A206036 -id:A206036 - cp's OEIS frontend

A206421 Corresponding values of sigma(m) of numbers in A206036.

12, 18, 12, 24, 24, 31, 18, 42, 32, 24, 60, 31, 42, 56, 72, 32, 48, 54, 48, 60, 56, 90, 42, 96, 84, 72, 48, 124, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 96, 104, 84, 144, 126, 96, 144, 72, 114, 124, 140, 96, 168, 80, 186, 126, 84, 224, 108, 132, 120, 180

Offset: 1

Jaroslav Krizek, Feb 07 2012

nonn

			a(1) = 12 because sigma(A206036(1)) = sigma(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A159886 (values k such that sigma(x)= k has more than one solution), A206036 (numbers m such that sigma(m) = sigma(k) has solution for distinct k).

list(lim) = my(s); for(k = 1, lim, s = sigma(k); if(invsigmaNum(s) > 1, print1(s, ", "))); \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

a(n) = A000203(A206036(n)). - Amiram Eldar, Dec 15 2024

A286603 Restricted growth sequence computed for sigma, A000203.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 14, 17, 21, 22, 23, 24, 18, 25, 26, 27, 26, 28, 29, 20, 22, 30, 17, 31, 32, 33, 34, 24, 26, 35, 36, 37, 24, 38, 27, 39, 24, 39, 40, 30, 20, 41, 42, 31, 43, 44, 33, 45, 46, 47, 31, 45, 24, 48, 49, 50, 35, 51, 31, 41, 40, 52, 53, 47, 33, 54, 55, 56, 39, 57, 30, 58, 59, 41, 60

Offset: 1

Antti Karttunen, May 11 2017

nonn

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A000203, because for all i, j it holds that: a(i) = a(j) <=> A000203(i) = A000203(j) <=> A286358(i) = A286358(j).

Note that the latter equivalence indicates that this is also the restricted growth sequence of A286358.

			Construction: we start with a(1)=1 for sigma(1)=1 (where sigma = A000203), and then after, for all n > 1, whenever the value of sigma(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever sigma(n) = sigma(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, sigma(2) = 3, not encountered before, thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n=3, sigma(3) = 4, not encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=4, sigma(4) = 7, not encountered before, thus we allot for a(4) the least so far unused number, which is 4, thus a(4) = 4.
For n=5, sigma(5) = 6, not encountered before, thus we allot for a(5) the least so far unused number, which is 5, thus a(5) = 5.
For n=6, sigma(6) = 12, not encountered before, thus we allot for a(6) the least so far unused number, which is 6, thus a(6) = 6.
And this continues for n=7..10 because also for those n sigma obtains fresh new values, so here a(n) = n up to n = 10.
But then comes n=11, where sigma(11) = 12, a value which was already encountered at n=6 for the first time, thus we set a(11) = a(6) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A206036, A211656, A286358.

Cf. also A101296, A286605, A286610, A286619, A286621, A286622, A286626, A286378.

With[{nn = 93}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

A000203(n) = sigma(n);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(10000,n,A000203(n))),"b286603.txt");

A211656 Numbers k such that the value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 19, 22, 27, 29, 32, 36, 37, 43, 45, 49, 50, 61, 64, 67, 72, 73, 81, 91, 98, 100, 101, 106, 109, 121, 128, 129, 133, 134, 137, 146, 148, 149, 152, 157, 162, 163, 169, 171, 173, 192, 193, 197, 199, 200, 202, 211, 217, 218, 219

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Values of sigma(n) in increasing order are in A007370. Corresponding values of sigma(a(n)) is in A211657(n).
Complement of A206036 (numbers n such that sigma(n) = sigma(k) has solution for distinct numbers n and k).
Union of A066076 (primes p such that value of sigma(p) is unique) and A211658 (nonprimes p such that value of sigma(p) is unique).

			Number 36 is in sequence because sigma(36) = 91 and there is no other number m with sigma(m) = 91.
Number 6 is not in the sequence because sigma(6) = 12 and 12 is also sigma(11).

Robert Israel, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A007370, A066076, A211657, A211658, A211659, A211660, A206036.

N:= 1000: # to get terms < the least m with sigma(m) > N
S:= map(numtheory:-sigma, [$1..N-1]):
m:=min(select(t -> S[t]>N, [$1..N-1]))-1:
select(n->numboccur(S[n],S)=1, [$1..m]); # Robert Israel, Jul 04 2019

nn = 300; mx = Max[DivisorSigma[1, Range[nn]]]; d = DivisorSigma[1, Range[mx]]; t = Transpose[Select[Sort[Tally[d]], #[[1]] <= mx && #[[2]] == 1 &]][[1]]; Select[Range[nn], MemberQ[t, d[[#]]] &] (* T. D. Noe, Apr 20 2012 *)

isok(k) = invsigmaNum(sigma(k)) == 1; \\ Amiram Eldar, Jan 11 2025, using Max Alekseyev's invphi.gp

A211657 Sigma(k) of numbers k such that value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 15, 13, 28, 14, 39, 20, 36, 40, 30, 63, 91, 38, 44, 78, 57, 93, 62, 127, 68, 195, 74, 121, 112, 171, 217, 102, 162, 110, 133, 255, 176, 160, 204, 138, 222, 266, 150, 300, 158, 363, 164, 183, 260, 174, 508, 194, 198, 200, 465, 306, 212, 256, 330

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

			For n = 4, a(n) = 7 because A211656(4) = 4; sigma (4) = 7.

