A038550 -id:A038550 - cp's OEIS frontend

A267697 Numbers with 7 odd divisors.

729, 1458, 2916, 5832, 11664, 15625, 23328, 31250, 46656, 62500, 93312, 117649, 125000, 186624, 235298, 250000, 373248, 470596, 500000, 746496, 941192, 1000000, 1492992, 1771561, 1882384, 2000000, 2985984, 3543122, 3764768, 4000000, 4826809, 5971968, 7086244, 7529536, 8000000, 9653618

Offset: 1

Omar E. Pol, Apr 03 2016

Positive integers that have exactly seven odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 7 subparts. - Omar E. Pol, Dec 28 2016
Numbers that can be formed in exactly 6 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018
Numbers of the form p^6 * 2^k where p is an odd prime. - David A. Corneth, Aug 14 2018

David A. Corneth, Table of n, a(n) for n = 1..10000

Column 7 of A266531.
Cf. A001227, A030516, A038547, A085966, A236104, A237593, A279387.
Numbers with k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, this sequence, A267891, A267892, A267893.

isok(n) = sumdiv(n, d, (d%2)) == 7; \\ Michel Marcus, Apr 03 2016

upto(n) = {my(res = List()); forprime(p = 3, sqrtnint(n, 6), listput(res, p^6)); q = #res; for(i = 1, q, odd = res[i]; for(j = 1, logint(n \ odd, 2), listput(res, odd <<= 1))); listsort(res); res} \\ David A. Corneth, Aug 14 2018

from sympy import integer_log, primerange, integer_nthroot
def A267697(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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-sum(integer_log(x//p**6,2)[0]+1 for p in primerange(3,integer_nthroot(x,6)[0]+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

A001227(a(n)) = 7.

Sum_{n>=1} 1/a(n) = 2 * P(6) - 1/32 = 0.00289017370127..., where P(6) is the value of the prime zeta function at 6 (A085966). - Amiram Eldar, Sep 16 2024

More terms from Michel Marcus, Apr 03 2016

A267891 Numbers with 8 odd divisors.

Original entry on oeis.org

105, 135, 165, 189, 195, 210, 231, 255, 270, 273, 285, 297, 330, 345, 351, 357, 375, 378, 385, 390, 399, 420, 429, 435, 455, 459, 462, 465, 483, 510, 513, 540, 546, 555, 561, 570, 594, 595, 609, 615, 621, 627, 645, 651, 660, 663, 665, 690, 702, 705, 714, 715, 741, 750, 756, 759, 770, 777, 780, 783, 795, 798, 805, 837

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly eight odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 8 subparts. - Omar E. Pol, Dec 29 2016
Numbers n such that A000265(n) has prime signature {7} or {3,1} or {1,1,1}, i.e., is in A092759 or A065036 or A007304. - Robert Israel, Mar 15 2018
Numbers that can be formed in exactly 7 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

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

Column 8 of A266531.
Cf. A000265, A001227, A007304, A038547, A065036, A092759, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, this sequence, A267892, A267893.

[n: n in [1..1000] | #[d: d in Divisors(n) | IsOdd(d)] eq 8]; // Bruno Berselli, Apr 04 2016

filter:= proc(n) local r;
  r:= n/2^padic:-ordp(n,2);
  numtheory:-tau(r)=8
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 15 2018

Select[Range@ 840, Length@ Select[Divisors@ #, OddQ] == 8 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 8; \\ after Michel Marcus

A001227(a(n)) = 8.

A267892 Numbers with 9 odd divisors.

