A262744 Remainder when sum of first n primes is divided by n-th triangular number.

0, 2, 4, 7, 13, 20, 2, 5, 10, 19, 28, 41, 56, 71, 88, 109, 134, 159, 188, 9, 19, 32, 46, 63, 85, 108, 130, 153, 175, 198, 232, 267, 305, 342, 386, 429, 475, 524, 574, 627, 683, 738, 800, 861, 923, 984, 1054, 1133, 1213, 17, 46, 77, 106, 141, 178

Offset: 1

Author

Altug Alkan, Sep 29 2015

Keywords

• nonn
• easy
• look

Comments

Sequence is interesting because a(n)-a(n-1) < 0 in certain points such as n=7 and n=20, although a(n)-a(n-1) > 0 for other points, generally.

Old name was: a(n) = (Sum_{k=1..n} prime(k)) mod (Sum_{k=1..n} k).

Examples

			a(1) = prime(1) mod 1 = 0.
a(2) = (prime(1) + prime(2)) mod (1+2) = 2.
a(3) = (prime(1) + prime(2) + prime(3)) mod (1+2+3) = 4.
a(4) = (prime(1) + prime(2) + prime(3) + prime(4)) mod (1+2+3+4) = 7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007504, A000217, A090396.

Programs

Maple

s:= proc(n) option remember; ithprime(n)+`if`(n>1, s(n-1), 0) end:
a:= n-> irem(s(n), n*(n+1)/2):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 01 2015

Mathematica

Table[Mod[Sum[Prime@ k, {k, n}], Sum[k, {k, n}]], {n, 60}] (* Michael De Vlieger, Sep 30 2015 *)
Module[{nn=60,pr,tr},pr=Accumulate[Prime[Range[nn]]];tr=Accumulate[ Range[ nn]];Mod[#[[1]],#[[2]]]&/@Thread[{pr,tr}]] (* Harvey P. Dale, Aug 02 2020 *)

PARI

a(n) = sum(k=1, n, prime(k)) % (n*(n+1)/2);
vector(500, n, a(n))

Formula

a(n) = (Sum_{k=1..n} prime(k)) mod (n*(n+1)/2).

a(n) = A007504(n) mod A000217(n).

Extensions

New name from Altug Alkan, Feb 06 2017, following a suggestion from N. J. A. Sloane