A024934 -id:A024934 - cp's OEIS frontend

A065132 Arithmetic mean of first n terms of A008472 is an integer.

2, 13, 134, 167, 2239, 62268, 75255, 135681, 439867, 18139940, 23671044, 40892256, 312083956, 724031017, 1990127567, 2144843867, 2588619526, 7439533243, 15054156002, 54892225873, 69959798320, 79760490898, 282311798922

Offset: 1

Labos Elemer, Oct 15 2001

nonn

			Sum of first 13 terms of A008472 gives A024924(13) = 65 which is divisible by n = 13, so 13 is here: 0+2+3+2+5+5+7+2+3+7+11+5+13 = 65 = 13*5 = A024924(13).

Cf. A024924, A008472.

s=0; Do[s=s+sp[n]; If[IntegerQ[n/25000], Print[n]]; If[IntegerQ[s/n], Print[{n, s, s/n}]], {n, 2, 4000000}] where sp[n]=A008472(n).

Integers n that divide A024924(n)=A008472(1)+A008472(2)+...+A008472(n).
Also, integers n that divide A024934(n).
Prime terms are listed in A143851.

a(10)-a(19) from Donovan Johnson, Nov 22 2009

a(20)-a(23) from Donovan Johnson, Aug 31 2010

A101336 Alternating addition and subtraction of the residues of the primes less than the number.

Original entry on oeis.org

0, 0, 0, 1, -1, -1, 1, 2, 0, 3, -4, -4, -2, -1, 4, 2, 0, 0, 2, 3, -4, 6, -7, -7, -5, -9, 3, 7, 13, 14, 10, 10, 9, 2, -16, -13, -11, -10, 7, 23, 16, 16, 25, 26, 13, 11, -14, -14, -12, -4, -10, -23, -11, -10, -9, -25, -20, 2, 29, 29, 26, 27, -6, 4, 2, 10, 0, 0, -18, -37, -36, -35, -34, -34, 2, 1, 19, 16, 31, 32, 25, 28, -15, -15, -6, -27, 15

Offset: 0

Gordon Hamilton, Dec 24 2004

The amplitude and periodicity of fluctuations increase... for example a(813) through a(836) are all positive and a(914) through a(937) all are positive except for a(922).

			a(10) = -4 because 10 (mod 2) - 10 (mod 3) + 10 (mod 5) - 10 (mod 7) = 0-1+0-3.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A024934.

a:= n-> add(-irem(n, ithprime(i))*(-1)^i, i=1..numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 27 2015

Table[Total[Times@@@Partition[Riffle[Mod[n,Prime[Range[PrimePi[n]]]],{1,-1},{2,-1,2}],2]],{n,0,90}] (* Harvey P. Dale, Jun 03 2017 *)

a(0)=0 prepended by Alois P. Heinz, Jul 27 2015

A233131 Sum of remainders of n modulo all smaller composite numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 2, 5, 9, 14, 9, 15, 21, 28, 24, 33, 27, 37, 33, 44, 56, 69, 52, 66, 81, 88, 87, 105, 92, 111, 102, 122, 143, 165, 139, 163, 187, 212, 196, 223, 209, 237, 239, 244, 274, 305, 266, 298, 296, 330, 335, 371, 347, 384, 368, 407, 447, 488, 432, 474, 516, 529, 513, 558, 543, 590, 599, 647, 637, 687, 620

Offset: 0

Max Alekseyev, Dec 07 2013

nonn

Cf. A143853, A233344

Table[Total[Mod[n,Select[Range[n-1],CompositeQ]]],{n,0,80}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 01 2018 *)

a(n) = A004125(n) - A024934(n).

A383752 Product of nonzero remainders n mod p, over all primes p < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 6, 8, 3, 8, 10, 36, 24, 8, 30, 288, 420, 1920, 2268, 640, 270, 2880, 9240, 13824, 7560, 19200, 17820, 120960, 64064, 362880, 5054400, 1881600, 475200, 165888, 464100, 6386688, 4082400, 1228800, 2120580, 34836480, 23474880, 217728000

Offset: 1

Darío Clavijo, May 28 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000040, A024934, A013939, A102647, A309912.

A383752[n_] := Times @@ DeleteCases[Mod[n, Prime[Range[PrimePi[n - 2]]]], 0];
Array[A383752, 50] (* Paolo Xausa, Jun 05 2025 *)

a(n) = vecprod(select(x->(x!=0), apply(lift, apply(x->Mod(n, x), primes([2,n-1]))))); \\ Michel Marcus, May 28 2025

from sympy import primerange
def a(n):
    s = 1
    for p in primerange(0, n):
        if p > (n >> 1): s *= (n-p)
        elif (x:= n % p) > 0: s*= x
    return s
print([a(n) for n in range(1,41)])

a(p) = A102647(p) if p prime.

A024925 Sum of remainders of n mod prime(k), for k = 1,2,3,...,n.

Original entry on oeis.org

1, 2, 4, 9, 13, 19, 25, 38, 52, 64, 74, 92, 104, 122, 143, 172, 188, 218, 236, 268, 299, 329, 351, 393, 437, 473, 523, 569, 597, 646, 676, 737, 788, 836, 893, 959, 995, 1049, 1110, 1182, 1222, 1293, 1335, 1409, 1490, 1556, 1602, 1692, 1782, 1874, 1955, 2043, 2095, 2197, 2290

Offset: 1

Clark Kimberling

nonn

Cf. A004125, A024934.

