A372562 -id:A372562 - 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

A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

Original entry on oeis.org

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279

Offset: 1

Reinhard Zumkeller, Jun 29 2001

The theorem that a(n) > 0 for all n is known as "Bertrand's Postulate", and was proved by Tchebycheff in 1852.

The analog for Ramanujan primes is Paksoy's theorem that 2*R(n) - R(n+1) > 0 for n > 1. See A233822. - Jonathan Sondow, Dec 16 2013

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Harry J. Smith)

Cf. A000040, A001223, A215808 (prime terms), A233822.
Cf. A098764, A100484, A258383 (run lengths), A258432, A258469, A257762, A258449, A257892, A257951.
When negated, forms the left edge of irregular triangle A252750, and also the leftmost column of square array A372562.

a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015

a:= n-> (p-> 2*p(n)-p(n+1))(ithprime):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2022

Table[2*Prime[n]-Prime[n+1],{n,60}] (* James C. McMahon, Apr 27 2024 *)
2#[[1]]-#[[2]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Jul 29 2024 *)
ListConvolve[{-1, 2}, Prime[Range[100]]] (* Paolo Xausa, Nov 02 2024 *)

a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009

a(n) = A000040(n) - A001223(n). - Zak Seidov, Sep 07 2012
a(n) = 2*A000040(n) - A000040(n+1). - Zak Seidov, May 12 2020
a(n) = A098764(n) - A000040(n). - Anthony S. Wright, Feb 19 2024

Edited by N. J. A. Sloane, Feb 24 2023

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Antti Karttunen, Dec 12 2024

nonn

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
Positions of 0's in A378981.
Subsequence of A263837.
Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
Cf. also A378983.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);

A252750 a(n) = A003961(A005940(n+1)) - 2 * A005940(n+1).

Original entry on oeis.org

-1, -1, -1, 1, -3, 3, 7, 11, -3, 1, 5, 21, -1, 39, 71, 49, -9, 5, 13, 23, 7, 45, 85, 87, 23, 47, 95, 153, 93, 267, 463, 179, -9, -5, -1, 43, -19, 81, 149, 109, -11, 91, 175, 195, 189, 345, 605, 309, -73, 167, 311, 241, 357, 435, 775, 531, 645, 529, 965, 909, 1151, 1551, 2639, 601, -15, -1, 7, 29, -11, 63, 127, 185, 5, 53, 125, 327, 87, 573, 997, 407, -65, 121, 253, 413, 231

Offset: 0

Antti Karttunen, Dec 21 2014

From Antti Karttunen, May 21 2024: (Start)
Like A005940 itself, also this irregular table derived from it can be represented as a binary tree:
                                     -1
                                      |
                   ................. -1 ..................
                 -1                                       1
       -3 ......./ \....... 3                   7 ......./ \....... 11
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   -3       1           5       21         -1       39         71       49
 -9  5    13 23        7 45   85  87     23  47   95  153    93  267 463  179
etc.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A252743 (characteristic function for positive terms), A252751 (partial sums of sequence b(0) = 0, b(n) = a(n), for n>0).
Cf. A003961, A005940, A252745, A252748, A249820, A246282.
Cf. A062234 (when negated forms the left edge apart from the initial term), A003063 (right edge).
Cf. also A372562 (apart from the initial term, same data in square array).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A252750(n) = (A003961(A005940(1+n)) - (2 * A005940(1+n))); \\ Antti Karttunen, May 21 2024

(define (A252750 n) (- (A003961 (A005940 (+ 1 n))) (* 2 (A005940 (+ 1 n)))))

a(n) = A003961(A005940(n+1)) - 2 * A005940(n+1).
a(n) = A252748(A005940(n+1)).
Other identities. For all n >= 1:
sgn(a(n)) = (-1)^(1+A252743(n)).