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

Cf. A000203, A066076, A211656, A211658, A211659, A211660, A206036.

Cf. A007370 (sorted version of this sequence).

a(n) = sigma(A211656(n)).

A380303 Numbers m such that GCD of the solutions x to sigma(x) = sigma(m) are setwise coprime.

Original entry on oeis.org

1, 6, 10, 11, 14, 15, 16, 17, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 44, 46, 47, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 69, 70, 71, 74, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 99, 102, 103, 105, 107, 108, 110, 111, 112, 113, 114, 115, 117, 118, 119, 120, 123, 124, 125, 126, 127, 130, 131, 132, 135, 136, 138, 139, 140, 141, 142, 143, 145, 147, 150

Offset: 1

Max Alekseyev, Jan 19 2025

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A206036, A211656, A241480, A241481, A241625, A380304.

isok(k) = if(k == 1, 1, my(v = invsigma(sigma(k))); #v > 1 && gcd(v) == 1); \\ Amiram Eldar, May 28 2025, using Max Alekseyev's invphi.gp (see links).

Union of {1} and the set difference of A206036 and A241481.

A206447 Composite numbers n such that sigma(n) = sigma(d) has solution for some other composite number d.

Original entry on oeis.org

14, 15, 16, 20, 24, 25, 26, 28, 30, 33, 35, 38, 39, 40, 42, 44, 46, 48, 51, 54, 55, 56, 58, 60, 62, 65, 66, 68, 69, 70, 75, 77, 78, 80, 82, 84, 87, 88, 90, 92, 94, 95, 96, 99, 102, 104, 105, 108, 110, 112, 114, 115, 116, 118, 119, 120, 122, 123, 124, 125

Offset: 1

Jaroslav Krizek, Feb 07 2012

nonn

			Composite numbers 14 and 15 are in sequence because sigma(14) = sigma(15) = 24.

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

Cf. A206036, A158913, A066073.

N:= 500:
Res:= {}: Q:= {}:
for n from 4 to N do
  if isprime(n) then next fi;
  s:= numtheory:-sigma(n);
  if not assigned(V[s]) then
     V[s]:= n;
     if s > N then Q:= Q union {n} fi;
  else
     Res:= Res union {n,V[s]};
     if s > N then Q:= Q minus {V[s]} fi;
  fi
od:
convert(select(`<`,Res, min(Q)),list); # Robert Israel, Dec 17 2017

t2 = Table[If[PrimeQ[n], 0, DivisorSigma[1, n]], {n, 1000}]; Select[Range[132], ! PrimeQ[#] && Length[Position[t2, t2[[#]]]] > 1 &] (* T. D. Noe, Feb 27 2012 *)

A380304 a(n) = sigma(A380303(n)).

Original entry on oeis.org

1, 12, 18, 12, 24, 24, 31, 18, 42, 32, 24, 60, 31, 42, 56, 72, 32, 48, 54, 48, 60, 56, 90, 42, 96, 84, 72, 48, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 96, 104, 84, 144, 96, 144, 72, 114, 140, 96, 168, 80, 84, 224, 108, 132, 120, 180, 90, 234, 168, 128, 144, 120, 252, 98, 156, 216, 104, 192, 108, 280, 216, 152, 248, 114, 240, 144, 182, 180, 144, 360, 168, 224, 156, 312, 128, 252, 132, 336, 240, 270, 288, 140, 336, 192

Offset: 1

Max Alekseyev, Jan 19 2025

nonn

Except for a(1)=1, subsequence of A206421.

Cf. A000203, A206036, A211656, A241480, A241481, A241625, A380303.

a(n) = sigma(A380303(n)) = A000203(A380303(n)).

A231364 Numbers n such that antisigma(n) = antisigma(k) has solution for distinct numbers n and k.

Original entry on oeis.org

1, 2, 5, 6, 8585, 8586, 16119, 16020, 29886159, 29886160

Offset: 1

Jaroslav Krizek, Nov 10 2013

nonn

Antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n. a(9) > 10^5. Conjecture: a(9) = 29886159, a(10) = 29886160.

			5 and 6 are in the sequence because antisigma(5) = antisigma(6) = 9.

Cf. A067816 (numbers n such that antisigma(n) = antisigma(n+1)).
Cf. A206036 (numbers n such that sigma(n) = sigma(k) has solution for distinct numbers n and k).
Cf. A225775 (numbers k such that antisigma(x) = antisigma(x+1) = k has solution).
Cf. A231365, A231366, A231367, A231368, A231369.

a(9)-a(10) from Donovan Johnson, Nov 12 2013

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A206421 Corresponding values of sigma(m) of numbers in A206036.

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A286603 Restricted growth sequence computed for sigma, A000203.

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A211656 Numbers k such that the value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

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A211657 Sigma(k) of numbers k such that value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

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A380303 Numbers m such that GCD of the solutions x to sigma(x) = sigma(m) are setwise coprime.

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A206447 Composite numbers n such that sigma(n) = sigma(d) has solution for some other composite number d.

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A380304 a(n) = sigma(A380303(n)).

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A231364 Numbers n such that antisigma(n) = antisigma(k) has solution for distinct numbers n and k.

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