Original entry on oeis.org

225, 441, 450, 882, 900, 1089, 1225, 1521, 1764, 1800, 2178, 2450, 2601, 3025, 3042, 3249, 3528, 3600, 4225, 4356, 4761, 4900, 5202, 5929, 6050, 6084, 6498, 6561, 7056, 7200, 7225, 7569, 8281, 8450, 8649, 8712, 9025, 9522, 9800, 10404, 11858, 12100, 12168, 12321, 12996, 13122, 13225, 14112, 14161, 14400, 14450, 15129

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly nine odd divisors.
Numbers k such that the symmetric representation of sigma(k) has 9 subparts. - Omar E. Pol, Dec 29 2016
From Robert Israel, Dec 29 2016: (Start)
Numbers k such that A000265(k) is in A030627.
Numbers of the form 2^j*p^8 or 2^j*p^2*q^2 where p and q are distinct odd primes. (End)
Numbers that can be formed in exactly 8 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

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

Column 9 of A266531.
Cf. A001227, A038547, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, this sequence, A267893.
Cf. A000265, A030627.

A:=List([1..16000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
a:=Filtered([1..Length(B)],i->Length(B[i])=9); # Muniru A Asiru, Aug 14 2018

N:= 10^5: # to get all terms <= N
P:= select(isprime, [seq(i,i=3..floor(sqrt(N)/2),2)]);
Aodd:= select(`<=`,map(t -> t^8, P) union {seq(seq(P[i]^2*P[j]^2,i=1..j-1),j=1..nops(P))}, N):
A:= map(t -> seq(2^j*t,j=0..ilog2(N/t)), Aodd):
sort(convert(A,list)); # Robert Israel, Dec 29 2016

Select[Range[5^6], Length[Divisors@ # /. d_ /; EvenQ@ d -> Nothing] == 9 &] (* Michael De Vlieger, Apr 04 2016 *)
Select[Range[16000],Total[Boole[OddQ[Divisors[#]]]]==9&] (* Harvey P. Dale, May 12 2019 *)

isok(n) = sumdiv(n, d, (d%2)) == 9; \\ after Michel Marcus.

A001227(a(n)) = 9.

Sum_{n>=1} 1/a(n) = (P(2)-1/4)^2 - P(4) + 2*P(8) + 7/128 = 0.026721189882055998428..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 16 2024

A267893 Numbers with 10 odd divisors.

Original entry on oeis.org

405, 567, 810, 891, 1053, 1134, 1377, 1539, 1620, 1782, 1863, 1875, 2106, 2268, 2349, 2511, 2754, 2997, 3078, 3240, 3321, 3483, 3564, 3726, 3750, 3807, 4212, 4293, 4375, 4536, 4698, 4779, 4941, 5022, 5427, 5508, 5751, 5913, 5994, 6156, 6399, 6480, 6642, 6723, 6875, 6966, 7128, 7203, 7209, 7452, 7500, 7614, 7857, 8125

Offset: 1

Omar E. Pol, Apr 03 2016

nonn

Positive integers that have exactly 10 odd divisors.
Numbers n such that the symmetric representation of sigma(n) has 10 subparts. - Omar E. Pol, Dec 29 2016
Numbers that can be formed in exactly 9 ways by summing sequences of 2 or more consecutive positive integers. - Julie Jones, Aug 13 2018

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Column 10 of A266531.
Cf. A001227, A038547, A237593, A279387.
Numbers with exactly k odd divisors (k = 1..10): A000079, A038550, A072502, apparently A131651, A267696, A230577, A267697, A267891, A267892, this sequence.

A:=List([1..10000],n->DivisorsInt(n));; B:=List([1..Length(A)],i->Filtered(A[i],IsOddInt));;
a:=Filtered([1..Length(B)],i->Length(B[i])=10); # Muniru A Asiru, Aug 14 2018

Select[Range@ 8125, Length@ Select[Divisors@ #, OddQ] == 10 &] (* Michael De Vlieger, Dec 30 2016 *)

isok(n) = sumdiv(n, d, (d%2)) == 10; \\ after Michel Marcus

A001227(a(n)) = 10.