Term a(0) = -1 prepended by Antti Karttunen, May 21 2024

A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

Original entry on oeis.org

1, 1, 2, 0, 5, 4, 1, 6, 19, 6, 2, 14, 22, 41, 10, -4, 10, 94, 58, 109, 12, 4, 12, 38, 286, 118, 155, 16, 1, 18, 102, 70, 1198, 190, 271, 18, -3, 41, 46, 394, 158, 2014, 286, 341, 22, -2, 26, 469, 94, 1284, 214, 4606, 394, 505, 28, 8, 22, 148, 2001, 178, 2452, 350, 6478, 614, 811, 30, -12, 22, 178, 630, 13177, 262, 4842, 502, 11614, 838, 929, 36

Offset: 1

Antti Karttunen, Dec 13 2024

Each column is strictly increasing.

For all k >= 1, A(1+A378985(k), k) > 0, and it is the topmost positive number of the column k.

			The top left corner of the array:
k=  |  1    2    3     4    5     6    7      8     9    10   11     12
2k= |  2    4    6     8   10    12   14     16    18    20   22     24
----+-------------------------------------------------------------------
  1 |  1,   1,   0,    1,   2,   -4,   4,     1,   -3,   -2,   8,   -12,
  2 |  2,   5,   6,   14,  10,   12,  18,    41,   26,   22,  22,    30,
  3 |  4,  19,  22,   94,  38,  102,  46,   469,  148,  178,  62,   502,
  4 |  6,  41,  58,  286,  70,  394,  94,  2001,  630,  476, 106,  2746,
  5 | 10, 109, 118, 1198, 158, 1284, 178, 13177, 1522, 1720, 218, 14110,
  6 | 12, 155, 190, 2014, 214, 2452, 262, 26181, 3216, 2762, 334, 31858,
  7 | 16, 271, 286, 4606, 350, 4842, 446, 78301, 5416, 5926, 478, 82294,

Cf. A033879, A246278, A336835, A355927, A372562, A372563.
Cf. A006093 (column 1), A306190 (column 2), A378978 (row 1), A378985 (row index of the topmost positive term in column n).
Cf. also arrays A341605, A341606 and A341607.
Cf. also A324055.

up_to = 78;
A033879(n) = (n+n-sigma(n));
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));
A378979sq(row,col) = A033879(A246278sq(row,col));
A378979list(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] = A378979sq(col,(a-(col-1))))); (v); };
v378979 = A378979list(up_to);
A378979(n) = v378979[n];

A(n, k) = A033879(A246278(n, k)) = 2*A246278(n, k) - A355927(n, k).

A(n, k) = A372563(n,k) - A372562(n, k).

A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

0, 2, 1, 3, 12, 1, 12, 11, 18, 3, 3, 85, 29, 64, 1, 17, 23, 187, 47, 36, 3, 9, 97, 19, 931, 53, 106, 1, 50, 17, 291, 75, 733, 71, 54, 3, 36, 504, 35, 889, 31, 2533, 77, 148, 5, 21, 121, 1620, 65, 1011, 111, 1639, 187, 288, 1, 3, 171, 505, 11840, 59, 2197, 119, 4927, 179, 90, 5

Offset: 1

Antti Karttunen, May 21 2024

			The top left corner of the array:
k=   1    2    3      4    5      6    7       8      9     10   11      12
2k=  2    4    6      8   10     12   14      16     18     20   22      24
---+-------------------------------------------------------------------------
1  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,

Cf. A000203, A003961, A246278, A286385, A355927.
Cf. A046933 (column 1).
Cf. also A355924, A372562.

up_to = 66;
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));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A286385(n) = (A003961(n)-sigma(n));
A372563sq(row,col) = A286385(A246278sq(row,col));
A372563list(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] = A372563sq(col,(a-(col-1))))); (v); };
v372563 = A372563list(up_to);
A372563(n) = v372563[n];

A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).

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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A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

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A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A252750 a(n) = A003961(A005940(n+1)) - 2 * A005940(n+1).

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A378979 Square array A(n, k) = 2*A246278(n, k) - sigma(A246278(n, k)), read by falling antidiagonals. Deficiency applied to the prime shift array.

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A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

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