A007506 -id:A007506 - cp's OEIS frontend

A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

1, 3, 20, 31464, 22096548, 1483892396791177

Offset: 1

a(6) > pi(10^12) = 37607912018. - Jon E. Schoenfield, Sep 11 2008
a(6) > pi(10^14) = 3204941750802. - Giovanni Resta, Jan 09 2014
a(7) > 6.5*10^15. - Paul W. Dyson, Sep 27 2022

			The third prime, 5, divides 2 + 3 + 5 = 10, so 3 is in the sequence.
2 + 3 + 5 + 7 = 17, which is not divisible by the fourth prime, 7, so 4 is not in the sequence.

Cf. A007506, A028581, A028582, A071089.

s = 0; For[i = 1, i <= 5 * 10^7, i++, s = s + Prime[i]; If[Mod[s, Prime[i + 1]] == 0, Print[i + 1]]]
With[{prs = Prime[Range[221000000]]}, PrimePi /@ Transpose[Select[ Thread[ {Accumulate[prs], prs}], Divisible[#[[1]], #[[2]]] &]][[2]]] (* Harvey P. Dale, Jul 23 2013 *)
nMax = 50000; primeSums = Accumulate[Prime[Range[nMax]]]; Select[Range[nMax], Divisible[primeSums[[#]], Prime[#]] &] (* Alonso del Arte, Nov 11 2019 *)

s=0; t=0; for(w=2,1000000000,if(isprime(w),s=s+w; t=t+1; if(s%w,print(t)),))

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

a(6) from Paul W. Dyson, Apr 16 2022

A071089 Remainder when sum of first n primes is divided by n-th prime.

Original entry on oeis.org

0, 2, 0, 3, 6, 2, 7, 1, 8, 13, 5, 12, 33, 23, 46, 10, 27, 13, 32, 0, 55, 1, 44, 73, 90, 50, 28, 87, 63, 11, 69, 17, 70, 42, 41, 11, 72, 139, 75, 146, 44, 8, 9, 164, 88, 48, 7, 201, 121, 79, 224, 92, 46, 57, 170, 26, 145, 95, 216, 112, 58, 71, 293, 185, 129, 13, 255, 81, 128

Offset: 1

Randy L. Ekl, May 26 2002

Conjecture: Every nonnegative integer can appear in the sequence at most finitely many times. - Thomas Ordowski, Jul 22 2013
I conjecture the opposite. Heuristically a given number should appear log log x times below x. - Charles R Greathouse IV, Jul 22 2013
In the first 10000 terms, one sees a(n) = n for n=2,7,12. Does this ever happen again? - J. M. Bergot, Mar 26 2018
Yes, it happens for n = 83408, too. - Michel Marcus, Mar 27 2018

			a[5] = 6 because s[5] = 2+3+5+7+11 = 28, p[5]=11 and q[5]= floor(28/11)=2, so a[5] = 28-11*2 = 6.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A007506, A024011.

P:=Filtered([1..1000],IsPrime);
a:=List([1..70],i->Sum(P{[1..i]}) mod P[i]); # Muniru A Asiru, Mar 27 2018

s:= proc(n) option remember; `if`(n=0, 0, ithprime(n)+s(n-1)) end:
a:= n-> irem(s(n), ithprime(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 27 2018

f[n_] := Mod[ Sum[ Prime[i], {i, 1, n - 1}], Prime[n]]; Table[ f[n], {n, 1, 70}] or
a[1] = 0; a[n_] := Block[{s = Sum[Prime[i], {i, 1, n}]}, s - Prime[n]*Floor[s/Prime[n]]]; Table[ f[n], {n, 1, 70}]
f[n_] := Mod[Plus @@ Prime@ Range@ n, Prime@ n]; Array[f, 70] (* Robert G. Wilson v, Nov 12 2016 *)
Module[{nn=70,t},t=Accumulate[Prime[Range[nn]]];Mod[#[[1]],#[[2]]]&/@ Thread[ {t,Prime[Range[nn]]}]] (* Harvey P. Dale, Sep 19 2019 *)

for(n=1,100,s=sum(i=1,n, prime(i)); print1(s-prime(n)*floor(s/prime(n)),","))

a(n) = vecsum(primes(n)) % prime(n); \\ Michel Marcus, Mar 27 2018

a(n) = s[n] - p[n]*q[n], where s[n] = sum of first n primes, p[n] is n-th prime and q[n] is floor(s[n]/p[n]).

a(A024011(n)) = 0. - Michel Marcus, Jan 22 2015

Edited by Robert G. Wilson v and Benoit Cloitre, May 30 2002

A028581 Quotients associated with A024011.

