A033955 -id:A033955 - cp's OEIS frontend

A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

2, 5, 5, 17, 28, 41, 58, 77, 50, 129, 16, 197, 119, 281, 164, 127, 440, 501, 568, 71, 356, 791, 46, 321, 530, 1161, 1264, 457, 1480, 1593, 344, 1851, 284, 2127, 2276, 809, 2584, 2747, 1457, 441, 1633, 1149, 3638, 3831, 1007, 4227, 4438

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= a(n)/A165660(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165660.

a1(n)=sum(i=1, n, prime(i));
b1(n)=sum(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, numerator(a1(n)/b1(n)));
for(n=1, 50, print1(a(n) ", "))

Typo in definition corrected by Creighton Dement, Oct 09 2009

A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

Original entry on oeis.org

1, 3, 2, 8, 13, 18, 27, 29, 23, 56, 7, 74, 44, 98, 67, 49, 171, 200, 217, 28, 137, 309, 17, 116, 209, 448, 471, 174, 571, 629, 137, 739, 111, 793, 853, 318, 997, 1002, 560, 164, 610, 446, 1419, 1466, 385, 1615, 1573, 1633, 1707, 1825, 946, 662, 2221, 781, 1198

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= A165659(n)/a(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165659

a1(n)=sum(i=1, n, prime(i)); b1(n)=sum(i=1, n, prime(n+1)%prime(i)); a(n)=if(n<0, 0, denominator(a1(n)/b1(n))); for(n=1, 50, print1(a(n) ", "))

Terms corrected by Creighton Dement, Oct 03 2009
Removed a conjecture - R. J. Mathar, Oct 09 2009
Typo in definition corrected by Creighton Dement, Oct 09 2009

A033956 Add prime(n) to A033955.

Original entry on oeis.org

2, 4, 8, 11, 19, 26, 35, 46, 52, 75, 87, 107, 115, 131, 145, 187, 206, 232, 267, 288, 325, 353, 392, 412, 445, 519, 551, 578, 631, 684, 756, 816, 876, 916, 942, 1004, 1111, 1160, 1169, 1293, 1327, 1401, 1529, 1612, 1663, 1739, 1826, 1796, 1860, 1936, 2058

Offset: 1

N. J. A. Sloane, Armand Turpel armandt(AT)unforgettable.com

More terms from Erich Friedman.

a(20) onward corrected by Sean A. Irvine, Jul 22 2020

A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st prime.

Original entry on oeis.org

1, 1, 2, 2, 8, 36, 288, 1920, 2880, 120960, 362880, 6386688, 34836480, 217728000, 3881779200, 275904921600, 1785411403776, 28217548800000, 608662978560000, 3492203839488000, 964122158039040000, 2224367550332928000, 1314079960596480000000, 3758268687305932800000

Offset: 1

Hans Boelens (h.p.m.boelens(AT)pl.hanze.nl), Feb 02 2005

nonn

			Prime(6) = 13, 13 mod 2 = 1, 13 mod 3 = 1, 13 mod 5 = 3, 13 mod 7 = 6, 13 mod 11 = 2 so a(6) = 1*1*3*6*2 = 36.

Robert Israel, Table of n, a(n) for n = 1..413

Cf. A033955, A062347.

f:= proc(n) local p,i;
  p:= ithprime(n);
  mul(p mod ithprime(i),i=1..n-1)
end proc:
map(f, [$1..25]); # Robert Israel, Jan 12 2021

f[n_] := Times @@ Mod[ Prime[n], Table[ Prime[i], {i, n - 1}]]; Table[ f[n], {n, 22}] (* Robert G. Wilson v, Feb 04 2005 *)
Join[{0},Table[Times@@Mod[Prime[n],Prime[Range[n-1]]],{n,2,30}]] (* Harvey P. Dale, May 16 2019 *)

a(n) = my(pr = 1, pn = prime(n)); forprime (q=1, precprime(pn-1), pr *= (pn % q)); pr; \\ Michel Marcus, Jan 12 2021

More terms from Robert G. Wilson v, Feb 04 2005

a(1) (an empty product, therefore 1 by standard convention) corrected by N. J. A. Sloane, Jan 11 2021

