A211656 -id:A211656 - cp's OEIS frontend

A286603 Restricted growth sequence computed for sigma, A000203.

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");

A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

Original entry on oeis.org

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, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

			6 and 11 are in the sequence because sigma(6) = sigma(11) = 12.
7 is not on the list because sigma(7) = 8 and there is no other integer for which sigma(n) = 8.

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

Complement of A211656.

Cf. A000203, A054973, A066074, A158913.

max = 9000; sigmaList = Table[DivisorSigma[1, n], {n, Prime[PrimePi[max]]}]; Select[Range[Floor[Sqrt[max]]], Count[sigmaList, sigmaList[[#]]] > 1 &] (* Alonso del Arte, Feb 06 2012 *)

is(k) = invsigmaNum(sigma(k)) > 1; \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A241625 Smallest number m such that the GCD of the x's that satisfy sigma(x)=m is n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6187272, 8, 15, 13, 196602, 8105688, 28, 14

Offset: 1

Michel Marcus, Apr 26 2014

This sequence is a sequel to A240667.
Some large known terms: a(16)=2031554, a(25)=1355816, a(31)=8880128, a(80)=11532, a(97)=5488.
a(14) > 10^9. - Michel Marcus, May 09 2014
a(n) is a multiple of A353783(n). Some further terms: a(15) = 497943732, a(17) = 962949708, a(20) = 612372264, a(48) = 12692888, a(53) = 39887316. - Max Alekseyev, Jan 19 2025

			a(2) = 3, because the only x such that sigma(x)=3 is 2.
a(6) = 6187272, because the x's that satisfy sigma(x)=6187272 are [2651676, 2855646] and their GCD is 6.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A211656, A240667, A353783, A380303, A380304.

lista() = {lim = 12000000; nn = 100; out = "a241625.txt"; v = vector(lim, i, sigma(i)); w = vector(lim); for (i=1, lim, vi = v[i]; if (vi <= lim, if (w[vi] == 0, w[vi] = i, w[vi] = concat(w[vi], i)););); for (i=1, nn, got = 0; write1(out, i, " "); for (j=1, #w, wj = w[j]; if (gcd(wj) == i, got = 1; write(out, j);break;);); if (! got, write(out, );););}

a(n) = my(m=1); while(gcd(invsigma(m)) != n, m++); m; \\ Michel Marcus, Jan 16 2025; using Max Alekseyev's invphi.gp

For n in A211656, a(n) = sigma(n).

A211660 Numbers k such that k and k+2 both have unique values of sigma(k) and sigma(k+2); sigma(k) = A000203(k) = sum of divisors of k.

Original entry on oeis.org

1, 2, 3, 5, 7, 27, 43, 98, 146, 169, 171, 197, 200, 217, 241, 257, 281, 331, 347, 379, 386, 409, 448, 461, 487, 505, 507, 509, 547, 554, 576, 577, 641, 800, 821, 829, 841, 857, 907, 937, 1117, 1250, 1261, 1283, 1289, 1322, 1352, 1359, 1387, 1415, 1601, 1621

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Subsequence of A211656. Number k is in sequence iff k and k+2 are in A211656.

Supersequence of A211767 (lesser of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)).

			Number 27 is in sequence because sigma(27) = 40, sigma(29) = 30 and there are no other numbers m, n with sigma(m) = 40 or sigma(n) = 30.

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

Cf. A000203, A211656, A211657, A211658, A211659, A211767, A211768, A211769.

A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 1, 9, 13, 8, 0, 0, 1, 0, 19, 0, 0, 0, 1, 0, 0, 0, 12, 0, 29, 1, 1, 0, 0, 0, 22, 0, 37, 18, 27, 0, 1, 0, 43, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 49, 0, 0, 1, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 1, 0, 73, 0, 0, 0, 45, 0, 1, 0, 0

Offset: 1

Michel Marcus, Apr 10 2014

nonn

From n = 1 to 5, the least integers such that a(x) = n, depending on if singletons (see A007370 and A211656) are accepted or not, are 1, 3, 4, 7, 6 or 12, 126, 124, 210, 22152.

Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.

			There are no integers such that sigma(x) = 2, so a(2) = 0.
There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.

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

Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.

