A240471(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2014

Better description from Frank Ellermann, Jun 15 2001

A178193 Smith numbers of order 4.

Original entry on oeis.org

3777, 7773, 17418, 30777, 53921, 66111, 97731, 111916, 119217, 122519, 128131, 133195, 135488, 138878, 145229, 178814, 180174, 198581, 257376, 269636, 281179, 296396, 317686, 358256, 362996, 366514, 394114, 435777, 457377, 469552, 475856, 502960, 513833

Offset: 1

Paul Weisenhorn, Dec 19 2010

Composite numbers n not in A176670 such that the sum of the 4th power of the digits of n equals the sum of the 4th power of the digits of the prime factors of n (with multiplicity). A176670 lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

			3777 = 3*1259 is composite; sum of 4th power of the digits is 3^4 + 7^4 + 7^4 + 7^4 = 7284. Sum of 4th power of the digits of the prime factors 3, 1259 is 3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284. The sums are equal, so 3777 is in the sequence.
17418 = 2*3*2903 is composite; sum of 4th power of the digits is 1^4 + 7^4 + 4^4 + 1^4 + 8^4 = 6755. Sum of 4th power of the digits of the prime factors 2, 3, 2903 is 2^4 + 3^4 + 2^4 + 9^4 + 0^4 + 3^4 = 6755. The sums are equal, so 17418 is in the sequence.
269636 = 2*2*67409 is composite; sum of 4th power of the digits is 2^4 + 6^4 + 9^4 + 6^4 + 3^4 + 6^4 = 10546. Sum of 4th power of the digits of the prime factors 2, 2, 67409 (with multiplicity) is 2^4 + 2^4 + 6^4 + 7^4 + 4^4 + 0^4 + 9^4 = 10546. The sums are equal, so 269636 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178203, A178204.

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^4) == Plus @@ (fid^4)]; k = 1; lst = {}; While[k < 10^6, If[f Q@ k, AppendTo[lst, k]; Print@ k]; k++]; lst

A178213 Smith numbers of order 3.

Original entry on oeis.org

6606, 8540, 13086, 16866, 21080, 26637, 27468, 33387, 34790, 35364, 35377, 40908, 44652, 48154, 48860, 52798, 54814, 55055, 57726, 57894, 66438, 67297, 67356, 67594, 69549, 72465, 72598, 73026, 74371, 74785, 77485, 78745, 81546, 83175, 85927, 90174, 91208

Offset: 1

Paul Weisenhorn, Dec 19 2010

Composite numbers n not in A176670 such that the sum of the cubes of the digits of n equals the sum of the cubes of the digits of the prime factors of n (with multiplicity). A176670 lists composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

			6606 = 2*3*3*367 is composite; sum of cubes of the digits is 6^3+6^3+0^3+6^3 = 648. Sum of cubes of the digits of the prime factors 2, 3, 3, 367 (with multiplicity) is 2^3+3^3+3^3+3^3+6^3+7^3 = 648. The sums are equal, so 6606 is in the sequence.
21080 = 2*2*2*5*17*31 is composite; sum of cubes of the digits is 2^3+1^3+0^3+8^3+0^3 = 521. Sum of cubes of the digits of the prime factors 2, 2, 2, 5, 17, 31 (with multiplicity) is 2^3+2^3+2^3+5^3+1^3+7^3+3^3+1^3 = 521. The sums are equal, so 21080 is in the sequence.

Ely Golden and Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms 1 to 1000 by Donovan Johnson)
Ely Golden, Smith number sequence generator optimized for order 3
Eric W. Weisstein, Smith Number

Cf. A006753 (Smith numbers), A174460, A176670, A178193, A178203, A178204.

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]]; !PrimeQ@ n && id != fid && Plus @@ (id^3) == Plus @@ (fid^3)]; k = 2; lst = {}; While[k < 22002, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++]; lst
With[{k = 3}, Select[Range[10^5], Function[n, And[Total@ Map[#^k &, IntegerDigits@ n] == Total@ Map[#^k &, Flatten@ IntegerDigits[#]], Not[Sort@ DeleteCases[#, 0] &@ IntegerDigits@ n == Sort@ DeleteCases[#, 0] &@#]] &@ Flatten@ Map[IntegerDigits@ ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]]]] (* Michael De Vlieger, Dec 10 2016 *)

A178203 Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 of its prime factors without the numbers in A176670 that have the same digits as its prime factors (without the zero digits).

Original entry on oeis.org

414966, 443166, 454266, 1274664, 1371372, 1701856, 1713732, 1734616, 1771248, 1858436, 1858616, 2075664, 2624976, 3606691, 3771031, 3771301, 4266914, 4414866, 4461786, 4605146, 4670576, 4710739, 5209663, 5281767, 5434572, 5836565, 5861712, 5871968, 6046357

Offset: 1

Paul Weisenhorn, Dec 19 2010

			a(10) = 1858436 = 2*2*29*37*433;
1^5 + 3^5 + 4^5 + 5^5 + 6^5 + 2*8^5 = 3*2^5 + 3*3^5 + 4^5 + 7^5 + 9^5 = 77705.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
S. S. Gupta, Smith Numbers.
Eric Weisstein's World of Mathematics, Smith number.
Wikipedia, Smith number.
A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.