Original entry on oeis.org

1, 2, 9, 15001, 10736033, 731778483019656

Offset: 1

G. L. Honaker, Jr.

Cf. A024011, A007506, A028582.

lst={}; s=0; Do[p=Prime[n]; s=s+p; r=s/p; If[r==IntegerPart[r], AppendTo[lst, r]], {n, 1, 10^6}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

a(6) from Paul W. Dyson, Apr 16 2022

A028582 Dividends associated with A024011.

Original entry on oeis.org

2, 10, 639, 5537154119, 4456255064711219, 40753506500253984833129731848456

Offset: 1

G. L. Honaker, Jr.

Cf. A024011, A007506, A028581.

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

a(6) from Paul W. Dyson, Apr 16 2022

A274649 a(n) is the smallest odd prime that divides n + the sum of all smaller primes, or 0 if no such prime exists.

Original entry on oeis.org

5, 3, 30915397, 11339869, 3, 5, 859, 3, 41, 233, 3, 7, 4175194313, 3, 307, 5, 3, 1459, 7, 3, 5, 9907, 3, 647, 13, 3, 31, 11, 3, 193, 5, 3, 7, 2939, 3, 5, 3167, 3, 11, 7, 3, 1321, 86629, 3, 17, 5, 3

Offset: 0

Neil Fernandez, Nov 10 2016

From David A. Corneth, Nov 12 2016: (Start)
a(n) is the smallest odd prime p such that p|(n + A007504(primepi(p) - 1)) or zero if no such p exists.
If a(n) = p then a(n + p) <= p. (End)
If n is congruent to 1 (mod 3), then a(n)=3.
a(2), a(3) and a(12) were found by Jack Brennen.
From Robert G. Wilson v, Nov 13 2016: (Start)
If n == 1 (mod 3) then a(n) = 3;
If n == 0 (mod 5) then a(n) = 5;
If n == 4 (mod 7) then a(n) = 7;
if n == 5 (mod 11) then a(n) = 11;
if n == 11 (mod 13) then a(n) = 13;
if n == 10 (mod 17) then a(n) = 17;
if n == 18 (mod 19) then a(n) = 19;
if n == 23 (mod 23) then a(n) = 23;
in that order, i.e., from smallest to greatest prime modulus, etc.
First occurrence of p > 2: 1, 0, 11, 27, 24, 44, 56, 84, 161, ..., .
a(47) > 10^11. (End)

			a(6) = 859 because 859 is the smallest odd prime that divides the sum of 6 + (sum of all primes smaller than itself).
a(8) = 41 because 8+2+3+5+7+11+13+17+19+23+29+31+37+41 = 246 and 246/41 = 6.

Michael S. Branicky, Alternate Python program.
Michael S. Branicky, n and a(n) for n = 0..10000 or 0 of no such value is known, search limit = 9*10^9.
Robert G. Wilson v, n and a(n) for n = 0..10000 or 0 if no such value is known.

Cf. A007504, A007506, A024011, A274995.

Cf. A016777 (n==1 (mod 3))

f[n_] := Block[{p = 3, s = n +2}, While[ Mod[s, p] != 0, s = s + p; p = NextPrime@ p]; p]; Array[f, 47, 0] (* Robert G. Wilson v, Nov 12 2016 *)

# see link for an alternate that searches in parallel to a limit
from sympy import nextprime
def a(n):
  psum, p = 2, 3
  while (n + psum)%p: psum, p = psum + p, nextprime(p)
  return p
for n in range(12):
  print(a(n), end=", ") # Michael S. Branicky, May 03 2021

A274995 a(n) is the smallest odd prime that divides (-n) + the sum of all smaller primes, or 0 if no such prime exists.

Original entry on oeis.org

5, 19, 3, 7, 82811, 3, 11, 17, 3, 191, 5, 3, 37, 29, 3, 5, 69431799799, 3, 1105589, 28463, 3, 431, 2947308589, 3, 7, 5, 3, 59, 11, 3, 5, 7, 3, 41

