A211656 -id:A211656 - cp's OEIS frontend

A241480 Numbers k such that the GCD of the x's that satisfy sigma(x) = sigma(k) is not equal to 1.

2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 19, 22, 27, 29, 32, 36, 37, 43, 45, 48, 49, 50, 61, 64, 67, 68, 72, 73, 75, 80, 81, 82, 91, 98, 100, 101, 104, 106, 109, 116, 121, 122, 128, 129, 133, 134, 137, 144, 146, 148, 149, 152, 156, 157, 160, 162, 163, 169, 171, 173

Offset: 1

Michel Marcus, Apr 23 2014

nonn

Apart from 1, all terms of A211656 belong here since the solutions to sigma(x)=sigma(n) form a singleton and thus their GCD is n itself.

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

Cf. A000203, A211656, A241479.

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
isok(n) = (gcd(sigv(sigma(n))) != 1);

is(k) = gcd(invsigma(sigma(k))) > 1; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

Numbers k such that A241479(k) is not equal to 1.

A211678 Twin primes p, p+2 with unique values of sigma(p) and sigma(p+2); sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

3, 5, 7, 197, 199, 281, 283, 347, 349, 461, 463, 641, 643, 821, 823, 857, 859, 1289, 1291, 1697, 1699, 1721, 1723, 1787, 1789, 1877, 1879, 2081, 2083, 2141, 2143, 2381, 2383, 2549, 2551, 2801, 2803, 3257, 3259, 3539, 3541, 3557, 3559, 3929, 3931, 4019, 4021

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

			Twin primes 197 and 199 are in sequence because sigma(197) = 198, sigma(199) = 200 and there are no other numbers m, n with sigma(m) = 198 or sigma(n) = 200.

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

Subsequence of A211656 and A211660.
Cf. A211767 (lesser of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)), A211769 (greater of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)).
Cf. A000203.

d = DivisorSigma[1, Range[4100]]; t = Transpose[Select[Tally[Sort[d]], #[[2]] == 1 && #[[1]] <= Length[d] &]][[1]]; t2 = Sort[Flatten[Table[Position[d, i], {i, t}]]]; t3 = Select[t2, PrimeQ]; tp = {}; Do[If[t3[[i + 1]] - t3[[i]] == 2 && DivisorSigma[1, t3[[i]]] != DivisorSigma[1, t3[[i + 1]]], AppendTo[tp, t3[[i]]]; AppendTo[tp, t3[[i]] + 2]], {i, Length[t3] - 1}]; Union[tp] (* T. D. Noe, Apr 26 2012 *)

is(k) = isprime(k) && invsigmaNum(sigma(k)) == 1 && ((isprime(k+2) && invsigmaNum(sigma(k+2)) == 1) || (isprime(k-2) && invsigmaNum(sigma(k-2)) == 1)); \\ Amiram Eldar, Aug 08 2024, using Max Alekseyev's invphi.gp

A241481 Numbers such that the GCD of the x's that satisfy sigma(x) = sigma(n) is not equal to 1 while the number of such x's is not 1 either.

Original entry on oeis.org

48, 68, 75, 80, 82, 104, 116, 122, 144, 156, 160, 189, 196, 212, 225, 237, 242, 273, 279, 291, 309, 328, 342, 356, 364, 403, 490, 513, 524, 531, 592, 597, 614, 640, 651, 679, 684, 688, 712, 784, 788, 804, 808, 810, 822, 833, 889, 898, 903, 922, 925, 927, 954

Offset: 1

Michel Marcus, Apr 23 2014

nonn

Subsequence of A241480, restricted to those terms that do not belong to A211656.
Is it possible, for each term of A211656, to find a corresponding term in the present sequence such that the corresponding GCD is equal to the initial A211656 term?
The first 11 terms of A211656 are: 2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 19.
For these, we have 68, 48, 104, 12735, 364, 7848, 144, 9984, 273, 1764, 1197 in the present sequence.
For instance for m = 9984, the x's are [9984, 12252], with gcd = 12.
Is it possible to find a term here with corresponding gcd = 22, the 12th term of A211656?

			48 is in the sequence because sigma(48)=124 and the x's such that sigma(x) = 124 are 48 and 75, with gcd(48, 75) not equal to 1.

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

Cf. A000203, A211656, A241479, A241480.