[&+[n mod NthPrime(k): k in [1..n]]:n in [1..55]]; // Marius A. Burtea, Jul 16 2019

a(n) = sum(k=1, n, n % prime(k)); \\ Michel Marcus, Jul 18 2019

a(n) = my(s=0); forprime(p=2, prime(n), s += n%p); s; \\ Michel Marcus, Jul 18 2019

G.f.: x * (1 + x)/(1 - x)^3 - (1/(1 - x)) * Sum_{k>=1} prime(k) * x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jul 16 2019

A302867 a(n) is the sum of remainders n mod p, over primes p for which n falls between p and p+p^2.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 3, 6, 6, 4, 7, 8, 11, 8, 7, 11, 15, 20, 25, 26, 25, 20, 26, 33, 35, 29, 36, 36, 43, 46, 53, 61, 58, 49, 50, 58, 66, 56, 52, 61, 70, 73, 83, 83, 94, 82, 93, 105, 110, 122, 117, 116, 128, 141, 143, 149, 142, 125, 137, 150, 163, 146, 160, 174

Offset: 1

Meir-Simchah Panzer, May 06 2018

nonn

"Jubilees". Motivation: 7 years are counted 7 times and capped off with a 50th year, the Jubilee (Leviticus 25:8); similarly, 7 days are counted 7 times and capped off with "Chag ha-Atzeret" (The Festival of Stopping) in the Omer-counting cycle (ibid 23:15); and these iterative cycles overlay other iterative cycles, like the lunar cycle nested not-quite-evenly within the solar year. This sequence idealizes the overlaying of multiple cycles. Each prime p generates a "swell" of p waves each with max amplitude = p-1, a kind of wavelet that is added into the total signal that is the sequence (e.g., the swell generated by 3 is (3^2)+1 terms in length, running for n=3,...,12 and has values n mod 3 = 0,1,2,0,1,2,0,1,2,0).

			For n = 12, we sum over primes 3, 5, 7, 11: a(12) = 12 mod 3 + 12 mod 5 + 12 mod 7 + 12 mod 11 = 0 + 2 + 5 + 1 = 8. In contrast with A024934, the sum does not include 12 mod 2 since 12 > 2+2^2.

Similar to A024934, but waves generated by primes are wavelets.

a(n) = sum(k=1, n, (n % k)*isprime(k)*(n <= (k^2+k))); \\ Michel Marcus, May 14 2018

a(n) = Sum_{primes p, sqrt(n) - 1/2 < p <= n} (n mod p).

A384310 Numbers k such that A383844(k) and A383844(k+1) are nonzero.

Original entry on oeis.org

0, 3, 6, 7, 12, 20, 26, 27, 28, 53, 56, 61, 74, 88, 145, 146, 252, 289, 299, 308, 320, 323, 340, 471, 577, 578, 739, 1240, 1517, 1568, 1579, 1857, 2638, 3042, 3043, 3133, 3455, 3565, 4910, 8683, 8684, 8857, 8858, 9291, 14549, 17913, 18117, 20005, 21989, 32552, 37902, 42514, 44869, 47877, 49942

Offset: 1

Miles Englezou, Jun 04 2025

nonn

a(n) is the lesser term of a pair of consecutive nonzero terms in A383844.

Triplets of consecutive nonzero terms can also be found in A383844 and are represented here as pairs. Up to n = 83354 there are 8 such triplets, the least terms of each being 6, 26, 27, 145, 577, 3042, 8683, 8857.

			26 is a term since A383844(26) and A383844(27) are nonzero.
27 is a term since A383844(27) and A383844(28) are nonzero.
252 is a term since A383844(252) and A383844(253) are nonzero.
61890 is a term since A383844(61890) and A383844(61891) are nonzero.

Cf. A024934, A383844.

isok(n) = (count(n) = my(f, S=[], b);(f(m)=my(r=0); forprime(p=2, m, r+=m%p); return(r)); if(n<=21, b=26); if(n>21, b=n); if(n>=250, b=n^0.8); if(n>=6000, b=n^0.7); if(n>=21000, b=n^0.68); if(n>=43000, b=n^0.67); for(k=0, b, if(f(k)==n, S=concat(S,k))); return(S)); if(n==0 || (n>1 && count(n)<>[] && count(n+1)<>[]), return(1), return(0))

A384584 Numbers k such that A383844(k) = 2.

Original entry on oeis.org

4, 46, 62, 119, 145, 180, 200, 247, 305, 522, 707, 900, 1235, 1504, 1532, 1540, 2396, 3140, 4181, 4231, 6419, 9066, 9885, 14292, 17914, 22696, 33924, 35933, 38951, 80602

Offset: 1

Miles Englezou, Jun 04 2025

Numbers k such that there are exactly two m such that Sum_{i=1..t} m mod prime(i) for prime(t) <= m < prime(t+1) is equal to k (see A024934).

A383844(s) <= 3 for s <= 82000, with A383844(s) = 3 only for s = 0, 1, 8, 37, 781.

			4 is a term since 7 and 10 are the only numbers r such that A024934(r) = 4.
46 is a term since 29 and 30 are the only numbers r such that A024934(r) = 46.
9066 is a term since 552 and 566 are the only numbers r such that A024934(r) = 9066.

Cf. A024934, A383844.

isok(n) = (count(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)); if(count(n)==2, return(1), return(0))

cp's OEIS Frontend

A065132 Arithmetic mean of first n terms of A008472 is an integer.

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A101336 Alternating addition and subtraction of the residues of the primes less than the number.

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Maple

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A233131 Sum of remainders of n modulo all smaller composite numbers.

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A383752 Product of nonzero remainders n mod p, over all primes p < n.

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A024925 Sum of remainders of n mod prime(k), for k = 1,2,3,...,n.

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Magma

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A302867 a(n) is the sum of remainders n mod p, over primes p for which n falls between p and p+p^2.

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A384310 Numbers k such that A383844(k) and A383844(k+1) are nonzero.

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A384584 Numbers k such that A383844(k) = 2.

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PARI