A306612 -id:A306612 - cp's OEIS frontend

A306582 a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.

0, 2, 4, 34, 52, 194, 502, 1138, 4042, 5794, 5794, 62488, 798298, 5314448, 41592688, 483815692, 483815692, 5037219688, 18517814158, 18517814158, 19566774820732, 55249201504132, 1257253598786974, 6743244322196288, 24165921989926702, 24165921989926702, 5346711077171356252, 47449991406350138602, 278545375679341352084, 5604477496256287791854

Offset: 1

Charlie Neder, Jun 03 2019

If "strictly increasing" is replaced with "nondecreasing", this sequence becomes A000004.

Trivially, a(n) <= A002110(n)-2. Equality only holds for n = 0.

			  a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
  ==== ==================================================
     0        0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0, ...
     2        0, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2, ...
     4        0, 1, 4, 4,  4,  4,  4,  4,  4,  4,  4, ...
    34        0, 1, 4, 6,  1,  8,  0, 15, 11,  5,  3, ...
    52        0, 1, 2, 3,  8,  0,  1, 14,  6, 23, 21, ...
   194        0, 2, 4, 5,  7, 12,  7,  4, 10, 20,  8, ...
   502        0, 1, 2, 5,  7,  8,  9,  8, 19,  9,  6, ...
  1138        0, 1, 3, 4,  5,  7, 16, 17, 11,  7, 22, ...
  4042        0, 1, 2, 3,  5, 12, 13, 14, 17, 11, 12, ...
  5794        0, 1, 4, 5,  8,  9, 14, 18, 21, 23, 28, ...

Cf. A000004, A002110, A306612, A325057.

isok(k, n) = {my(last = -1, cur); for (i=1, n, cur = k % prime(i); if (cur <= last, return (0)); last = cur;); return (1);}
a(n) = {my(k=0); while(!isok(k, n), k++); k;} \\ Michel Marcus, Jun 04 2019

from sympy import prime
def A306582(n):
    plist, rlist, x = [prime(i) for i in range(1,n+1)], [0]*n, 0
    while True:
        for i in range(n-1):
            if rlist[i] >= rlist[i+1]:
                break
        else:
            return x
        for i in range(n):
            rlist[i] = (rlist[i] + 1) % plist[i]
        x += 1 # Chai Wah Wu, Jun 15 2019

a(16)-a(20) from Daniel Suteu, Jun 03 2019
a(21)-a(23) from Giovanni Resta, Jun 16 2019
a(24)-a(27) from Bert Dobbelaere, Jun 22 2019
a(28)-a(30) from Bert Dobbelaere, Sep 05 2019

A325057 Number of positive integers k <= prime(n)# so that (k mod p_1) < (k mod p_2) < ... < (k mod p_n).

Original entry on oeis.org

1, 2, 3, 7, 19, 94, 381, 2217, 10248, 64082, 572741, 3590815, 33731134, 291308123, 1896596488, 14675287694, 147847569839, 1642854121867, 12717640104203, 134707566446733, 1285768348848054, 9334472487460317, 97284913917125312, 922382339920122509, 10370484766702974615

Offset: 0

Bert Dobbelaere, Sep 04 2019

nonn

This sequence emerges during computation of A306582 and A306612.

			a(3) = 7:
  Solutions for k that have increasing remainders modulo the first 3 primes:
  k   modulo  2   3   5
  =====================
  22          0 < 1 < 2
  28          0 < 1 < 3
   4          0 < 1 < 4
   8          0 < 2 < 3
  14          0 < 2 < 4
  23          1 < 2 < 3
  29          1 < 2 < 4

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002110, A306582, A306612.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-1, j-1), j=1..min(i, ithprime(n))))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=0..24);  # Alois P. Heinz, Jan 04 2023

from sympy import prime
def f(k, r, n):
    if k==n: return prime(k)-r
    global cache ; args = (k, r)
    if args in cache: return cache[args]
    rv = f(k+1, r+1, n)
    if r < (prime(k)-1): rv += f(k, r+1, n)
    cache[args]=rv ; return rv
def A325057(n):
    global cache ; cache = {}
    return f(1, 0, n)

Name edited and a(0)=1 prepended by Alois P. Heinz, Jan 04 2023

A327408 Smallest integer > 0 so that its remainders modulo the first n primes are less than half their respective moduli.

Original entry on oeis.org

2, 4, 6, 10, 16, 16, 70, 136, 210, 210, 442, 786, 786, 786, 6450, 53110, 53110, 247690, 303810, 303810, 813450, 3443146, 5889382, 9327220, 10068256, 63916062, 63916062, 63916062, 285847290, 285847290, 285847290, 285847290, 370793956, 370793956, 370793956, 370793956

Offset: 1

Bert Dobbelaere, Sep 07 2019

nonn

			a(6) = 16.
16 mod 2 = 0 < 2/2
16 mod 3 = 1 < 3/2
16 mod 5 = 1 < 5/2
16 mod 7 = 2 < 7/2
16 mod 11 = 5 < 11/2
16 mod 13 = 3 < 13/2
16 is the smallest integer > 0 satisfying these inequalities for the first 6 primes.