A171565 Number of partitions of n into odd divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 1, 5, 3, 2, 5, 2, 3, 14, 1, 2, 12, 2, 5, 18, 3, 2, 9, 7, 3, 23, 5, 2, 54, 2, 1, 26, 3, 26, 35, 2, 3, 30, 9, 2, 72, 2, 5, 286, 3, 2, 17, 9, 18, 38, 5, 2, 93, 38, 9, 42, 3, 2, 275, 2, 3, 493, 1, 44, 108, 2, 5, 50, 110, 2, 117, 2, 3, 698, 5, 50, 126, 2, 17, 239, 3, 2, 375, 56

Offset: 0

Reinhard Zumkeller, Dec 11 2009

nonn

a(2*n+1) = A018818(2*n+1), a(A005408(n))=A018818(A005408(n));
a(2^k) = 1, a(A000079(n))=1;
for odd primes p: a(p*2^k) = 2^k + 1,
especially for n>1: a(A000040(n))=2, a(A100484(n))=3, a(A001749(n))=5.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A038550, A065091.

with(numtheory):
a:= proc(n) option remember; local b, l; l, b:= sort(
      [select(x-> is(x:: odd), divisors(n))[]]),
      proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
        b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
      end; b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 30 2017

a[0] = 1; a[n_] := a[n] = Module[{b, l}, l = Select[Divisors[n], OddQ]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i < 1, 0, b[m, i-1] + If[l[[i]] > m, 0, b[m - l[[i]], i]]]]; b[n, Length[l]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 11 2017, after Alois P. Heinz *)

a(n) = f(n,n,1) with f(n,m,k) = if k<=m then f(n,m,k+2)+f(n,m-k,k)*0^(n mod k) else 0^m.

A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 81, 98, 100, 121, 144, 162, 169, 196, 200, 225, 242, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1058, 1089, 1152, 1156, 1225, 1250, 1296, 1352, 1369

Offset: 1

Benoit Cloitre, Apr 18 2002

Previous name: sum(d|n,6d/(2+mu(d))) is odd, where mu(.) is the Moebius function, A008683.
From Peter Munn, Jul 06 2020: (Start)
Numbers that have an odd number of odd nonsquarefree divisors.
[Proof of equivalence to the name, where m denotes a positive integer:
(1) These properties are equivalent: (a) m has an even number of odd squarefree divisors; (b) m has a nontrivial odd part.
(2) These properties are equivalent: (a) m has an odd number of odd divisors; (b) the odd part of m is square.
(3) m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are both true or both false.
(4) The trivial odd part, 1, is a square, so (1)(b) and (2)(b) cannot both be false, which (from (1), (2)) means (1)(a) and (2)(a) cannot both be false.
(5) From (3), (4), m satisfies the condition at the start of this comment if and only if (1)(a) and (2)(a) are true.
(6) m satisfies the condition in the name if and only if (1)(b) and (2)(b) are true, which (from (1), (2)) is equivalent to (1)(a) and (2)(a) being true, and hence from (5), to m satisfying the condition at the start of this comment.]
(End)
Numbers whose sum of non-unitary divisors (A048146) is odd. - Amiram Eldar, Sep 16 2024

			To determine the odd part of 18, remove all factors of 2, leaving 9. 9 is a nontrivial square, so 18 is in the sequence. - _Peter Munn_, Jul 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd part.

A000265, A008683 are used in definitions of this sequence.
Lists of numbers whose odd part satisfies other conditions: A028982 (square), A028983 (nonsquare), A029747 (less than 6), A029750 (less than 8), A036349 (even number of prime factors), A038550 (prime), A070776 U {1} (power of a prime), A072502 (square of a prime), A091067 (has form 4k+3), A091072 (has form 4k+1), A093641 (noncomposite), A105441 (composite), A116451 (greater than 4), A116882 (less than or equal to even part), A116883 (greater than or equal to even part), A122132 (squarefree), A229829 (7-rough), A236206 (11-rough), A260488\{0} (has form 6k+1), A325359 (proper prime power), A335657 (odd number of prime factors), A336101 (prime power).
Cf. A048146, A091476.

