A358683 - OEIS

A358683

a(n) is the sum of all divisors of all positive integers k where A182986(n) < k <= prime(n), n >= 1.

%I #61 Dec 21 2022 20:47:28

%S 4,4,13,20,58,42,97,59,134,259,104,342,248,140,282,498,542,230,623,

%T 438,269,722,517,854,1256,646,320,672,390,730,2767,815,1348,428,2361,

%U 524,1564,1553,1002,1729,1670,728,2980,702,1227,668,4125,4172,1477,790,1500,2246,986,3859,2601,2470,2630

%N a(n) is the sum of all divisors of all positive integers k where A182986(n) < k <= prime(n), n >= 1.

%C a(n) has a symmetric representation as shown in the example.

%F a(n) = Sum_{k=1+A182986(n)..A182986(n+1)} A000203(k), n >= 1.

%e a(1) = sigma(1) + sigma(2) = 1 + 3 = 4.

%e a(2) = sigma(3) = 4.

%e a(3) = sigma(4) + sigma(5) = 7 + 6 = 13.

%e a(4) = sigma(6) + sigma(7) = 12 + 8 = 20.

%e a(5) = sigma(8) + sigma(9) + sigma(10) + sigma(11) = 15 + 13 + 18 + 12 = 58.

%e a(6) = sigma(12) + sigma(13) = 28 + 14 = 42.

%e ...

%e a(40) = sigma(168) + sigma(169) + sigma(170) + sigma(171) + sigma(172) + sigma(173) = 480 + 183 + 324 + 260 + 308 + 174 = 1729.

%e Illustration of initial terms using the Dyck paths described in A237593:

%e .

%e . n prime(n) a(n) Diagram

%e . _ _ _ _ _ _ _ _ _ _ _ _ _

%e . | | | | | | |

%e . 1 2 4 |_ _|_| | | | |

%e . 2 3 4 |_ _| _ _| | | |

%e . | | _ _| | |

%e . 3 5 13 |_ _ _| _| | |

%e . | | _ _ _| |

%e . 4 7 20 |_ _ _ _| _| _ _ _|

%e . | _| |

%e . | | _|

%e . | | _ _|

%e . 5 11 58 |_ _ _ _ _ _| |

%e . | |

%e . 6 13 42 |_ _ _ _ _ _ _|

%e .

%e The diagram of a(40) = 1729 is too large to include.

%t {Total@ DivisorSigma[1, Range[2]]}~Join~Array[Total@ DivisorSigma[1, Range[Prime[# - 1] + 1, Prime[#]]] &, 56, 2] (* _Michael De Vlieger_, Nov 29 2022 *)

%o (PARI) A358683(n) = sum(k=if(1==n,1,1+prime(n-1)),prime(n),sigma(k)); \\ _Antti Karttunen_, Nov 29 2022

%Y Partial sums give A244583.

%Y Cf. A000040, A000203, A001235, A024916, A182986, A237270, A237591, A237593, A299763.

%K nonn

%O 1,1

%A _Omar E. Pol_, Nov 26 2022