A246259 -id:A246259 - cp's OEIS frontend

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

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)).

Original entry on oeis.org

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

A246261 Numbers n such that A003961(n) is of the form 4k+1.

Original entry on oeis.org

1, 3, 4, 9, 10, 11, 12, 13, 14, 16, 23, 25, 27, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 47, 48, 49, 52, 56, 58, 59, 64, 69, 71, 75, 81, 82, 83, 85, 86, 89, 90, 92, 93, 95, 97, 99, 100, 102, 105, 106, 107, 108, 109, 110, 111, 114, 117, 119, 120, 121, 122, 124, 126, 130, 131, 132, 133, 134, 136, 139, 140, 141, 143, 144

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Equally: numbers n for which A048673(n) is odd.

Also, after 1, numbers n for which A243062(n) is even.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A246263.
Positions of zeros in A246271.
Inverse function: A246262.
The first row of array A246259.
Cf. A003961, A048673, A243062, A246265.

{1}~Join~Select[Range[144], Mod[Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]], 4] == 1 &] (* Michael De Vlieger, Mar 12 2021 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
print([n for n in range(1, 201) if a003961(n)%4==1]) # Indranil Ghosh, Jun 12 2017

For all n >= 1, A246262(a(n)) = n.

A246263 Numbers n such that A003961(n) is of the form 4k+3.

Original entry on oeis.org

2, 5, 6, 7, 8, 15, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 32, 41, 43, 45, 46, 50, 51, 53, 54, 55, 57, 60, 61, 62, 63, 65, 66, 67, 68, 70, 72, 73, 74, 76, 77, 78, 79, 80, 84, 87, 88, 91, 94, 96, 98, 101, 103, 104, 112, 113, 115, 116, 118, 123, 125, 127, 128, 129, 135, 137, 138, 142, 149, 150

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Equally: numbers n for which A048673(n) is even.
Also, after the first term (2), numbers n for which A243062(n) is odd.
Numbers that occur below the topmost row of square array A246259.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A246261.
Positions of nonzeros in A246271.
Inverse function: A246264.
Cf. A003961, A048673, A246259, A243062, A246267.

A246271 Starting from A003961(n), the number of additional iterations of A003961 required for the result to be of the form 4k+1.

Original entry on oeis.org

0, 1, 0, 0, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 6, 1, 0, 2, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 7, 0, 0, 0, 1, 1, 0, 1, 2, 5, 0, 2, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 2, 0, 3, 0, 1, 2, 2, 0, 1, 1, 1, 1, 2, 0, 0, 0, 6, 0, 0, 6, 1, 0, 0, 4, 0, 0, 3, 0, 2, 0, 1, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

			a(5) = 2, because exactly two additional iterations of A003961 are needed before A003961(5) = 7 is of the form 4k+1; as A003961(7) = 11 and A003961(11) = 13. (We have 7 = 3 mod 4, 11 = 3 mod 4 and 13 = 1 mod 4.)

Antti Karttunen, Table of n, a(n) for n = 1..10001

A246261 gives the positions of zeros, A246263 the positions of nonzeros.
A246280 the positions where n occurs for the first time, A246167 the positions of new distinct values.
Cf. A003961, A246260, A246272, A246259, A246277.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A246271(n) = {my(i); i=0; n = A003961(n); while(((n%4)!=1), i++; n = A003961(n)); i};
for(n=1, 10001, write("b246271.txt", n, " ", A246271(n)));
(Scheme, two different variants, the second one employing memoizing definec-macro)
(define (A246271 n) (let loop ((i 0) (n n)) (let ((next (A003961 n))) (if (= 1 (modulo next 4)) i (loop (+ i 1) next)))))
(definec (A246271 n) (if (= 1 (A246260 n)) 0 (+ 1 (A246271 (A003961 n)))))

If A246260(n) = 1, a(n) = 0, otherwise 1 + a(A003961(n)).

A246280 a(n) = the smallest starting value for k such that n will be the exact number of additional iterations of A003961 needed when we start from A003961(k) before we first reach a number of the form 4k+1; the least k such that A246271(k) = n.

Original entry on oeis.org

1, 2, 5, 66, 91, 55, 21, 46, 1362, 1654, 1419, 574, 6463, 5263, 4607, 3497, 589843, 430261, 574823, 567583, 554111, 545869, 20490043, 14635735, 8781429

Offset: 0

Antti Karttunen, Aug 22 2014

nonn

The sequence is infinite (well-defined for all n), provided that A246271 is not bounded.
All the terms are squarefree (see the comment in A246259).
From 2 onward their prime factorizations are: 2, 5, 2*3*11, 7*13, 5*11, 3*7, 2*23, 2*3*227, 2*827, 3*11*43, 2*7*41, 23*281, 19*277, 17*271, 13*269, 571*1033, 13*23*1439, 563*1021, 557*1019, 547*1013, 541*1009, 7*2927149, 5*2927147, 3*2927143. (Note that 2927149 = A000040(211943)).
Note how A003961(21) = 55 and A003961(55) = 91. Also A003961(545869) = 554111, A003961(554111) = 567583, A003961(567583) = 574823.
Similarly: A003961(8781429) = 14635735 and A003961(14635735) = 20490043.
Apart from those descending subsections, the growth rate of the sequence should be roughly a(n) ~ 2^n, assuming that the distribution of 4k+1 (A002144) and 4k+3 primes (A002145) among the primes is even and essentially random.

			a(0) = 1, because 1 is the first such number >= 1 that no iterations are needed before A003961(1) (= 1) is of the form 4k+1.
a(1) = 2, because 2 is the first such number >= 1 that exactly one additional iteration of A003961 is needed before A003961(2) = 3 is of the form 4k+1; as A003961(3) = 5.
a(2) = 5, because 5 is the first such number that exactly two additional iterations of A003961 are needed before A003961(5) = 7 is of the form 4k+1; as A003961(7) = 11 and A003961(11) = 13.

Leftmost column of array A246259.
A246167 gives the same sequence sorted into ascending order.
Cf. A003961, A002144, A002145, A246271, A048672.

;; Other code as in A246259:
(define (A246280 n)  (A246259bi n 1))

cp's OEIS Frontend

A246258 Transpose of array A246259.

Views

Author

Keywords

Comments

Crossrefs

Programs

Scheme

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Perl

Python

Formula

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)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

Extensions

A246261 Numbers n such that A003961(n) is of the form 4k+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A246263 Numbers n such that A003961(n) is of the form 4k+3.

Views

Author

Keywords

Comments

Links

Crossrefs

A246271 Starting from A003961(n), the number of additional iterations of A003961 required for the result to be of the form 4k+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A246280 a(n) = the smallest starting value for k such that n will be the exact number of additional iterations of A003961 needed when we start from A003961(k) before we first reach a number of the form 4k+1; the least k such that A246271(k) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Scheme