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Numbers k such that the sum of the numbers from 1 to k and that from 1 to k+1 share the same sum of divisors.
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%I #22 Oct 14 2024 12:52:40

%S 28,33,412,1914,2668,2728,2913,3268,4187,5370,5948,7169,8728,9359,

%T 12565,15085,21461,24881,25019,26609,29682,37746,38716,40290,46863,

%U 49225,49914,51835,53963,59987,67996,73132,77057,81055,85636,101101,113128,127585,129330,143369

%N Numbers k such that the sum of the numbers from 1 to k and that from 1 to k+1 share the same sum of divisors.

%C The sequence gives the index of the first element of a pair of consecutive triangular numbers with the same sum of divisors.

%C The even terms of this sequence are exactly twice the terms of A002961. - _Amiram Eldar_, Sep 16 2024

%F A000203(A000217(a(n))) = A000203(A000217(a(n)+1)).

%e Sum of the numbers from 1 to 28 -> 406; from 1 to 29 -> 435;

%e sigma(406) = sigma(435) = 720.

%p with(numtheory): P:=proc(q) local a,j,n,t; j:=[]; t:=0;

%p for n from 1 to q do a:=n*(n+1)/2; if sigma(a)=t then j:=[op(j),n-1]; fi;

%p t:=sigma(a); od; op(j); end: P(143370);

%t Select[Range[145000],DivisorSigma[1,#(#+1)/2]==DivisorSigma[1,(#+1)(#+2)/2] &] (* _Stefano Spezia_, Sep 16 2024 *)

%o (Python)

%o from itertools import count, islice

%o from sympy import divisor_sigma

%o def A375819_gen(): # generator of terms

%o s, a = 1, 1

%o return (k-1 for k in count(2) if a==(a:=divisor_sigma(s:=s+k)))

%o A375819_list = list(islice(A375819_gen(),10)) # _Chai Wah Wu_, Oct 14 2024

%Y Cf. A000203, A000217, A002961.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Sep 16 2024