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 1-3 of 3 results.

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.

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

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 1-3 of 3 results.

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