A344027 -id:A344027 - 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

A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).

Original entry on oeis.org

1, 4, 5, 12, 7, 16, 9, 32, 21, 24, 13, 44, 15, 32, 31, 80, 19, 60, 21, 68, 41, 48, 25, 112, 45, 56, 81, 92, 31, 92, 33, 192, 61, 72, 59, 156, 39, 80, 71, 176, 43, 124, 45, 140, 123, 96, 49, 272, 77, 140, 91, 164, 55, 216, 87, 240, 101, 120

Offset: 1

Reinhard Zumkeller, Feb 26 2002

Terms are either odd or multiples of 4. - Antti Karttunen, Jul 31 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Bruno Berselli)

Cf. A003415, A005843, A068720, A068721, A260624, A327862, A327864.
Second diagonal (without the initial 1) in A084890.
Row 1 of A344027.

Ad:=func; [Ad(2*n): n in [1..60]]; // Bruno Berselli, Oct 22 2013

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := ad[2*n]; Array[a, 100] (* Amiram Eldar, Apr 11 2025 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068719(n) = (n + 2*A003415(n)); \\ Antti Karttunen, Jul 31 2022

a(n) = A003415(A005843(n)).

A344026 Arithmetic derivative applied to the Doudna sequence: a(n) = A003415(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 6, 12, 1, 7, 8, 16, 10, 21, 27, 32, 1, 9, 10, 24, 12, 31, 39, 44, 14, 45, 55, 60, 75, 81, 108, 80, 1, 13, 14, 32, 16, 41, 51, 68, 18, 59, 71, 92, 95, 123, 162, 112, 22, 77, 91, 140, 119, 185, 240, 156, 147, 275, 350, 216, 500, 297, 405, 192, 1, 15, 16, 48, 18, 61, 75, 92, 20, 87, 103, 124, 135, 165, 216

Offset: 0

Antti Karttunen, May 07 2021

Coincides with A344028 on Fibbinary numbers, A003714.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000079 (positions of ones), A003415, A003714, A005940.

Cf. also A344027, A344028, A344182.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344026(n) = A003415(A005940(1+n));

a(2^n) = 1 for all n >= 0.

A343221 Number of iterations of x -> A003961(x) needed until A003415(x) < x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0

Offset: 1

Antti Karttunen, Apr 08 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A083347 (positions of zeros).

Cf. A003415, A003961, A343222, A344027.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A343221(n) = if(A003415(n)A343221(A003961(n)));

A343222 Number of iterations of x -> A003961(x) needed until A003415(x) <= x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0

Offset: 1

Antti Karttunen, Apr 08 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A003961.
Positions of zeros: Union of A051674 and A083347.
Cf. also A343221, A344027.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A343222(n) = if(A003415(n)<=n,0,1+A343222(A003961(n)));

A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

1, 4, 2, 7, 12, 3, 12, 19, 30, 4, 11, 54, 41, 56, 5, 20, 26, 225, 79, 110, 6, 15, 87, 58, 588, 131, 156, 7, 32, 37, 310, 94, 1815, 193, 238, 8, 33, 216, 69, 861, 162, 3042, 269, 304, 9, 32, 140, 1500, 117, 2156, 218, 6069, 355, 414, 10, 21, 120, 427, 5488, 183, 3835, 314, 8664, 491, 580, 11, 52, 44, 455, 1254, 26620, 255, 6834, 422, 14283, 629, 682, 12

Offset: 1

Antti Karttunen, Jul 29 2022

Each column is strictly monotonic.

			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
-----+--------------------------------------------------------------------------
n= 1 |  1,   4,   7,    12,  11,    20,  15,     32,    33,    32,  21,      52,
   2 |  2,  12,  19,    54,  26,    87,  37,    216,   140,   120,  44,     351,
   3 |  3,  30,  41,   225,  58,   310,  69,   1500,   427,   455,  86,    2075,
   4 |  4,  56,  79,   588,  94,   861, 117,   5488,  1254,  1022, 132,    8183,
   5 |  5, 110, 131,  1815, 162,  2156, 183,  26620,  2561,  2717, 214,   31581,
   6 |  6, 156, 193,  3042, 218,  3835, 255,  52728,  4828,  4316, 304,   67093,
   7 |  7, 238, 269,  6069, 314,  6834, 373, 137564,  7695,  8075, 404,  154615,
   8 |  8, 304, 355,  8664, 422, 10241, 457, 219488, 12098, 12426, 524,  261003,
   9 |  9, 414, 491, 14283, 532, 17296, 609, 438012, 20909, 18653, 668,  535877,
  10 | 10, 580, 629, 25230, 718, 27231, 787, 975560, 29388, 31552, 836, 1050409,

Cf. A000027 (column 1), A097240 (column 3), A246278, A258851.

Cf. also A344027.

up_to = 78;
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));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A356155sq(row,col) = A258851(A246278sq(row,col));
A356155list(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] = A356155sq(col,(a-(col-1))))); (v); };
v356155 = A356155list(up_to);
A356155(n) = v356155[n];

A353484 a(1) = 0; and for n > 1, a(n) = A165560(n) + a(A064989(n)), where A165560 is the parity of arithmetic derivative, and A064989 shifts the prime factorization of its argument one step toward lower primes.

