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 1-7 of 7 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

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)).

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.

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Subsequence of A211656, A211660, A211678.

Examples

			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.

Links

Crossrefs

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)).

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Subsequence of A211656, A211660, and A211678.

Examples

			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.

Links

Crossrefs

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)).

Programs

  • Mathematica
    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 *)

Extensions

A-number corrected by Jaroslav Krizek, Mar 17 2013

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

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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
Showing 1-7 of 7 results.

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