Select[Range[1000], (odd = #/2^IntegerExponent[#, 2]) > 1 && IntegerQ @ Sqrt[odd] &] (* Amiram Eldar, Sep 29 2020 *)

upto(n) = { my(res = List()); forstep(i = 3, sqrtint(n), 2, for(j = 0, logint(n\i^2, 2), listput(res, i^2<David A. Corneth, Sep 28 2020

Sum_{n>=1} 1/a(n) = 2 * Sum_{k>=1} 1/(2*k+1)^2 = Pi^2/4 - 2 = A091476 - 2 = 0.467401... - Amiram Eldar, Feb 18 2021

New name from Peter Munn, Jul 06 2020

A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 14, 17, 20, 24, 28, 31, 34, 40, 48, 56, 62, 68, 80, 96, 112, 124, 127, 136, 160, 192, 224, 248, 254, 257, 272, 320, 384, 448, 496, 508, 514, 544, 640, 768, 896, 992, 1016, 1028, 1088, 1280, 1536, 1792, 1984, 2032, 2056, 2176, 2560, 3072, 3584, 3968, 4064, 4112, 4352, 5120, 6144, 7168, 7936, 8128, 8191

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers k such that A000265(k) is either in A000668 or in A019434.

Product of any two terms (whether distinct or not) can be found in A335912.

Row 1 of A335910.
Union of A334101 and A335431. Subsequence of A038550.
Cf. A000668, A000668, A335885, A335912.
Cf. A141453 (after its initial 2, gives the primes present in this sequence).

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335911(n) = (1==A335885(n));

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2, n-1));
isA335911(n) = (isA000668(A000265(n))||isA019434(A000265(n)));

A336897 Infinite sum of the natural numbers, compacted (see comments for an explanation).

Original entry on oeis.org

3, 7, 11, 24, 46, 29, 376, 134, 73, 158, 13504, 1388, 718, 734, 373, 758, 328192, 1667, 55456, 3602, 123712, 2063, 4138, 8324, 68896, 4442, 3831808, 3579392, 8017, 521408, 66328, 16622, 8317, 540608, 1130368, 18182, 36412, 73016, 36604, 9161, 295264, 9293, 74488, 74744, 150256, 37724, 5357056, 11489, 348602368

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 07 2020

nonn

The sequence is a subset of A038550, numbers that can be expressed as the sum of k>1 consecutive positive integers in only one way. If the successive terms of the present sequence are expressed as the sum of k>1 consecutive integers and added, the result will be 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11... (conjectured to extend ad infinitum).
Any sequence with this property has distinct positive terms. This is the lexicographically earliest sequence with this property.
The sequence behaves in a strange way: most of its terms are the sum of 2 or 3 consecutive integers, but sometimes huge "gaps" appear. The term a(65) = 34404982325248 is equal to 92911 + 92912 + 92913 + ... + 8295695 + 8295696 + 8295697; but a(66) = 16591397 is the sum of the next 2 consecutive terms only, 8295698 and 8295699. The term a(71) = 7893201574690816 is the sum of more than 10^8 consecutive integers(!):  8299930 + 8299931 + 8299932 + ... + 125917798.
Further indices of records are 77, 81, 89, 91, 101, 102, 106, 145, 149, 153, 157, 169, 173, 181, 191, 201, ... with a(201) ~ 10^562. - M. F. Hasler, Aug 29 2020

			The 1st term is 3 = 1 + 2.
The 2nd term is 7 = 3 + 4.
The 3rd term is 11 = 5 + 6.
The 4th term is 24 = 7 + 8 + 9.
The 5th term is 46 = 10 + 11 + 12 + 13.
The 6th term is 29 = 14 + 15, etc.

M. F. Hasler, Table of n, a(n) for n = 1..200

Cf. A038550 (numbers that can be expressed as the sum of k>1 consecutive integers in only one way).

(A336897_vec(N,s=0)=vector(N,n,my(o=s++);while(!is_A038550(o+=s++),);o)) (60) \\ slow for N >= 65. - M. F. Hasler, Aug 29 2020

A267895 Numbers whose number of odd divisors is prime.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 56, 58, 59, 61, 62, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 109

Offset: 1

Omar E. Pol, Apr 04 2016

nonn

All odd primes are in the sequence.

			The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The odd divisors of 36 are 1, 3, 9. There are 3 odd divisors of 36 and 3 is prime, so 36 is in the sequence.

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

Complement of A267894.

