A352996 -id:A352996 - cp's OEIS frontend

A352997 Numbers k such that A352996(k) is prime.

4, 9, 18, 20, 24, 25, 30, 42, 49, 50, 57, 65, 66, 69, 70, 75, 76, 78, 80, 85, 93, 96, 98, 99, 102, 104, 110, 112, 114, 121, 129, 133, 141, 145, 152, 153, 169, 177, 186, 189, 190, 192, 198, 213, 217, 228, 238, 242, 249, 252, 258, 261, 266, 272, 273, 275, 282, 286, 289, 290, 292, 294, 297, 305, 309

Offset: 1

J. M. Bergot and Robert Israel, Apr 14 2022

nonn

Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k is prime.

Contains p^2 for prime p.

			a(3) = 18 is a term because A352996(18) = A000217(18) mod A001414(18) = 171 mod 8 = 3 is prime.

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

Cf. A000217, A001414, A352996.

filter:= proc(n) local t; isprime((n*(n+1)/2) mod add(t[1]*t[2],t=ifactors(n)[2])) end proc:
select(filter, [$2..500]);

Select[Range[300], PrimeQ[Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]]] &] (* Amiram Eldar, Apr 14 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 *)

A353002 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 the same prime.

Original entry on oeis.org

93, 2653, 30433, 1922113, 15421122, 28776673, 240409057, 611393953, 2713190397, 5413336381

Offset: 1

J. M. Bergot and Robert Israel, Apr 15 2022

			a(1) = 93 is a term because 93*94/2 = 4371, A000217(93) = 128, A001414(93) = 34, and 4371 mod 128 = 4371 mod 34 = 19, which is prime.

Cf. A000217, A001414, A232324, A352996, A353001.

filter:= proc(n) local a,b,c,t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2],t=ifactors(n)[2]);
  t:= a mod b; if not isprime(t) then return false fi;
  c:= numtheory:-sigma(n);
  a mod c = t
end proc:
select(filter, [$2..2*10^7]);

Select[Range[2*10^6], (r = Mod[#*(# + 1)/2, DivisorSigma[1, #]]) == Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] && PrimeQ[r] &] (* Amiram Eldar, Apr 15 2022 *)

a(8) from Amiram Eldar, Apr 15 2022

a(9)-a(10) from Daniel Suteu, May 12 2022

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A352997 Numbers k such that A352996(k) is prime.

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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.

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Maple

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A353002 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 the same prime.

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