A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
seq(A240667(n), n=1..82); # Peter Luschny, Apr 13 2014

a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
Array[a, 100] (* Jean-François Alcover, Jul 30 2018 *)

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}

a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

a(A007369(n)) = 0.

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)).

A211658 Nonprime 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, 4, 8, 9, 12, 18, 22, 27, 32, 36, 45, 49, 50, 64, 72, 81, 91, 98, 100, 106, 121, 128, 129, 133, 134, 146, 148, 152, 162, 169, 171, 192, 200, 202, 217, 218, 219, 243, 256, 259, 262, 268, 274, 288, 289, 292, 301, 314, 324, 333, 338, 343, 361, 381, 386, 388

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Complement of A066076 with respect to A211656.

			Number 36 is in the 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).

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

Cf. A000203, A066076, A211656, A211657, A211659, A211660.

A211659 Numbers k such that k and k+1 both have unique values of sigma(k) and sigma(k+1); sigma(k) = A000203(k) = sum of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 12, 18, 36, 49, 72, 100, 128, 133, 148, 162, 192, 199, 217, 218, 256, 288, 313, 337, 400, 421, 457, 511, 547, 548, 562, 576, 577, 578, 652, 661, 676, 721, 841, 842, 871, 876, 1058, 1093, 1152, 1171, 1191, 1200, 1227, 1233, 1249, 1282, 1306

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Subsequence of A211656. Number k is in sequence iff k and k+1 are in A211656.

			Number 36 is in sequence because sigma(36) = 91, sigma(37) = 38 and there are no other numbers m, n with sigma(m) = 91 or sigma(n) = 38.

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

Cf. A000203, A211656, A211657, A211658, A211660.

A211767 Lesser of 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, 197, 281, 347, 461, 641, 821, 857, 1289, 1697, 1721, 1787, 1877, 2081, 2141, 2381, 2549, 2801, 3257, 3539, 3557, 3929, 4019, 4241, 4637, 4721, 5441, 5477, 5501, 5657, 6449, 6689, 6701, 6761, 6827, 6947, 7457, 7589, 7877, 8009, 8387, 8537, 8597, 8627

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Subsequence of A211656, A211660, A211678.

			Prime 197 is in sequence because 197 and 199 are twin primes, 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

Cf. A211678 (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)).

d = DivisorSigma[1, Range[10000]]; 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, AppendTo[tp, t3[[i]]]], {i, Length[t3] - 1}]; tp (* T. D. Noe, Apr 26 2012 *)

A-number corrected by Jaroslav Krizek, Mar 17 2013

A211769 Greater of 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

5, 7, 199, 283, 349, 463, 643, 823, 859, 1291, 1699, 1723, 1789, 1879, 2083, 2143, 2383, 2551, 2803, 3259, 3541, 3559, 3931, 4021, 4243, 4639, 4723, 5443, 5479, 5503, 5659, 6451, 6691, 6703, 6763, 6829, 6949, 7459, 7591, 7879, 8011, 8389, 8539, 8599, 8629

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Subsequence of A211656, A211660, and A211678.

			Prime 199 is in sequence because 197 and 199 are twin primes, 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

Cf. A211678 (twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)), A211767 (lesser of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)).

d = DivisorSigma[1, Range[10000]]; 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, AppendTo[tp, t3[[i + 1]]]], {i, Length[t3] - 1}]; tp (* T. D. Noe, Apr 26 2012 *)

A-number corrected by Jaroslav Krizek, Mar 17 2013

cp's OEIS Frontend

A286603 Restricted growth sequence computed for sigma, A000203.

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A206036 Numbers m such that sigma(m) = sigma(k) has solution for distinct numbers m and k.

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A241625 Smallest number m such that the GCD of the x's that satisfy sigma(x)=m is n.

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A211660 Numbers k such that k and k+2 both have unique values of sigma(k) and sigma(k+2); sigma(k) = A000203(k) = sum of divisors of k.

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A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

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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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A211658 Nonprime numbers k such that value of sigma(k) is unique; sigma(k) = A000203(k) = sum of divisors of k.

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A211659 Numbers k such that k and k+1 both have unique values of sigma(k) and sigma(k+1); sigma(k) = A000203(k) = sum of divisors of k.

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