Cf. A006753 (Smith numbers), A176670, A174460, A178213, A178193, A178204.

a(21) corrected by Donovan Johnson, Jan 02 2013

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).

Original entry on oeis.org

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

A178156 Numbers m such that (m-1)! is not divisible by m^2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163

Offset: 1

Reinhard Zumkeller, Dec 17 2010

nonn

Union of {8, 9} and A001751.

G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1972), Part Eight, Chap. 3, Sect. 1, Problem 133b.

Cf. A046022, A000142, A174460.

Cf. A001751.

import Data.List (insert)
a178156 n = a178156_list !! (n-1)
a178156_list = insert 9 $ insert 8 a001751_list
-- Reinhard Zumkeller, Oct 14 2014

Select[Range[200],!Divisible[(#-1)!,#^2]&] (* Harvey P. Dale, Mar 06 2016 *)

for(m=1,3e2,if((m-1)!%m^2,print1(m", "))) \\ Charles R Greathouse IV, Aug 21 2011

from sympy import primepi
def A178156(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-primepi(x)-primepi(x>>1)-(x>=8)-(x>=9))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024

Entries corrected by Charles R Greathouse IV, Aug 21 2011

A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

Original entry on oeis.org

0, 1, 10, 35, 100, 152, 350, 377, 452, 539, 709, 1000, 1299, 1398, 1439, 1519, 1520, 1569, 1591, 1679, 1965, 2599, 2838, 3332, 3500, 3598, 3770, 4520, 4586, 4754, 4854, 5390, 5501, 5835, 5857, 6388, 6595, 6735, 6861, 6951, 7090, 7349, 7887, 8395, 9795, 10000, 10056, 10159, 10389, 11055, 11091, 12990, 12999

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, r >= 1.

			377^2 = 142129, A003132(377) = 3^2 + 7^2 + 7^2 = 107, A003132(142129) = 1^2 + 4^2 + 2^2 + 1^2 + 2^2 + 9^2 = 107.

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

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309884.

[0] cat [k:k in [1..13000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^2)]]; // Marius A. Burtea, Aug 24 2019

filter:= proc(n) local t;
  add(t^2, t = convert(n,base,10)) = add(t^2, t = convert(n^2,base,10))
end proc:
select(filter, [$0..20000]); # Robert Israel, Apr 30 2023

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 13000], digSum[#] == digSum[#^2] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 30000, if(norml2(digits(i^2)) == norml2(digits(i)), print1(i, ", ")))

def A003132(n):
    s = 0
    while n > 0:
        s, n = s+(n%10)**2, n//10
    return s
n, a = 0, 0
while n < 50:
    if A003132(a) == A003132(a*a):
        n = n+1
        print(n,a)
    a = a+1 # A.H.M. Smeets, Aug 23 2019

A309884 Numbers k such that A003132(k^3) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

Original entry on oeis.org

0, 1, 10, 74, 100, 740, 1000, 3488, 7400, 10000, 23658, 30868, 34880, 47508, 48517, 52187, 58947, 59468, 67685, 68058, 74000, 76814, 78368, 78845, 84878, 100000, 108478, 145877, 149217, 163871, 179685, 186884, 188647, 218977, 219878, 236580, 238758, 248967, 278638, 292597, 308680

Offset: 1

Antonio Roldán, Aug 21 2019

If k is in the sequence, then so are k*10^r, with r >= 1.

			74^3 = 405224, A003132(74) = 7^2 + 4^2 = 65, A003132(405224) = 4^2 + 0^2 + 5^2 + 2^2 + 2^2 + 4^2 = 65.

Cf. A003132, A058369, A070276, A174460, A176670, A178213, A307735, A309883.

[0] cat [k:k in [1..310000]| &+[c^2: c in Intseq(k)] eq &+[c^2: c in Intseq(k^3)]]; // Marius A. Burtea, Aug 26 2019

digSum[n_] := Total[IntegerDigits[n]^2]; Select[Range[0, 310000], digSum[#] == digSum[#^3] &] (* Amiram Eldar, Aug 22 2019 *)

for(i = 0, 400000, if(norml2(digits(i^3)) == norml2(digits(i)), print1(i, ", ")))

cp's OEIS Frontend

A046022 Primes together with 1 and 4.

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A178193 Smith numbers of order 4.

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A178213 Smith numbers of order 3.

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A178203 Smith numbers of order 5; composite numbers n such that sum of digits^5 equal sum of digits^5 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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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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A178156 Numbers m such that (m-1)! is not divisible by m^2.

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A309883 Numbers k such that A003132(k^2) = A003132(k), where A003132(n) is the sum of the squares of the digits of n.

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