Offset: 0

Neil Fernandez, Nov 11 2016

From Robert G. Wilson v, Nov 15 2016: (Start)
If n == 2 (mod 3) then a(n) = 3;
If n == 0 (mod 5) then a(n) = 5;
If n == 3 (mod 7) then a(n) = 7;
If n == 6 (mod 11) then a(n) = 11;
If n == 2 (mod 13) then a(n) = 13;
If n == 7 (mod 17) then a(n) = 17;
If n == 1 (mod 19) then a(n) = 19;
If n == 8 (mod 23) then a(n) = 23;
in that order; i.e., from smaller to greater prime modulus, etc.
First occurrence of p>2: 2, 0, 3, 6, 54, 7, 1, 123, 13, 36, 12, 33, 453, 46, ..., .
(End)
The congruence classes in the above list, modulo the prime bases, namely 2, 0, 3, 6, 2, ..., are given by A071089, in which each term is the remainder when the sum of the first n primes is divided by the n-th prime. - Neil Fernandez, Nov 23 2016

			a(1) = 19 because 19 is the smallest odd prime that divides the sum of (-1) + (sum of all primes smaller than itself), that is, -1 + 58 = 57.
a(7) = 17 because -7 + 2 + 3 + 5 + 7 + 11 + 13 + 17 = 49 and 49/7 = 7.

Robert G. Wilson v, n and a(n), or 0 if no such value is known, for n=0..10000

Cf. A007504, A007506, A071089, A274649.

f[n_] := Block[{p = 3, s = 2 - n}, While[ Mod[s, p] != 0, s = s + p; p = NextPrime@ p]; p]; Array[f, 16, 0] (* Robert G. Wilson v, Nov 15 2016 *)

sump(n) = s = 0; forprime(p=2, n-1, s+=p); s;
a(n) = {my(p=3); while ((sump(p)-n) % p, p = nextprime(p+1)); p;} \\ Michel Marcus, Nov 12 2016

a(n)=my(s=2); forprime(p=3,, if((s-n)%p==0, return(p)); s+=p) \\ Charles R Greathouse IV, Nov 15 2016

a(16)-a(33) from Charles R Greathouse IV, Nov 15 2016

A330580 Composite numbers n that divide the sum of all composite numbers <= n.

Original entry on oeis.org

4, 9, 604, 1849, 7123, 29588, 71056, 317238, 945414, 1347895, 3567705, 4624025, 8518502, 1523051853, 4353560959, 7537630621, 24916885136, 41483524658, 513849020166, 1021228997529, 1137756574012, 1185967711155, 1581191501535, 1705826324526

Offset: 1

Rémy Sigrist, Dec 18 2019

Cf. A002808, A007506 (prime variant), A330579.

s=0; k=0; forcomposite (c=4, oo, k++; s+=c; if (s%c==0, print1 (c", ")))

a(n) = A002808(A330579(n)).

a(15)-a(24) from Giovanni Resta, Dec 20 2019

A350878 Integers m that divide the sum of values dp < m, where d is a divisor of m, p is a prime, and dp does not divide m.

Original entry on oeis.org

1, 2, 5, 10, 18, 24, 32, 60, 71, 100, 512, 2990, 9910, 10031, 12618, 32674, 53586, 153878, 223500, 312608, 369119, 386110, 466569, 4491817, 7068356, 8765871, 65311881

Offset: 1

Devansh Singh, Jan 20 2022

Conjecture: The sum of values d*p < m in the definition of the sequence is equal to m for m = 5 only. True for m <= 15000.
A007506 is the subsequence of the prime terms of this sequence. - Amiram Eldar, Jan 20 2022
a(28) > 10^8. - David A. Corneth, Jan 21 2022

David A. Corneth, PARI program
Jon E. Schoenfield, Magma program

Cf. A334800, A000040, A007506.

q[n_] := Module[{ds = Divisors[n], s = 0, r}, Do[r = n/d; ps = Select[Range[2, r], PrimeQ[#] && ! Divisible[n, d*#] &]; s += Total[d*ps], {d, ds}]; Divisible[s, n]]; Select[Range[3000], q] (* Amiram Eldar, Jan 20 2022 *)

isok(m) = {my(d=divisors(m), s=0); forprime(p=2, m, for(k=1, #d, my(x=d[k]*p); if ((x < m) && (m % x), s+=x););); (s % m) == 0;} \\ Michel Marcus, Jan 21 2022

\\ See Corneth link \\ David A. Corneth, Jan 21 2022

import sympy
A350878=[]
for m in range(1,15001):
    sum=0
    primes_lessthan_m_by2 = list(sympy.primerange(2,-(m//-2)))
    primes_between_m_by2_and_m = list(sympy.primerange(m//2+1,m))
    divisors_of_m=sympy.divisors(m,generator=False)
    divisors_of_m.remove(m)
    if m%2==0:
        divisors_of_m.remove(m//2)
    for p in primes_between_m_by2_and_m:
        sum+=p
    for p in primes_lessthan_m_by2:
        for d in divisors_of_m:
            if p< m//d and m%(d*p)!=0:
                sum+=d*p
    if sum%m==0:
       A350878.append(m)
print(A350878)

a(16)-a(20) from Amiram Eldar, Jan 21 2022

a(21)-a(27) from David A. Corneth, Jan 21 2022

A173663 Numbers k that divide the k-th partial sum of all semiprimes.

