A024934 -id:A024934 - cp's OEIS frontend

A143851 Primes p that divide the sum of their residues modulo all smaller primes (=A024934(p)).

2, 13, 167, 2239, 439867, 724031017, 1990127567, 54892225873

Offset: 1

Neil Fernandez, Sep 03 2008

Also, primes p such that p divides A024924(p). The prime terms of A065132.

			13 is congruent to 1,1,3,6 and 2, modulo 2,3,5,7 and 11 respectively. 1+1+3+6+2=13, which is a multiple of the original number, 13. So the original number, is in the sequence.

Cf. A065132

For[n = 1, n < 1000001, n++, p = Prime[n]; m = Mod[Sum[Mod[p, Prime[i]], {i, 1, n - 1}], p]; If[m == 0, Print[p]]]

a(6)-a(8) from Max Alekseyev, Feb 10 2012

A383844 a(n) is the number of occurences of n in A024934.

Original entry on oeis.org

3, 3, 0, 1, 2, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1

Offset: 0

Miles Englezou, May 12 2025

nonn

Every k in A024934 is the sum of a tuple (x_1, ..., x_t) for prime(t) <= k < prime(t+1), where x_i = k mod prime(i). The tuples can be seen combinatorially as sets of t counters where the i-th counter cycles through 0 to prime(i)-1.
Since A024934(n) > n for n > 21, the set of numbers k for which A024934(k) = n is bounded above by n for those n, though smaller bounds are possible.
It is interesting to compare A024934 to A049802 (and likewise the current sequence with A383327). Every m in A049802 is also the sum of a tuple of congruences except the moduli are ascending powers of 2. Ordered by divisibility, the set of moduli in A049802 therefore form a chain (i.e., they are totally ordered) whilst the set of moduli in A024934 form an antichain (no one modulus divides any other). These opposed orders mean the two sequences behave quite differently.
Comparison of A049802 and A383327 vs. A024934 and {a(n)}:
A049802 and A383327:
 - 0 appears infinitely many times in A049802, for every m = 2^n. Therefore 0 is not a term of A383327.
 - A049802(2^n+1) = n. Therefore every n appears at least once in A049802.
 - It is likely that every n appears at least once in A383327, however this is currently conjectural.
 - If A049802(k) = m for 2^r-2^(r-2) <= k < 2^r, and if the rightmost summand in the tuple of m is x, then for s >= 0, A049802(k+2^((r-1)+s)) = m + x*(s+1).
A024934 and {a(n)}:
 - 0 appears 3 times in A024934, for n = 0, 1, 2. Therefore a(0) = 3.
 - It is not true that every n appears at least once in A024934 (e.g., 2 and 5 are not terms), and this is likely to be the case for infinitely many n, meaning it is likely that a(n) = 0 for infinitely many n.
 - It appears to be unlikely that a(k) = n for every n: for 0 < n < 3500, a(n) <= 3 (and a(n) = 3 only for n = 0, 1, 8, 37, 781).

			 n | a(n) | k such that A024934(k) = n
---+------+---------------------------
 0 |  3   | {0, 1, 2}
 1 |  3   | {3, 4, 6}
 2 |  0   | {}
 3 |  1   | {5}
 4 |  2   | {7, 10}
 5 |  0   | {}
 6 |  1   | {8}
 7 |  1   | {9}
 8 |  3   | {11, 12, 15}
 9 |  0   | {}
10 |  1   | {14}
11 |  0   | {}
12 |  1   | {16}
13 |  1   | {13}
14 |  0   | {}
15 |  0   | {}
16 |  0   | {}
17 |  0   | {}
18 |  1   | {17}
19 |  0   | {}
20 |  1   | {18}
--------------------------------------
Illustration of some tuples
 n | A024934(n) |     tuple of n
---+------------+---------------------
 0 |     0      | ()
 1 |     0      | ()
 2 |     0      | (0)
 3 |     1      | (1 0)
 4 |     1      | (0 1)
 5 |     3      | (1 2 0)
 6 |     1      | (0 0 1)
 7 |     4      | (1 1 2 0)
 8 |     6      | (0 2 3 1)
 9 |     7      | (1 0 4 2)
10 |     4      | (0 1 0 3)
11 |     8      | (1 2 1 4 0)
12 |     9      | (0 0 2 5 1)
13 |    13      | (1 1 3 6 2 0)
14 |    10      | (0 2 4 0 3 1)
15 |     8      | (1 0 0 1 4 2)
16 |    12      | (0 1 1 2 5 3)
17 |    18      | (1 2 2 3 6 4 0)
18 |    20      | (0 0 3 4 7 5 1)
19 |    27      | (1 1 4 5 8 6 2 0)
20 |    28      | (0 2 0 6 9 7 3 1)

Cf. A024934, A049802, A383327.

a(n) = my(f, S=[], b); (f(m) = my(r=0); forprime(p=2, m, r+=m%p); return(r)); if(n<=21, b=26, b=n); for(k=0, b, if(f(k)==n, S=concat(S, k))); return(#S)

A072267 Duplicate of A024934.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 4, 6, 7, 4, 8, 8, 13, 10, 8, 12, 18, 20, 27, 28, 26, 21, 29, 33, 37, 31, 37, 37

Offset: 2

dead

A024924 a(n) = Sum_{k=1..n} prime(k)*floor(n/prime(k)).

