cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

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).

Examples

			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).

Links

Crossrefs

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

Programs

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

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

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)).

Examples

			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.

Links

Crossrefs

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

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Complement of A066076 with respect to A211656.

Examples

			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).

Links

Crossrefs

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

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

Views

Author

Jaroslav Krizek, May 12 2019

Keywords

Comments

A211656 is the sorted version of this sequence.

Examples

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

Links

Crossrefs

Cf. A000203, A007370, A211656, A211657.

Programs

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

Formula

A000203(a(n)) = A007370(n) for all n.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.