Original entry on oeis.org

1, 2, 9, 19, 29, 44, 632, 11829, 19262, 25286, 26606, 29824, 247273, 310556, 491240, 1419166, 1601984, 9509238, 113333959, 220531559, 1034662494, 8323088842, 13102043650, 14053673678, 23505911647

Offset: 1

Jonathan Vos Post, Nov 24 2010

a(26) > 3*10^10. - Donovan Johnson, Nov 26 2010

			a(1) = 1 because 1 divides the first semiprime 4, trivially also the first partial sum of all semiprimes.
a(2) = 2 because A062198(2) = A001358(1) + A001358(2) = 4 + 6 = 10 is divisible by 2.
a(3) = 9 because A062198(9) = 126 = 2 * 3^2 * 7 is divisible by 9.
a(4) = 19 because A062198(19) = 532 = 2^2 * 7 * 19 is divisible by 19.
a(5) = 29 because A062198(29) = 1247 = 29 * 43 is divisible by 29.
a(6) = 44 because A062198(44) = 2904 = 44 * 66.

Cf. A001358, A007506.

SemiprimeQ[n_Integer] := If[Abs[n]<2, False, (2==Plus@@Transpose[FactorInteger[Abs[n]]][[2]])]; nn=10^6; sm=0; cnt=0; Reap[Do[If[SemiprimeQ[n], cnt++; sm=sm+n; If[Divisible[sm, cnt], Sow[cnt]]], {n, nn}]][[2, 1]]

s=0; p=0; for(n=1, 1e9, until(bigomega(p++)==2,); (s+=p)%n || print1(n", ")) \\ M. F. Hasler, Nov 24 2010

{k: k | Sum_{i=1..k} A001358(i)}.

Extended by T. D. Noe, Nov 24 2010
a(1)-a(17) double-checked and a(18) from M. F. Hasler, Nov 25 2010
a(19) from Ray Chandler, Nov 25 2010
a(20)-a(25) from Donovan Johnson, Nov 26 2010

A234540 Nonprimes k that divide the sum of the nonprimes up to k.

Original entry on oeis.org

1, 22, 25, 118, 414, 2745, 10623, 63201, 1039161, 1693864, 2285297, 52962021, 66998865, 232974402, 315796805, 336125877, 834972771, 1233903294, 1309750075, 1617454215, 2056836133, 5455816485, 8589896196, 9030058217, 10622144467, 12324258770, 33725308558

Offset: 1

Vicente Izquierdo Gomez, Dec 27 2013

nonn

Standard heuristics suggest that this sequence is infinite. - Charles R Greathouse IV, Dec 29 2013

			a(2) = 22 because 1 + 4 + 6 + 8 + 9 + 10 + ... + 22 = 176 and 176/22 = 8.

Cf. A007506.

s = 0; Do[If[PrimeQ[k], Continue[]]; s += k; If[Mod[s, k] == 0, Print[k]], {k, 10^10}] (* Gomez *)
Select[Range[10^4], Not[PrimeQ[#]] && Divisible[Sum[Boole[Not[PrimeQ[m]]]m, {m, #}], #] &] (* Alonso del Arte, Dec 29 2013 *)

v=List([s=1]); forcomposite(n=4,1e9,if(s%n==0, listput(v,n)); s+=n); Vec(v) \\ Charles R Greathouse IV, Dec 29 2013

a(14)-a(27) from Donovan Johnson, Dec 30 2013

cp's OEIS Frontend

A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

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A071089 Remainder when sum of first n primes is divided by n-th prime.

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A028581 Quotients associated with A024011.

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A028582 Dividends associated with A024011.

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A274649 a(n) is the smallest odd prime that divides n + the sum of all smaller primes, or 0 if no such prime exists.

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A274995 a(n) is the smallest odd prime that divides (-n) + the sum of all smaller primes, or 0 if no such prime exists.

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A330580 Composite numbers n that divide the sum of all composite numbers <= n.

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PARI

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Extensions

A350878 Integers m that divide the sum of values d*p < m, where d is a divisor of m, p is a prime, and d*p does not divide m.

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A350878 Integers m that divide the sum of values dp < m, where d is a divisor of m, p is a prime, and dp does not divide m.