A352908 -id:A352908 - cp's OEIS frontend

A353265 Partial sums of A208981.

0, 0, 3, 3, 4, 8, 20, 20, 35, 37, 47, 52, 57, 70, 83, 83, 91, 107, 123, 126, 127, 138, 149, 155, 174, 180, 287, 301, 315, 329, 431, 431, 453, 462, 471, 488, 505, 522, 552, 556, 661, 663, 688, 700, 712, 724, 824, 831, 851, 871, 891, 898, 905, 1013, 1121, 1136, 1164, 1179, 1207, 1222, 1237, 1340, 1443, 1443

Offset: 1

Omar E. Pol, Apr 09 2022

nonn

Cf. A208981, A347270 (gives all 3x+1 sequences).

Cf. A006370, A006877, A014682, A347272, A347519, A352907, A352908, A352939.

Accumulate @ Table[-1 + Length @ NestWhileList[If[OddQ[#], 3*# + 1, #/2] &, n, ! IntegerQ @ Log[2, #] &], {n, 1, 64}] (* Amiram Eldar, Apr 09 2022 *)

ispow2(n)=n>>=valuation(n, 2); n==1;
f(n) = my(s); while(!ispow2(n), n=if(n%2, 3*n+1, n/2); s++); s; \\ A208981
a(n) = sum(i=1, n, f(i)); \\ Michel Marcus, Apr 13 2022

A353001 Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k, and the k-th triangular number mod the sum of divisors of k, are both prime.

Original entry on oeis.org

4, 57, 70, 93, 129, 217, 322, 381, 417, 453, 513, 565, 597, 646, 682, 781, 813, 921, 925, 1057, 1081, 1102, 1137, 1165, 1197, 1261, 1317, 1393, 1405, 1558, 1582, 1641, 1750, 1798, 1846, 1857, 1918, 1929, 2073, 2101, 2110, 2173, 2181, 2305, 2329, 2361, 2482, 2506, 2569, 2577, 2626, 2649, 2653

Offset: 1

Robert Israel, Apr 14 2022

nonn

Numbers k such that A232324(k) and A352996(k) are prime.

			a(3) = 70 is a term because 70*71/2 = 2485, A000217(70) = 144, A001414(70) = 14, and both 2485 mod 144 = 37 and 2485 mod 14 = 7 are prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A352908 and A352997.

Cf. A000217, A001414, A232324, A352996.

filter:= proc(n) local a,b,c,t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2],t=ifactors(n)[2]);
  if not isprime(a mod b) then return false fi;
  c:= numtheory:-sigma(n);
  isprime(a mod c)
end proc:
select(filter, [$2..3000]);

Select[Range[3000], And @@ PrimeQ[Mod[#*(# + 1)/2, {DivisorSigma[1, #], Plus @@ Times @@@ FactorInteger[#]}]] &] (* Amiram Eldar, Apr 15 2022 *)

cp's OEIS Frontend

A353265 Partial sums of A208981.

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