Original entry on oeis.org

0, 0, 2, 5, 7, 12, 17, 24, 26, 29, 36, 47, 52, 65, 74, 82, 84, 101, 106, 125, 132, 142, 155, 178, 183, 188, 203, 206, 215, 244, 254, 285, 287, 301, 320, 332, 337, 374, 395, 411, 418, 459, 471, 514, 527, 535, 560, 607, 612, 619, 626, 646, 661, 714, 719, 735, 744, 766, 797, 856

Offset: 0

Clark Kimberling

nonn

For n > 2, sum of all distinct prime factors composing numbers from 2 to n.

M. Kalecki, On certain sums extended over primes or prime factors (in Polish), Prace Mat., Vol. 8 (1963/64), pp. 121-129.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Plot of a(n) / ((Pi^2/12) * n^2/log(n)) for n = 1..10^8

Partial sums of A008472.

Cf. A000720, A024934, A034387.

[0] cat [ &+[ NthPrime(k)*Floor(n/NthPrime(k)): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Jul 28 2019

Join[{0}, Table[Sum[Prime[k] Floor[n / Prime[k]], {k, 1, n}], {n, 1, 60}]] (* Vincenzo Librandi, Jul 28 2019 *)
Join[{0}, Accumulate[Table[Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}]]] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = sum(k=1, n, prime(k)*(n\prime(k))); \\ Michel Marcus, Mar 01 2015

a(n)=my(s); forprime(p=2,n, s+=n\p*p); s \\ Charles R Greathouse IV, Jun 26 2020

from sympy import prime
def A024924(n): return sum((p:=prime(k))*(n//p) for k in range(1,n+1)) # Chai Wah Wu, Sep 18 2023

a(n) = n*A000720(n) - A024934(n). - Max Alekseyev, Feb 10 2012
a(n) = A034387([n/1]) + A034387([n/2]) + ... + A034387([n/n]). Terms can be computed efficiently with the following formula: a(n) = A034387([n/1]) + ... + A034387([n/m]) - m*A034387([n/m]) + Sum_{prime p<=n/m} p*[n/p], where m = [sqrt(n)]. - Max Alekseyev, Feb 10 2012
G.f.: Sum_{k >=1} (prime(k)*x^prime(k)/(1-x^prime(k)))/(1-x). - Vladeta Jovovic, Aug 11 2004
a(n) ~ ((Pi^2 + o(1))/12) * n^2/log(n) (Kalecki, 1963/64). - Amiram Eldar, Mar 04 2021

a(0)=0 prepended by Max Alekseyev, Feb 10 2012

A033955 a(n) = sum of the remainders when the n-th prime is divided by primes up to the (n-1)-th prime.

Original entry on oeis.org

0, 1, 3, 4, 8, 13, 18, 27, 29, 46, 56, 70, 74, 88, 98, 134, 147, 171, 200, 217, 252, 274, 309, 323, 348, 418, 448, 471, 522, 571, 629, 685, 739, 777, 793, 853, 954, 997, 1002, 1120, 1148, 1220, 1338, 1419, 1466, 1540, 1615, 1573, 1633, 1707, 1825, 1892, 1986

Offset: 1

Armand Turpel (armandt(AT)unforgettable.com)

Row sums of A207409. - Bob Selcoe, Apr 14 2014

			a(5) = 8. The remainders when the fifth prime 11 is divided by 2, 3, 5, 7 are 1, 2, 1, 4, respectively and their sum = 8.

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

Cf. A067435, A067436, A024934, A207409.

P:= [seq(ithprime(i),i=1..200)]:
f:= proc(n) local j;  add(P[n] mod P[j],j=1..n-1) end proc:
map(f, [$1..200]); # Robert Israel, Dec 29 2020

a[n_] := Sum[Mod[Prime[n], Prime[i]], {i, 1, n-1}]
Table[Total[Mod[Prime[n],Prime[Range[n-1]]]],{n,60}] (* Harvey P. Dale, Mar 07 2018 *)

{for(n=1, 200, print1(sum(k=1, n, prime(n)%prime(k)), ", "))}

from sympy import prime; {print(sum(prime(n)%prime(k) for k in range(1,n)), end =', ') for n in range(1,54)} # Ya-Ping Lu, May 05 2024

a(n) = Sum_{k=1..n-1} ( prime(n) mod prime(k) ).