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

A173655 Triangle read by rows: T(n,k) = prime(n) mod prime(k), 0 < k <= n.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 1, 4, 0, 1, 1, 3, 6, 2, 0, 1, 2, 2, 3, 6, 4, 0, 1, 1, 4, 5, 8, 6, 2, 0, 1, 2, 3, 2, 1, 10, 6, 4, 0, 1, 2, 4, 1, 7, 3, 12, 10, 6, 0, 1, 1, 1, 3, 9, 5, 14, 12, 8, 2, 0, 1, 1, 2, 2, 4, 11, 3, 18, 14, 8, 6, 0, 1, 2, 1, 6, 8, 2, 7, 3, 18, 12, 10, 4, 0

Offset: 1

Juri-Stepan Gerasimov, Nov 24 2010

			Triangle begins as:
  0;
  1, 0;
  1, 2, 0;
  1, 1, 2, 0;
  1, 2, 1, 4, 0;
  1, 1, 3, 6, 2,  0;
  1, 2, 2, 3, 6,  4,  0;
  1, 1, 4, 5, 8,  6,  2,  0;
  1, 2, 3, 2, 1, 10,  6,  4, 0;
  1, 2, 4, 1, 7,  3, 12, 10, 6, 0;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A001223 (2nd diagonal), A033955 (row sums), A102647 (row products excluding 0's), A031131 (3rd diagonal after first 3 terms).

Cf. A172662, A174497, A174947, A174996, A175620.

A173655:= func< n,k | NthPrime(n) mod NthPrime(k) >;
[A173655(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 10 2024

A173655 := proc(n,k) ithprime(n) mod ithprime(k) ;end proc:
seq(seq(A173655(n,k),k=1..n),n=1..20) ; # R. J. Mathar, Nov 24 2010

Flatten[Table[Mod[Prime[n], Prime[Range[n]]], {n, 15}]]

forprime(p=2,40,forprime(q=2,p,print1(p%q", "))) \\ Charles R Greathouse IV, Dec 21 2011

def A173655(n,k): return nth_prime(n)%nth_prime(k)
flatten([[A173655(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Apr 10 2024

A207409 Triangular array: T(n,k) = prime(n) mod prime(k), 1 <= k < n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 6, 2, 1, 2, 2, 3, 6, 4, 1, 1, 4, 5, 8, 6, 2, 1, 2, 3, 2, 1, 10, 6, 4, 1, 2, 4, 1, 7, 3, 12, 10, 6, 1, 1, 1, 3, 9, 5, 14, 12, 8, 2, 1, 1, 2, 2, 4, 11, 3, 18, 14, 8, 6, 1, 2, 1, 6, 8, 2, 7, 3, 18, 12, 10, 4, 1, 1, 3, 1, 10, 4, 9, 5, 20, 14, 12, 6, 2, 1

Offset: 2

Clark Kimberling, Feb 17 2012

Conjecture: For each row in the triangle, the maximum value occurs only once, and for n>2 it is never the first entry and the value previous to it in the row is always odd. - Mike Jones, Jul 12 2024

			Top 7 rows:
  n=2:  1............. 3 mod 2
  n=3:  1 2............5 mod 2, 5 mod 3
  n=4:  1 1 2..........7 mod 2, 7 mod 3, 7 mod 5
  n=5:  1 2 1 4
  n=6:  1 1 3 6 2
  n=7:  1 2 2 3 6 4
  n=8:  1 1 4 5 8 6 2

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Cf. A000040.

Cf. A001223 (right diagonal), A033955 (row sums), A039731 (row maxs).

P := select(isprime, [$1..100]):
seq(seq(P[n] mod P[k],k=1..n-1),n=1..nops(P)); # Robert Israel, May 01 2017

t = Table[Mod[Prime[n + 1], Prime[k]], {n, 1, 15}, {k, 1, n }];
Flatten[t]   (* this sequence *)
TableForm[t] (* this sequence as a triangle *)

row(n) = my(p=prime(n)); vector(n-1, k, p % prime(k)); \\ Michel Marcus, Jul 13 2024

A342173 a(n) = Sum_{j=1..n-1} floor(prime(n)/prime(j)).

Original entry on oeis.org

0, 1, 3, 6, 11, 14, 20, 23, 30, 39, 43, 53, 60, 64, 71, 81, 92, 96, 107, 115, 118, 130, 136, 148, 164, 171, 175, 183, 186, 194, 222, 229, 241, 245, 265, 269, 282, 293, 301, 313, 325, 329, 351, 354, 362, 366, 392, 417, 424, 428, 437, 450, 454, 473, 485, 498, 511

Offset: 1

J. M. Bergot and Robert Israel, Mar 03 2021

nonn

a(n) is the sum of the quotients in integer division of prime(n) by all smaller primes.

			a(4) = floor(7/2) + floor(7/3) + floor(7/5) = 6.

