cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A283363 Absolute values of first differences of A090396.

Original entry on oeis.org

1, 0, 0, 2, 2, 3, 3, 4, 8, 3, 1, 1, 3, 12, 0, 2, 0, 2, 2, 0, 2, 21, 3, 7, 7, 5, 5, 26, 2, 12, 12, 19, 11, 18, 14, 16, 20, 17, 21, 20, 24, 23, 23, 20, 18, 21, 15, 32, 20, 29, 21, 25, 33, 20, 36, 19, 23, 36, 24, 33, 23, 15, 47, 20, 22, 14, 54, 9, 13, 15, 15, 60, 12, 12, 14, 14, 12, 65
Offset: 1

Views

Author

Altug Alkan, Mar 06 2017

Keywords

Comments

See the beginning of helix in scatterplot link at Links section.

Examples

			a(5) = 2 because a(5) = abs(A090396(6) - A090396(5)) = abs(5 - 3) = 2.

Links

Crossrefs

Cf. A090396.

Programs

  • Mathematica
    A[n_] := Mod[Sum[Prime[k], {k, n}], n];  Table[Abs[A[n+1] - A[n]],{n, 78}] (* Indranil Ghosh, Mar 06 2017 *)
  • PARI
    a090396(n) = sum(k=1, n, prime(k)) % n;
a(n) = abs(a090396(n+1) - a090396(n));

Formula

a(n) = abs(A090396(n+1) - A090396(n)).

A045345 Numbers k such that k divides sum of first k primes A007504(k).

Original entry on oeis.org

1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601, 12255356398093, 153713440932055, 6361476515268337
Offset: 1

Views

Author

Jud McCranie

Keywords

Comments

a(10) and a(11) were found by Giovanni Resta (Nov 15 2004). He states that there are no other terms for primes p < 4011201392413. See link to Prime Puzzles, Puzzle 31 below. - Alexander Adamchuk, Aug 21 2006
a(13) > pi(2*10^13). - Donovan Johnson, Aug 23 2010
a(15) > 1.42*10^13. - Giovanni Resta, Jan 07 2020
a(16) > 1.55*10^14. - Bruce Garner, Mar 06 2021
a(17) > 6.5*10^15. - Paul W. Dyson, Sep 26 2022
Numbers k such that A090396(k) = 0. - Felix Fröhlich, May 05 2021

Examples

			23 is in the sequence because the sum of the first 23 primes is 874 and that's 23 * 38.
53 is in the sequence because the sum of the first 53 primes is 5830 and that's 53 * 110.
83 is not in the sequence because the sum of the first 83 primes is 15968, which leaves a remainder of 32 when divided by 83.
The sum of the first a(14) primes is equal to a(14)*196523412770096.

Links

Crossrefs

Cf. A007504, A090396, A171399.

Programs

  • Mathematica
    s = 0; t = {}; Do[s = s + Prime[n]; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t (* Alexander Adamchuk, Aug 21 2006 *)
nn = 4000000; With[{acpr = Accumulate[Prime[Range[nn]]]}, Select[Range[nn], Divisible[acpr[[#]], #] &]] (* Harvey P. Dale, Sep 14 2012 *)
Select[Range[100], Mod[Sum[Prime[i], {i, #}], #] == 0 &] (* Alonso del Arte, Mar 22 2014 based on Bill McEachen's Wolfram Alpha example *)
A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {, }][[All, 2]]; Select[Range[10^5], Divisible[A007504[[# + 1]], #] &] (* Robert Price, Mar 13 2020 *)
  • PARI
    s=0;n=0;forprime(p=2,1e7,s+=p;if(s%n++==0,print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011
  • PARI
    isok(n) = (vecsum(primes(n)) % n) == 0; \\ Michel Marcus, Nov 26 2020
  • Python
    from itertools import accumulate, count, islice
from sympy import prime
def A045345_gen(): return (i+1 for i, m in enumerate(accumulate(prime(n) for n in count(1))) if m % (i+1) == 0)
A045345_list = list(islice(A045345_gen(),5)) # Chai Wah Wu, Feb 23 2022

Formula

Matomäki proves that a(n) >> n^(24/19). - Charles R Greathouse IV, Jun 13 2012

Extensions

More terms from Alexander Adamchuk, Aug 21 2006
a(12) from Donovan Johnson, Aug 23 2010
a(13) from Robert Price, Mar 17 2013
a(14) from Giovanni Resta, Jan 07 2020
a(15) from Bruce Garner, Mar 06 2021
a(16) from Paul W. Dyson, Sep 26 2022

A060620 Average of the first n primes rounded down.

Original entry on oeis.org

2, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 60, 62, 65, 67, 69, 72, 74, 77, 79, 82, 84, 87, 89, 91, 94, 97, 99, 102, 104, 107, 110, 112, 115, 117, 120, 123, 125, 128, 130, 133, 136, 138, 141, 144
Offset: 1

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

Keywords

Comments

Closely related to the prime number theorem; a(n) has the asymptotic expression n * log(n) / 2.

Examples

			a(5) = floor((2 + 3 + 5 + 7 + 11)/5) = 5.

Links

Crossrefs

Cf. A007504, A090396.

Programs

  • Mathematica
    With[{prms=Prime[Range[500]]},Table[Floor[Mean[Take[prms,n]]],{n,70}]]  (* Harvey P. Dale, Mar 09 2011 *)
  • PARI
    { n=0; s=0; forprime (p=2, prime(1000), s+=p; write("b060620.txt", n++, " ", floor(s/n)); ) } \\ Harry J. Smith, Jul 08 2009
  • PARI
    a(n) = sum(k=1, n, prime(k))\n; \\ Michel Marcus, Jul 03 2018

Formula

a(n) = floor(A007504(n) / n).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001

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

Original entry on oeis.org

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

Views

Author

Altug Alkan, Sep 29 2015

Keywords

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

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

A334122 a(n) is the sum of all primes <= n, mod n.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 3, 1, 8, 7, 6, 4, 2, 13, 11, 9, 7, 4, 1, 17, 14, 11, 8, 4, 0, 22, 19, 16, 13, 9, 5, 0, 28, 24, 20, 16, 12, 7, 2, 37, 33, 28, 23, 17, 11, 5, 46, 40, 34, 28, 22, 16, 10, 3, 51, 45, 39, 33, 27, 20, 13, 5, 60, 53, 46, 39, 32, 24, 16, 8, 0, 63, 55
Offset: 1

Views

Author

Christoph Schreier, Apr 15 2020

Keywords

Examples

			a(7) = (2+3+5+7) mod 7 = 17 mod 7 = 3.

Links

Crossrefs

Cf. A000720, A034387, A090396.

Programs

  • Maple
    b:= proc(n) b(n):= `if`(n<2, 0, b(n-1)+`if`(isprime(n), n, 0)) end:
a:= n-> irem(b(n), n):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 15 2020
  • Mathematica
    Mod[Accumulate[(# * Boole @ PrimeQ[#]) & /@ (r = Range[100])], r] (* Amiram Eldar, Apr 15 2020 *)
  • PARI
    a(n) = my(np=primepi(n)); vecsum(primes(np)) % n; \\ Michel Marcus, Apr 16 2020
  • Python
    from sympy import primerange
a = lambda n: sum(primerange(n + 1)) % n # David Radcliffe, May 10 2025

Formula

a(n) = A034387(n) mod n.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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