Edited by Dean Hickerson, Mar 02 2002

A067439 a(n) = sum of all the remainders when n is divided by positive integers less than and coprime to n.

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 8, 6, 9, 5, 22, 8, 28, 15, 19, 20, 51, 20, 64, 30, 39, 33, 98, 33, 83, 56, 89, 55, 151, 46, 167, 95, 107, 95, 150, 71, 233, 120, 172, 106, 297, 92, 325, 163, 186, 162, 403, 144, 358, 189, 279, 217, 505, 173, 375, 230, 342, 276, 635, 165, 645, 338

Offset: 1

Amarnath Murthy, Jan 29 2002

nonn

			a(8) = 6. The remainders when 8 is divided by the coprime numbers 1, 3, 5 and 7 are 0, 2, 3 and 1, whose sum = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A004125, A067435, A067436, A024934, A033955, A111490.

Cf. A072514, A008683.

a := n -> add(ifelse(igcd(n, i) = 1, irem(n, i), 0), i = 1..n-1):
seq(a(n), n = 1..62);  # Peter Luschny, May 14 2025

a[n_] := Sum[If[GCD[i, n]>1, 0, Mod[n, i]], {i, 1, n-1}]
Table[Total[Mod[n,#]&/@Select[Range[n-1],CoprimeQ[#,n]&]],{n,70}] (* Harvey P. Dale, May 22 2012 *)

a(n)=sum(i=1,n-1,if(gcd(n,i)==1,n%i)) \\ Charles R Greathouse IV, Jul 17 2012

From Ridouane Oudra, May 14 2025: (Start)
a(n) = A004125(n) - A072514(n).
a(n) = Sum_{d|n} d*mu(d)*A004125(n/d).
a(n) = Sum_{d|n} mu(d)*f(n,d), where f(n,d) = Sum_{i=1..n/d} (n mod d*i).
a(p) = A004125(p), for p prime.
a(p^k) = A004125(p^k) - p*A004125(p^(k-1)), for p prime and k >= 0.
a(p^k) = A072514(p^(k+1))/p - A072514(p^k), for p prime and k >= 0. (End)

Edited by Dean Hickerson, Feb 15 2002

A136021 Sum of the proper prime divisors of all numbers up to 10^n.

Original entry on oeis.org

0, 19, 1047, 64373, 4481640, 340900331, 27436000061, 2292176360707, 196818634871899, 17246903703574357, 1534951275195670059, 138293592048140425181, 12583738258227621100170, 1154435823206834353336284, 106638384745041347295504523

Offset: 0

Enoch Haga, Dec 10 2007

The sum of the distinct prime factors less than k for all 1 <= k <= 10^n, as tabulated for the individual k in A105221.

			a(1)=19 because 10^1=10 and the factors to be summed are 2 for 4, added to 2 and 3 for 6, added to 2 for 8, added to 3 for 9, added to 2 and 5 for 10.

Cf. A136022, A136023, A136024, A136025.

A105221 := proc(n) local a,pfs,i ; a :=0 ; pfs := ifactors(n)[2] ; for i in pfs do if op(1,i) <> 1 and op(1,i) <> n then a := a+op(1,i) ; fi ; od: RETURN(a) ; end: A136021 := proc(n) add(A105221(i),i=2..10^n) ; end: for n from 1 do print(n,A136021(n)) ; od: # R. J. Mathar, Dec 12 2007

f[n_] := Plus @@ (First@# & /@ FactorInteger@ n); k = 2; s = 0; lst = {}; Do[While[k < 10^n + 1, If[ ! PrimeQ@k, s = s + f@k]; k++ ]; AppendTo[ lst, s]; Print[{n, s}], {n, 8}] (* Robert G. Wilson v, Aug 06 2010 *)

10 'distinct prime factors of composites <=10^n 20 S=0:N=N+1:Z=N\2 30 'print N; 40 for F=1 to Z:Q=N/F: if Q<>int(Q) then 60 50 S=S+F: if F=prmdiv(F) and F>1 then C=C+1:G=G+F 60 next F 70 'print C,G 80 if N=10^1 or N=10^2 or N=10^3 or N=10^4 or N=10^5 or N=10^6 or N=10^7 then print G:stop 90 C=0 100 goto 20

a(n) = Sum_{k=1..10^n} A105221(k). - R. J. Mathar, Dec 12 2007

a(n) = Sum_{prime p<10^n} p*floor((10^n-p)/p) = A006880(n)*10^n - A024934(10^n) - A046731(n). - Max Alekseyev, Jan 30 2012

One more term from R. J. Mathar, Dec 12 2007
Edited by R. J. Mathar, Apr 17 2009
a(7) & a(8) from Robert G. Wilson v, Aug 06 2010
a(9)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jun 29 2014

A274422 Numbers m such that there exists a j for which m = Sum_{k=1..j} (m mod k), where k runs through the largest j primes less than m.

