A102647 -id:A102647 - cp's OEIS frontend

A165657 Numerators of A002110 divided by A102647, starting from the second term of both.

2, 3, 15, 105, 385, 5005, 17017, 323323, 1062347, 30808063, 434113615, 35336848261, 1448810778701, 33545541876077, 266186053068611, 5426100312552455, 9156001667401012567, 42962777054727828199

Offset: 1

Creighton Dement, Sep 24 2009

Cf. A002110, A102647, A165658, A165659, A165660.

a1(n)=prod(i=1, n, prime(i)); b1(n)=prod(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, numerator(a1(n)/b1(n))); for(n=1, 20, print1(a(n) ", "))

A165658 Denominators of A002110 divided by A102647, starting from the second term of both.

Original entry on oeis.org

1, 1, 1, 4, 6, 48, 64, 96, 576, 1728, 13824, 165888, 1036800, 9953280, 119439360, 297271296, 134369280000, 222953472000, 75246796800, 32105299968000, 229323571200, 568865783808000000, 125150472437760000, 6129819058176000

Offset: 1

Creighton Dement, Sep 24 2009

Cf. A002110, A102647, A165657, A165659, A165660.

a1(n)=prod(i=1, n, prime(i));
b1(n)=prod(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, denominator(a1(n)/b1(n)));
for(n=1, 25, print1(a(n) ", "))

A062347 a(n) = (product of first n primes) modulo prime(n+1).

Original entry on oeis.org

1, 2, 1, 2, 1, 9, 8, 18, 15, 17, 19, 11, 6, 26, 35, 27, 15, 55, 30, 24, 28, 4, 47, 56, 28, 28, 75, 25, 105, 30, 9, 52, 111, 89, 46, 132, 53, 97, 104, 106, 42, 94, 158, 184, 155, 5, 6, 45, 87, 78, 86, 26, 120, 33, 242, 66, 86, 2, 214, 2, 198, 127, 14, 112, 7, 141, 77, 309

Offset: 0

Henry Bottomley, Jul 06 2001

nonn

			a(4)=1 since 2*3*5*7 = 210 = 19*11 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)

Cf. A000040, A002110, A051126, A102647.

Mod[ #[ [ 1 ] ], #[ [ 2 ] ] ]&/@ Transpose[ {FoldList[ Times, 1, Prime[ Range[ 70 ] ] ], Prime[ Range[ 71 ] ]} ]
Join[{1},Module[{nn=70,prs},prs=Prime[Range[nn]];Table[Mod[Fold[Times,Take[prs,n-1]],prs[[n]]],{n,2,nn}]]] (* Harvey P. Dale, Jun 30 2024 *)

{ n=-1; f=1; forprime (p=2, prime(1001), write("b062347.txt", n++, " ", f%p); f*=p ) } \\ Harry J. Smith, Aug 05 2009

from sympy import sieve, primorial
print([1] + [primorial(k) % sieve[k+1] for k in range(1, 71)])
# Karl-Heinz Hofmann, Jan 26 2022

a(n) = A051126(A002110(n), A000040(n+1)).

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

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

Original entry on oeis.org

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

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

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.

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A165657 Numerators of A002110 divided by A102647, starting from the second term of both.

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A165658 Denominators of A002110 divided by A102647, starting from the second term of both.

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A062347 a(n) = (product of first n primes) modulo prime(n+1).

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

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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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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.

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Maple

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

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Mathematica

PARI

Python

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