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.

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Examples

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

Links

Crossrefs

Cf. A000203, A066076, A211656, A211658, A211659, A211660, A206036.
Cf. A007370 (sorted version of this sequence).

Formula

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

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.

A333947 a(n) is the smallest k > 0 such that sigma(n+k) = sigma(n); if such k > 0 does not exist, then a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 0, 0, 0, 1, 8, 9, 0, 0, 0, 6, 10, 0, 0, 14, 0, 15, 0, 11, 0, 16, 0, 0, 2, 19, 12, 0, 0, 21, 0, 18, 0, 20, 0, 21, 0, 5, 0, 27, 0, 0, 4, 45, 0, 2, 16, 31, 22, 31, 0, 18, 0, 7, 40, 0, 18, 4, 0, 14, 8, 24, 0, 0, 0, 39, 0, 63, 0, 14, 0
Offset: 1

Views

Author

Bernard Schott, Apr 11 2020

Keywords

Comments

This sequence is inspired by A007365 where a(n) is the smallest k such that sigma(n+k) = sigma(k); indeed, n and k are switched between these two sequences.
There are three distinct cases for which a(n) = 0:
If n is prime then a(n) = 0,
If n is in A211658 then a(n) = 0,
If n is the largest number q_r of a sequence q_1 < q_2 < ... < q_r with q_r composite and sigma(q_1) = sigma(q_2) = ... = sigma(q_r) then a(n) = 0. The first two such examples are a(25) = 0 and a(39) = 0 with sigma(16) = sigma(25) = 31, and sigma(28) = sigma(39) = 56.

Examples

			sigma(9) = 13 and there is no k>0 such that sigma(9+k) = 13, then a(9) = 0.
sigma(14) = sigma(15) = sigma(23) = 24, so a(14) = 1 and a(15) = 8, and as 23 is prime, a(23) = 0.

Links

Crossrefs

Cf. A000203, A007365, A054973.
Cf. A002961 (a(n)=1).

Programs

  • Maple
    f:= proc(n) local s,k;
  s:= numtheory:-sigma(n);
for k from n+1 to s-1 do
  if numtheory:-sigma(k)=s then return k-n fi
od;
0
end proc:
map(f, [$1..100]); # Robert Israel, Apr 17 2020
  • Mathematica
    a[n_] := Module[{k = n+1, s = DivisorSigma[1, n]}, While[k < s && DivisorSigma[1, k] != s, k++];If[k >= s, 0, k-n]]; Array[a, 70] (* Amiram Eldar, Apr 12 2020 *)
  • PARI
    a(n) = {my(s=sigma(n)); for (k= n+1, s-1, if (sigma(k) == s, return (k-n));); return(0);} \\ Michel Marcus, Apr 11 2020
Showing 1-5 of 5 results.

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