Original entry on oeis.org

13, 17, 20, 23, 24, 34, 40, 82, 126, 200, 612, 1154, 1692, 2434, 2806, 3060, 3142, 4052, 4460, 4596, 5020, 5908, 6424, 7304, 7596, 8030, 8040, 9044, 11398, 12254, 12914, 13412, 13716, 16006, 16800, 18560, 22438, 23140, 24424, 24746, 25706, 28318, 29272, 30580

Offset: 1

Paolo P. Lava, Jun 21 2016

			13 mod 11 + 13 mod 7 + 13 mod 5 + 13 mod 3 + 13 mod 2 = 2 + 6 + 3 + 1 + 1 = 13;
40 mod 37 + 40 mod 31 + 40 mod 29 + 40 mod 23 = 3 + 9 + 11 + 17 = 40.

Paolo P. Lava, Table of n, a(n) for n = 1..250
Paolo P. Lava, First 200 terms with the number of primes j

Cf. A000040, A024934, A274423, A274424.

P:=proc(q) local a,b,k,n; for n from 3 to q do a:=0; b:=prevprime(n);
while n>a do a:=a+(n mod b); if b>2 then b:=prevprime(b); else break; fi; od;
if n=a then print(n); fi; od; end: P(10^9);

A274423 Let s(n,j) be Sum_{i=1..j} (prime(primepi(n) + i) mod n). Numbers n such that there exists j with s(n,j) = n.

Original entry on oeis.org

2, 3, 4, 6, 8, 44, 48, 66, 90, 108, 156, 206, 284, 854, 1002, 1188, 1194, 1212, 1320, 2234, 2460, 2792, 3336, 3478, 7096, 7164, 7218, 7236, 7752, 8304, 9164, 10272, 12090, 12594, 13322, 15530, 17038, 17162, 18026, 18212, 20412, 20494, 21966, 23374, 23518, 24664

Offset: 1

Paolo P. Lava, Jun 21 2016

			47 mod 44 + 53 mod 44 + 59 mod 44 + 61 mod 44 = 3 + 9 + 15 + 17 = 44.

Paolo P. Lava, Table of n, a(n) for n = 1..200
Paolo P. Lava, First 200 terms with the number of primes j

Cf. A000040, A024934, A274422, A274424.

P:=proc(q) local a,b,k,n; for n from 2 to q do a:=0;
b:=nextprime(n); while n>a do a:=a+(b mod n); b:=nextprime(b); od;
if n=a then print(n); fi; od; end: P(10^9);

Name corrected by David A. Corneth, Jun 22 2016

A274424 Numbers k such that there exists an m for which k = Sum_{j=1..m} (k mod prime(j)).

Original entry on oeis.org

13, 19, 48, 63, 67, 76, 94, 99, 123, 141, 143, 150, 179, 193, 247, 249, 285, 339, 404, 445, 517, 693, 711, 798, 969, 982, 1054, 1138, 1233, 1245, 1257, 1262, 1364, 1524, 1531, 1569, 1613, 1694, 1701, 1743, 1745, 1928, 2018, 2070, 2114, 2224, 2339, 2461, 2770

Offset: 1

Paolo P. Lava, Jun 21 2016

			48 mod 2 + 48 mod 3 + 48 mod 5 + 48 mod 7 + 48 mod 11 + 48 mod 13 + 48 mod 17 + 48 mod 19 + 48 mod 23 = 0 + 0 + 3 + 6 + 4 + 9 + 14 + 10 + 2 = 48, so 48 is a term.

Paolo P. Lava, Table of n, a(n) for n = 1..1500
Paolo P. Lava, First 1500 terms with the number of the first primes

Cf. A000040, A024934, A274422, A274423.

P:=proc(q) local a,b,k,n; for n from 2 to q do a:=0; b:=2;
while n>a do a:=a+(n mod b); b:=nextprime(b); od;
if n=a then  print(n); fi; od; end: P(10^9);

cp's OEIS Frontend

A143851 Primes p that divide the sum of their residues modulo all smaller primes (=A024934(p)).

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A383844 a(n) is the number of occurences of n in A024934.

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A072267 Duplicate of A024934.

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A024924 a(n) = Sum_{k=1..n} prime(k)*floor(n/prime(k)).

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A033955 a(n) = sum of the remainders when the n-th prime is divided by primes up to the (n-1)-th prime.

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A067439 a(n) = sum of all the remainders when n is divided by positive integers less than and coprime to n.

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A136021 Sum of the proper prime divisors of all numbers up to 10^n.

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