Cf. A000040, A001227, A028982, A028983, A038550, A065091, A072502, A266531, A267696, A267697.

Select[Range[100], PrimeQ[DivisorSigma[0, #/2^IntegerExponent[#, 2]]] &] (* Amiram Eldar, Dec 03 2020 *)

isok(n) = isprime(sumdiv(n, d, (d%2))); \\ Michel Marcus, Apr 04 2016

A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.

Original entry on oeis.org

6, 11, 12, 13, 23, 47, 192, 193, 383, 786432

Offset: 1

Juri-Stepan Gerasimov, Aug 22 2016

nonn

Conjecture: this sequence is finite.
Any further terms are greater than 10^11. - Charles R Greathouse IV, Aug 22 2016
From Robert Israel, Apr 27 2020: (Start)
Each term is either of the form 3*2^k with 3*2^k-1 and 3*2^k+1 prime, or 3*2^k-1 with 3*2^k-1 prime and 3*2^(k-1)-1 prime, or 3*2^k+1 with 3*2^k+1 prime and 3*2^(k-1)+1 prime.
Any further terms > 10^2000.
(End)

			6 is in this sequence because the largest odd divisor of 5 is 5, the largest odd divisor of 6 is 3 and the largest odd divisor of 7 is 7, and all three are prime.

Supersequence of A181493. Subsequence of A038550.

Cf. A001227, A181490, A181491, A181492, A275418.

[n: n in [2..3000000] | NumberOfDivisors(2*(n-1))- NumberOfDivisors(n-1)eq 2 and NumberOfDivisors(2(n))-NumberOfDivisors(n) eq 2 and NumberOfDivisors(2*(n+1))- NumberOfDivisors(n+1) eq 2];

Res:= 6:
for k from 2  while length(3*2^k-1)<1000 do
  if (isprime(3*2^k-1) and isprime(3*2^(k-1)-1)) then Res:= Res, 3*2^k-1
    fi;
  if (isprime(3*2^k-1) and isprime(3*2^k+1)) then Res:= Res, 3*2^k;
    fi;
  if (isprime(3*2^k+1) and isprime(3*2^(k-1)+1)) then Res:= Res, 3*2^k+1;
    fi;
od:
Res; # Robert Israel, Apr 27 2020

Select[Range[2, 10^6], Function[n, Times @@ Boole@ PrimeQ@ Map[First@ Reverse@ DeleteCases[Divisors@ #, d_ /; EvenQ@ d] &, n + Range[-1, 1]] == 1]] (* Michael De Vlieger, Aug 22 2016 *)
SequencePosition[Table[If[PrimeQ[Max[Select[Divisors[n],OddQ]]],1,0],{n,800000}],{1,1,1}][[;;,1]]+1 (* Harvey P. Dale, Jun 27 2023 *)

isA038550(n)=isprime(n>>valuation(n,2))
is(n)=isA038550(n-1) && isA038550(n) && isA038550(n+1) \\ Charles R Greathouse IV, Aug 22 2016

forprime(p=2,1e11, my(a=isA038550(p-1),b=isA038550(p+1)); if(a && isA038550(p-2), print1(p-1", ")); if(a && b, print1(p", ")); if(b && isA038550(p+2), print1(p+1", "))) \\ may print numbers several times, but won't skip numbers; Charles R Greathouse IV, Aug 22 2016

A038550(a(n-1)) + 1 = A038550(a(n)) = A038550(a(n+1)) - 1.

a(n) >> n log n. - Charles R Greathouse IV, Aug 22 2016

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A267697 Numbers with 7 odd divisors.

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A267891 Numbers with 8 odd divisors.

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A267892 Numbers with 9 odd divisors.

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A267893 Numbers with 10 odd divisors.

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A171565 Number of partitions of n into odd divisors of n.

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A069562 Numbers, m, whose odd part (largest odd divisor, A000265(m)) is a nontrivial square.

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A335911 Numbers of the form q*(2^k), where k >= 0 and q is either a Fermat prime or a Mersenne prime; Numbers k for which A335885(k) = 1.

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