A379011 -id:A379011 - cp's OEIS frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 11, 12, 21, 125, 77, 121, 13, 14, 45, 55, 343, 143, 169, 17, 16, 33, 175, 91, 1331, 221, 289, 19, 18, 81, 65, 539, 187, 2197, 323, 361, 23, 20, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 22, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31

Offset: 2

Antti Karttunen, Aug 21 2014

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
(End)

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)

(define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

A040976 a(n) = prime(n) - 2.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 65, 69, 71, 77, 81, 87, 95, 99, 101, 105, 107, 111, 125, 129, 135, 137, 147, 149, 155, 161, 165, 171, 177, 179, 189, 191, 195, 197, 209, 221, 225, 227, 231, 237, 239, 249, 255, 261

Offset: 1

Felice Russo

Numbers k such that k! reduced mod (k+2) is 1. - Benoit Cloitre, Mar 11 2002
The first a(n) numbers starting from 2 are divisible by primes up to prime(n-1). - Lekraj Beedassy, Jun 21 2006
The terms in this sequence are the cumulative sums of distances from one prime to another. For example for the distance from the first to 26th prime, 2 to 101, the cumulative sum of distances is 99, always the last prime, here 101, minus 2. - Enoch Haga, Apr 24 2006
The primes in this sequence are the initial primes of twin prime pairs. - Sebastiao Antonio da Silva, Dec 21 2008
Note that many, but not all, of these numbers satisfy x such that x^(x+1) = 1 mod (x+2). The first exception is 339. - Thomas Ordowski, Nov 27 2013
If this sequence had an infinite number of primes, the twin prime conjecture would follow. Sequence holds all primes in A001359. - John W. Nicholson, Apr 14 2014
From Bernard Schott, Feb 19 2023: (Start)
Equivalently, except for a(1)=0, all terms are odd integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.
For each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 2, so this unique AP is (2, 2+d) = (2, prime(m)) with m > 1; so, first examples are (2,3), (2,5), (2,7), (2,11), ... next elements should be respectively: 4, 8, 12, 20, ... that are all composite numbers.
Similar sequence with even common differences d is A360735.
This subsequence of A359408 corresponds to the first case: '2 is prime'; second case corresponding to the even common differences d is A360735. (End)

			a(13) = 39, because A000040(13) = 41.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002-2014.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A000040, A001223, A014689, A014692.
Equals A359408 \ A360735.
Cf. A123556, A342309.
First column of A086800, and of A379011, last diagonal of A090321, and of A162621.
See also irregular triangles A103728, A319148, A369497.

Filtered([1..10^2],IsPrime)-2; # Muniru A Asiru, Jan 31 2018

a040976 n = a000040 n - 2
a040976_list = map (subtract 2) a000040_list
-- Reinhard Zumkeller, Feb 22 2012

[NthPrime(n)-2: n in [1..60]]; // Vincenzo Librandi, Jan 31 2018

seq(ithprime(n)-2,n=1..100); # Muniru A Asiru, Jan 31 2018

Prime[Range[22]]-2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)-2 \\ Charles R Greathouse IV, Nov 20 2012

a(n) = A000040(n) - 2 = Sum_{i=1..n-1} A001223(i).
For n > 2: A092953(a(n)) = 1. - Reinhard Zumkeller, Nov 10 2012
If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(16) = 2 and 16 is not a term. - Bernard Schott, Feb 19 2023
a(n) = Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(A000720(k)/n)). [Ruiz and Sondow]. - Elias Alejandro Angulo Klein, Apr 09 2024

A379010 Square array A(n, k) = phi(A246278(n, k)), read by falling antidiagonals; Euler totient function applied to the prime shift array.