Bert Dobbelaere, Table of n, a(n) for n = 1..53

Companion sequence of A327409.

Cf. A002110, A306582, A306612.

isok(k, vp) = {for (i=1, #vp, if ((k % vp[i]) >= vp[i]/2, return (0));); return (1);}
a(n) = {my(k=1, vp = primes(n)); while (!isok(k, vp), k++); k;} \\ Michel Marcus, Sep 08 2019

A327409 Smallest integer > 0 so that its remainders modulo the first n primes are at least half their respective moduli.

Original entry on oeis.org

1, 5, 23, 53, 53, 293, 503, 713, 1439, 1439, 16673, 16673, 16673, 16673, 16673, 16673, 16673, 298583, 728153, 728153, 728153, 19420253, 19420253, 66663659, 207178199, 384974819, 384974819, 384974819, 546086693, 546086693, 8504041103, 22060162703, 60826761629, 60826761629

Offset: 1

Bert Dobbelaere, Sep 07 2019

nonn

			a(6) = 293.
293 mod 2 = 1 >= 2/2
293 mod 3 = 2 >= 3/2
293 mod 5 = 3 >= 5/2
293 mod 7 = 6 >= 7/2
293 mod 11 = 7 >= 11/2
293 mod 13 = 7 >= 13/2
293 is the smallest integer > 0 satisfying these inequalities for the first 6 primes.

Bert Dobbelaere, Table of n, a(n) for n = 1..50

Companion sequence of A327408.

Cf. A002110, A306582, A306612.

isok(k, vp) = {for (i=1, #vp, if ((k % vp[i]) < vp[i]/2, return (0));); return (1);}
a(n) = {my(k=1, vp = primes(n)); while (!isok(k, vp), k++); k;} \\ Michel Marcus, Sep 08 2019

A306613 First differences of A063990 (amicable numbers arranged in increasing order).

Original entry on oeis.org

64, 900, 26, 1410, 304, 2096, 544, 668, 136, 4376, 112, 1429, 2310, 2701, 1120, 44604, 3908, 64, 103, 2520, 1530, 4939, 3666, 7883, 1097, 11755, 21780, 103, 784, 1003, 15660, 1849, 646, 10866, 15554, 3126, 4416, 64, 4512, 4520, 11356, 5720, 988, 77108, 28080, 10930

Offset: 1

Conor Coons, Feb 28 2019

nonn

a(n) is the difference between the n-th and (n+1)-th amicable numbers when ordered by increasing value.

For 1 <= k <= 8, a(2k-1) is the difference between the larger and the smaller terms of the k-th amicable pair, and for 1 <= k <= 8, a(2k) is the difference between the smaller term of the (k+1)-th pair and the larger term of the k-th pair. Beginning with the 9th pair (63020,76084), the pairs ordered by their first element are no longer adjacent. - Bernard Schott, Mar 09 2019

			a(2) = amicable(3) - amicable(2) = 1184 - 284 = 900.
From _Bernard Schott_, Mar 10 2019: (Start)
a(1) = 284 - 220 = 64 is the difference between the larger and the smaller terms of the first amicable pair.
a(4) = 2620 - 1210 = 1410 is the difference between the smaller term of the third amicable pair and the larger term of the second amicable pair. (End)

Cf. A063990 (amicable numbers), A306612.
Cf. A002025, A002046.
Cf. A066539 (difference between larger and smaller terms of n-th amicable pair).
Cf. A139228 (first differences of perfect numbers).

clear
clc
A = zeros(100000,1);
parfor n = 1:1:100000
     f = find(rem(n, 1:floor(sqrt(n))) == 0);
     f = unique([1, n, f, fix(n./f)]);
     A(n) = sum(f) - n;
end
D = [];
d = 1;
for a = 1:1:100000
    for b = 1:1:100000
        if A(a) == b && A(b) == a && a~=b
            D(d) = a;
            d = d+1;
        end
    end
end
D
difference = diff(D)

a(n) = A063990(n+1) - A063990(n). - Michel Marcus, Apr 08 2019

More terms from Michel Marcus, Mar 09 2019

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A306582 a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.

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A325057 Number of positive integers k <= prime(n)# so that (k mod p_1) < (k mod p_2) < ... < (k mod p_n).

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A327408 Smallest integer > 0 so that its remainders modulo the first n primes are less than half their respective moduli.

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A327409 Smallest integer > 0 so that its remainders modulo the first n primes are at least half their respective moduli.

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PARI

A306613 First differences of A063990 (amicable numbers arranged in increasing order).

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