Robert Israel, Table of n, a(n) for n = 1..10000
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).

Cf. A033955, A006093, A013939, A308495.

f:= proc(n) local t,i,s;
  t:= ithprime(n);
  add(floor(t/ ithprime(i)),i=1..n-1)
end proc:
map(f, [$1..100]);

Table[Sum[Floor[Prime[n]/Prime[j]],{j,n-1}],{n,64}] (* Stefano Spezia, Mar 04 2021 *)

a(n) = sum(j=1, n-1, prime(n)\prime(j)); \\ Michel Marcus, Mar 04 2021

a(n) = A308495(n) - 2. - Hugo Pfoertner, Mar 04 2021

a(n) = A013939(A006093(n)). - Flávio V. Fernandes, Jan 03 2025

A143801 Primes with sum of remainders modulo all smaller primes which is smaller than this sum for the preceding prime.

Original entry on oeis.org

223, 359, 383, 449, 503, 547, 701, 797, 881, 1049, 1097, 1229, 1307, 1439, 1627, 1733, 1759, 1987, 1997, 2027, 2221, 2287, 2309, 2437, 2477, 2579, 2617, 2647, 2801, 2861, 2903, 2999, 3023, 3067, 3167, 3191, 3329, 3467, 3581, 3697, 3761, 3911, 3947, 4057

Offset: 1

Neil Fernandez, Sep 01 2008

nonn

These are the k-th primes, where A033955(k) < A033955(k-1)

			When divided by 2,3,5,7,11,...., the number 211 gives remainders 1,1,1,1,2, etc., which sum to 1615 and the number 223 gives remainders 1,1,3,6,3, etc., which sum to 1573. 1573 is smaller than 1615, so 223 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Prime[#]&/@(Flatten[Position[Differences[Table[Total[Mod[p,Prime[Range[PrimePi[p]]]]],{p,Prime[Range[600]]}]],?(#<0&)]]+1) (* _Harvey P. Dale, Jul 18 2025 *)

A340541 Primes p such that the product of (p mod q) for primes q < p is a multiple of the sum of (p mod q) for primes q < p.

Original entry on oeis.org

3, 11, 17, 31, 59, 67, 73, 101, 103, 173, 179, 193, 199, 211, 223, 349, 401, 463, 491, 499, 557, 563, 569, 571, 577, 587, 607, 613, 619, 631, 673, 709, 751, 757, 769, 797, 809, 857, 859, 877, 911, 919, 929, 967, 1009, 1033, 1039, 1049, 1151, 1153, 1193, 1201, 1237, 1249, 1259, 1289, 1297, 1303

Offset: 1

J. M. Bergot and Robert Israel, Jan 11 2021

nonn

Primes prime(n) such that A102647(n) is divisible by A033955(n).

			a(3) = 17 is a term since (17 mod q) for primes q=2,3,5,7,11,13 are 1,2,2,3,6,4, and 1*2*2*3*6*4 = 288 is divisible by 1+2+2+3+6+4 = 18.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A033955, A102647.

P:= [seq(ithprime(i),i=1..1000)]:
filter:= proc(n) local L,k;
  L:= [seq(P[n] mod P[k],k=1..n-1)];
  convert(L,`*`) mod convert(L,`+`) = 0
end proc:
S:=select(filter, [$2..1000]):
map(t -> P[t], S);

isok(p) = {if (isprime(p) && (p>2), my(s=0, t=1); forprime(q=2, p-1, my(x= p%q); s += x; t *= x;); !(t % s););} \\ Michel Marcus, Jan 11 2021

cp's OEIS Frontend

A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

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A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

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A033956 Add prime(n) to A033955.

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A102647 a(n) = product of the remainders when the n-th prime is divided by primes up to the (n-1)-st 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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A173655 Triangle read by rows: T(n,k) = prime(n) mod prime(k), 0 < k <= n.

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A207409 Triangular array: T(n,k) = prime(n) mod prime(k), 1 <= k < n.

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A342173 a(n) = Sum_{j=1..n-1} floor(prime(n)/prime(j)).

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