Original entry on oeis.org

1, 2, 2, 2, 6, 4, 4, 8, 20, 6, 4, 18, 24, 42, 10, 4, 12, 100, 60, 110, 12, 6, 24, 40, 294, 120, 156, 16, 8, 20, 120, 72, 1210, 192, 272, 18, 6, 54, 48, 420, 160, 2028, 288, 342, 22, 8, 40, 500, 96, 1320, 216, 4624, 396, 506, 28, 10, 36, 168, 2058, 180, 2496, 352, 6498, 616, 812, 30, 8, 24, 200, 660, 13310, 264, 4896, 504, 11638, 840, 930, 36

Offset: 1

Antti Karttunen, Dec 14 2024

Each column is strictly increasing.

			The top left corner of the array:
k=  |  1     2     3      4     5      6     7        8      9     10
2k= |  2     4     6      8    10     12    14       16     18     20
----+-------------------------------------------------------------------
1   |  1,    2,    2,     4,    4,     4,    6,       8,     6,     8,
2   |  2,    6,    8,    18,   12,    24,   20,      54,    40,    36,
3   |  4,   20,   24,   100,   40,   120,   48,     500,   168,   200,
4   |  6,   42,   60,   294,   72,   420,   96,    2058,   660,   504,
5   | 10,  110,  120,  1210,  160,  1320,  180,   13310,  1560,  1760,
6   | 12,  156,  192,  2028,  216,  2496,  264,   26364,  3264,  2808,
7   | 16,  272,  288,  4624,  352,  4896,  448,   78608,  5472,  5984,
8   | 18,  342,  396,  6498,  504,  7524,  540,  123462,  9108,  9576,
9   | 22,  506,  616, 11638,  660, 14168,  792,  267674, 17864, 15180,
10  | 28,  812,  840, 23548, 1008, 24360, 1120,  682892, 26040, 29232,
11  | 30,  930, 1080, 28830, 1200, 33480, 1260,  893730, 39960, 37200,
12  | 36, 1332, 1440, 49284, 1512, 53280, 1656, 1823508, 59040, 55944,

Cf. A000010, A246278, A379011.

Cf. A062570 (row 1), A006093 (column 1), A036689 (column 2), A083553 (column 3), A135177 (column 4).

up_to = 11325; \\ = binomial(150+1,2)
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A379010sq(row,col) = eulerphi(A246278sq(row,col));
A379010list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A379010sq(col,(a-(col-1))))); (v); };
v379010 = A379010list(up_to);
A379010(n) = v379010[n];

A378986 a(n) = 2phi(2n) - 2*n, where phi is Euler totient function.

Original entry on oeis.org

0, 0, -2, 0, -2, -4, -2, 0, -6, -4, -2, -8, -2, -4, -14, 0, -2, -12, -2, -8, -18, -4, -2, -16, -10, -4, -18, -8, -2, -28, -2, 0, -26, -4, -22, -24, -2, -4, -30, -16, -2, -36, -2, -8, -42, -4, -2, -32, -14, -20, -38, -8, -2, -36, -30, -16, -42, -4, -2, -56, -2, -4, -54, 0, -34, -52, -2, -8, -50, -44, -2, -48, -2, -4

Offset: 1

Antti Karttunen, Dec 14 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..32769

Even bisection of A083254, and of A297114.
First row of A379011.
Cf. A000010, A008683, A033879, A062570, A176095, A378978.
Cf. also A378987.

PARI
```
A378986(n) = (2*eulerphi(2*n)-(2*n));
```

a(n) = 2*A000010(2*n) - 2*n.
a(n) = A083254(2*n) = A297114(2*n).
a(n) = -2*A176095(n).
a(n) = Sum_{d|2n} A008683(d)*A033879(2*n/d).

cp's OEIS Frontend

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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A040976 a(n) = prime(n) - 2.

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A379010 Square array A(n, k) = phi(A246278(n, k)), read by falling antidiagonals; Euler totient function applied to the prime shift array.

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A378986 a(n) = 2*phi(2*n) - 2*n, where phi is Euler totient function.

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Formula

A378986 a(n) = 2phi(2n) - 2*n, where phi is Euler totient function.