Original entry on oeis.org

0, 1, 2, 0, 3, 2, 4, 0, 0, 3, 5, 1, 6, 4, 2, 0, 7, 1, 8, 2, 3, 5, 9, 1, 0, 6, 1, 3, 10, 3, 11, 0, 4, 7, 2, 0, 12, 8, 5, 2, 13, 4, 14, 4, 2, 9, 15, 1, 0, 1, 6, 5, 16, 1, 3, 3, 7, 10, 17, 2, 18, 11, 3, 0, 4, 5, 19, 6, 8, 3, 20, 0, 21, 12, 2, 7, 2, 6, 22, 2, 0, 13, 23, 3, 5, 14, 9, 4, 24, 2, 3, 8, 10, 15, 6, 1, 25

Offset: 1

Antti Karttunen, Apr 22 2022

a(n) counts the number of the terms of A235991 encountered [including also n itself if the arithmetic derivative of n is odd] when repeatedly prime shifting n down to 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A064989, A165560, A235991, A353485 (positions of zeros).

Cf. also A292375, A319703, A343221, A344027, A353515.

A000265(n) = (n>>valuation(n,2));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353484(n) = if(1==n, 0, (A003415(n)%2) + A353484(A064989(n)));

For n >= 1, a(A000040(n)) = n, a(n^2) = 0.

A353515 The length of the shortest path from n to 1 when using the transitions x -> A003415(x) and x -> A003961(x), or -1 if no 1 can ever be reached from n.

Original entry on oeis.org

0, 1, 1, 4, 1, 2, 1, 7, 3, 2, 1, 6, 1, 4, 7, 8, 1, 4, 1, 6, 3, 2, 1, 7, 3, 6, 7, 8, 1, 2, 1, 10, 5, 2, 6, 5, 1, 4, 4, 6, 1, 2, 1, 6, 5, 4, 1, 9, 4, 5, 5, 8, 1, 6, 8, 8, 3, 2, 1, 4, 1, 6, 5, 10, 5, 2, 1, 5, 4, 2, 1, 8, 1, 5, 6, 6, 5, 2, 1, 9, 7, 2, 1, 4, 3, 5, 7, 8, 1, 5, 7, 7, 3, 5, 4, 9, 1, 6, 7, 6, 1, 3, 1, 7, 2

Offset: 1

Antti Karttunen, Apr 23 2022

nonn

This is a variant of A327969 that seems to be less in need of an escape clause. Note that enough prime shifts with A003961 will eventually transform every term of A100716 (which is a subsequence of A099309) to a term of A048103, and that A051903(A003961(n)) = A051903(n). See also the array A344027.

Records 0, 1, 4, 7, 8, 10, 12, 13, 14, 15, 16, 19, ... occur at 1, 2, 4, 8, 16, 32, 128, 256, 768, 1024, 2048, 4096, ..., etc.

			From n = 4, we can reach 1 with just four steps as A003961(4) = 9, A003415(9) = 6, A003415(6) = 5 and A003415(5) = 1, and because there are no shorter paths we have a(4) = 4.
From n = 8, we can reach 1 with seven steps, as A003415(8) = 12, A003961(12) = 45, A003415(45) = 39, A003961(39) = 85, A003415(85) = 22, A003415(22) = 13, A003415(13) = 1, and because there are no shorter paths we have a(8) = 7.
For n = 15, as A003415(15) = 8, we know that a(15) is at most a(8)+1, i.e., 8. But we can do better, as A003961(15) = 35, A003961(35) = 77, A003415(77) = 18, A003415(18) = 21, A003415(21) = 10, A003415(10) = 7, A003415(7) = 1, and because there are no shorter paths we have a(15) = 7.
From n = 49, we can reach 1 in four steps, as A003961(49) = 121, A003415(121) = 22, A003415(22) = 13, A003415(13) = 1. Note that this is less than A099307(49)-1, as it would take 5 steps to reach 1 if using the arithmetic derivative only, 49 -> 14 -> 9 -> 6 -> 5 -> 1.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003961, A048103, A051903, A099307, A099308, A099309, A100716, A349905.

Cf. also A327969 and A343221, A344027, A353484.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A353515(n) = if(1==n,0,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A003415(u); if(1==a, return(k)); if(isprime(a), return(k+1)); b = A003961(u); newxs = setunion([a],newxs); newxs = setunion([b],newxs)); xs = newxs));

a(1) = 0, a(p^p) = 1 + a(A003961(p^p)) for primes p, and for other numbers, a(n) = 1 + min(a(A003415(n)), a(A003961(n))).
a(p) = 1 for all primes p.
a(n) < A099307(n), unless A099307(n) = 0. [I.e., for all n in A099308]

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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A068719 Arithmetic derivative of even numbers: a(n) = n+2*A003415(n).

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A344026 Arithmetic derivative applied to the Doudna sequence: a(n) = A003415(A005940(1+n)).

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A343221 Number of iterations of x -> A003961(x) needed until A003415(x) < x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

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A343222 Number of iterations of x -> A003961(x) needed until A003415(x) <= x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

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A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

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A353484 a(1) = 0; and for n > 1, a(n) = A165560(n) + a(A064989(n)), where A165560 is the parity of arithmetic derivative, and A064989 shifts the prime factorization of its argument one step toward lower primes.

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A353515 The length of the shortest path from n to 1 when using the transitions x -> A003415(x) and x -> A003961(x), or -1 if no 1 can ever be reached from n.

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