M:=1000: # to get all terms <= M
N:= 0:
for n from 1 to M do
  v:= numtheory:-sigma(n);
  N:= max(N,v);
  if assigned(R[v]) then R[v]:= igcd(R[v],n); S[v]:= S[v] union {n}
  else R[v]:= n; S[v]:= {n}
  fi;
od:
for n from M+1 to N do
  v:= numtheory:-sigma(n);
  if assigned(R[v]) then R[v]:= igcd(R[v],n);  S[v]:= S[v] union {n} fi;
od:
A:=
`union`(seq(S[v], v = select(t -> R[t]>1 and nops(S[t])>1, map(op,[indices(R)])))) intersect {$1..M}:
sort(convert(A,list)); # Robert Israel, Oct 24 2019

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
isok(n) = my(v = sigv(sigma(n))); ((gcd(v)!=1) && (#v != 1));

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

A325652 a(n) = the sum of numbers k such that sigma(k) = sigma(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 17, 7, 8, 9, 27, 17, 12, 13, 52, 52, 41, 27, 18, 19, 87, 52, 22, 52, 121, 41, 87, 27, 67, 29, 253, 52, 32, 115, 87, 115, 36, 37, 121, 67, 187, 87, 250, 43, 192, 45, 253, 115, 123, 49, 50, 253, 149, 87, 292, 253, 292, 136, 187, 121, 663, 61, 250

Offset: 1

Jaroslav Krizek, May 12 2019

a(n)=n if n is in A211656, otherwise a(n) > n. - Robert Israel, Jul 04 2019

			a(6) = 17 because sigma(6) = sigma(11) = 12; 6 + 11 = 17.

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

Cf. A000203, A211656, A275987, A325651.

See A070242 and A325653 for number and product of such numbers k.

[&+[k: k in[1..10000] | SumOfDivisors(k) eq SumOfDivisors(n)]: n in [1..100]];

N:= 1000: # to get a(n) before the first n with sigma(n) > N
S:= map(numtheory:-sigma, [$1..N-1]):
m:=min(select(t -> S[t]>N, [$1..N-1]))-1:
1,seq(convert(select(s -> S[s]=S[n], [$1..S[n]-1]), `+`), n=2..m); # Robert Israel, Jul 04 2019

a[n_] := Block[{s = DivisorSigma[1, n]}, Sum[Boole[s == DivisorSigma[1, k]] k, {k, s}]]; Array[a, 62] (* Giovanni Resta, Jul 03 2019 *)

a(n) = {my(s=sigma(n)); sum(k=1, s, (sigma(k)==s)*k);} \\ Michel Marcus, May 12 2019

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.

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)).

A308098 Numbers m such that sequence of their values of sigma(m) corresponds to sequence of unique values of function sigma(n) for n >= 1 in increasing order (A007370).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 13, 8, 19, 12, 29, 22, 37, 18, 27, 43, 49, 61, 32, 67, 73, 45, 36, 50, 101, 109, 91, 81, 64, 121, 137, 149, 157, 133, 106, 163, 98, 173, 129, 169, 193, 72, 197, 199, 134, 211, 100, 146, 229, 241, 128, 217, 257, 171, 148, 277, 281, 283, 219

Offset: 1

Jaroslav Krizek, May 12 2019

nonn

A211656 is the sorted version of this sequence.

			a(6) = 7 because A007370(6) = 8 and there is only one solution of equation sigma(x) = 8 for x = 7.

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

Cf. A000203, A007370, A211656, A211657.

[[m: m in [1..1000] | SumOfDivisors(m) eq n]:  n in [1..100] | #[#[m]: m in [1..1000] | SumOfDivisors(m) eq n] eq 1]

m = 500; v = Table[0, {m}]; Do[s = DivisorSigma[1, k]; If[s <= m ,  v[[s]] = If[ v[[s]] == 0, k, -1]], {k, 1, m - 1}]; Select[v, # > 0 &] (* Amiram Eldar, Jul 04 2019 *)

A000203(a(n)) = A007370(n) for all n.

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A241480 Numbers k such that the GCD of the x's that satisfy sigma(x) = sigma(k) is not equal to 1.

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A211678 Twin primes p, p+2 with unique values of sigma(p) and sigma(p+2); sigma(n) = A000203(n) = sum of divisors of n.

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A241481 Numbers such that the GCD of the x's that satisfy sigma(x) = sigma(n) is not equal to 1 while the number of such x's is not 1 either.

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A325652 a(n) = the sum of numbers k such that sigma(k) = sigma(n).

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

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A308098 Numbers m such that sequence of their values of sigma(m) corresponds to sequence of unique values of function sigma(n) for n >= 1 in increasing order (A007370).

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