A095195 -id:A095195 - cp's OEIS frontend

A349644 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest prime p = prime(k) such that all n-th differences of (prime(k), ..., prime(k+n+m)) are zero.

3, 251, 17, 9843019, 347, 347, 121174811, 2903, 2903, 41

Offset: 2

Pontus von Brömssen, Nov 23 2021

T(n,m) = prime(k), where k is the smallest positive integer such that A095195(j,n) = 0 for k+n <= j <= k+n+m.

Equivalently, T(n,m) is the smallest prime p = prime(k) such that there is a polynomial f of degree at most n-1 such that f(j) = prime(j) for k <= j <= k+n+m.

			Array begins:
  n\m|   0       1           2           3           4
  ---+------------------------------------------------
  2  |   3     251     9843019   121174811           ?
  3  |  17     347        2903       15373      128981
  4  | 347    2903       15373      128981    19641263
  5  |  41    8081      128981    19641263   245333213
  6  | 211  128981    19641263   245333213   245333213
  7  | 271  386471    81028373   245333213 27797667517
  8  |  23 2022971   245333213 27797667517           ?
  9  | 191 7564091 10246420463           ?           ?

Cf. A006560 (row n=2), A349642 (row n=3), A349643 (column m=0).

Cf. A095195.

from sympy import nextprime
def A349644(n,m):
    d = [float('inf')]*(n-1)
    p = [0]*(n+m)+[2]
    c = 0
    while 1:
        del p[0]
        p.append(nextprime(p[-1]))
        d.insert(0,p[-1]-p[-2])
        for i in range(1,n):
            d[i] = d[i-1]-d[i]
        if d.pop() == 0:
            if c == m: return p[0]
            c += 1
        else:
            c = 0

T(n,m) <= T(n-1,m+1).
T(n,m) <= T(n, m+1).
Sum_{j=0..n} (-1)^j*binomial(n,j)*prime(k+i+j) = 0 for 0 <= i <= m, where prime(k) = T(n,m).

A350001 Iterated differences of lucky numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A000959(k), T(n,k) = T(n-1,k+1) - T(n-1,k) for n > 0.

Original entry on oeis.org

1, 3, 2, 7, 4, 2, 9, 2, -2, -4, 13, 4, 2, 4, 8, 15, 2, -2, -4, -8, -16, 21, 6, 4, 6, 10, 18, 34, 25, 4, -2, -6, -12, -22, -40, -74, 31, 6, 2, 4, 10, 22, 44, 84, 158, 33, 2, -4, -6, -10, -20, -42, -86, -170, -328, 37, 4, 2, 6, 12, 22, 42, 84, 170, 340, 668

Offset: 0

Pontus von Brömssen, Dec 08 2021

			Array begins:
  n\k|    1    2    3    4    5    6    7     8    9    10   11   12
  ---+--------------------------------------------------------------
   0 |    1    3    7    9   13   15   21    25   31    33   37   43
   1 |    2    4    2    4    2    6    4     6    2     4    6    6
   2 |    2   -2    2   -2    4   -2    2    -4    2     2    0   -4
   3 |   -4    4   -4    6   -6    4   -6     6    0    -2   -4   14
   4 |    8   -8   10  -12   10  -10   12    -6   -2    -2   18  -32
   5 |  -16   18  -22   22  -20   22  -18     4    0    20  -50   56
   6 |   34  -40   44  -42   42  -40   22    -4   20   -70  106  -82
   7 |  -74   84  -86   84  -82   62  -26    24  -90   176 -188  102
   8 |  158 -170  170 -166  144  -88   50  -114  266  -364  290 -100
   9 | -328  340 -336  310 -232  138 -164   380 -630   654 -390   50
  10 |  668 -676  646 -542  370 -302  544 -1010 1284 -1044  440   78

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)
Index entries for sequences related to lucky numbers

Cf. A000959 (row n = 0), A031883 (row n = 1), A123593 (column k = 1).

Cf. A254967 (absolute differences), A095195 (iterated differences of primes), A350004 (iterated differences of ludic numbers).

T(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*A000959(k+j).

A229061 The (n+1)-th term of the n-th differences of the prime sequence.

Original entry on oeis.org

2, 2, 2, 4, 8, -2, -48, -70, 0, 56, 308, 1014, 798, -2072, -5126, -2820, 434, -1340, 62902, 398032, 1247046, 2834160, 5266626, 7862442, 9510040, 13829302, 37650208, 111410394, 260524940, 468110450, 626899146, 481007522, -490911164, -3217336656, -8570944960

Offset: 0

Jean-François Alcover, Sep 17 2013

sign

All terms are even. The only zero seems to be a(8), corresponding to A036269(9).

			The sequences of differences begin:
2,      3,   5,   7,  11,  13,   17,  19,  23,  29, ...
1,      2,   2,   4,   2,   4,    2,   4,   6,   2, ...
1,      0,   2,  -2,   2,  -2,    2,   2,  -4,   4, ...
-1,     2,  -4,   4,  -4,   4,    0,  -6,   8,  -6, ...
3,     -6,   8,  -8,   8,  -4,   -6,  14, -14,   6, ...
-9,    14, -16,  16, -12,  -2,   20, -28,  20,  -2, ...
23,   -30,  32, -28,  10,  22,  -48,  48, -22,  -6, ...
-53,   62, -60,  38,  12, -70,   96, -70,  16,  16, ...
115, -122,  98, -26, -82, 166, -166,  86,   0, -28, ...
etc.
Main diagonal begins:
2, 2, 2, 4, 8, -2, -48, -70, 0, 56, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..98 from Jean-François Alcover)

Cf. A095195, A036263-A036271, A007442.

T:= proc(n, k) option remember;
      `if`(k=0, ithprime(n), T(n+1, k-1)-T(n, k-1))
    end:
a:= n-> T(n+1, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 25 2013

max = 100; row[n_] := Differences[Prime /@ Range[max], n]; Table[row[n], {n, 0, max}] // Diagonal

a(n) = A095195(2*n+1,n).

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

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A349644 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest prime p = prime(k) such that all n-th differences of (prime(k), ..., prime(k+n+m)) are zero.

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A350001 Iterated differences of lucky numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A000959(k), T(n,k) = T(n-1,k+1) - T(n-1,k) for n > 0.

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A229061 The (n+1)-th term of the n-th differences of the prime sequence.

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A379542 Second term of the n-th differences of the prime numbers.

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