A246269 -id:A246269 - 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

A246272 Starting from n, the number of iterations of A003961 needed before the result has only prime factors of the form 4k+1 (a(1) = 0). [Where A003961(n) shifts the prime factorization of n one step towards larger primes].

Original entry on oeis.org

0, 2, 1, 2, 0, 5, 2, 2, 1, 9, 1, 5, 0, 2, 4, 2, 0, 5, 2, 9, 8, 2, 1, 5, 0, 6, 1, 2, 0, 23, 1, 2, 1, 5, 3, 5, 0, 2, 1, 9, 0, 49, 2, 2, 4, 9, 1, 5, 2, 9, 5, 6, 0, 5, 7, 2, 4, 2, 1, 23, 0, 2, 8, 2, 0, 5, 2, 5, 1, 9, 1, 5, 0, 6, 4, 2, 2, 23, 2, 9, 1, 5, 1, 49, 0, 2, 8, 2, 0, 23, 6, 9, 1, 6, 4, 5, 0, 2, 1, 9

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Among the first 10000 terms, of which 4371 are primes, there are 92 distinct values in total (of which 23 are primes), the most common of them being: 1600 x 2, 1324 x 5, 1131 x 1, 1074 x 0, 571 x 4, 557 x 6, 538 x 9, 409 x 23, 404 x 3, 378 x 11, 211 x 8, 197 x 15, 131 x 12, 130 x 24, 128 x 49, 119 x 10, 95 x 7, 95 x 76, 92 x 22, 80 x 32, 70 x 14, 53 x 20, 47 x 77, 44 x 28, 29 x 17, 27 x 58, 24 x 21, 24 x 64, 23 x 13, 22 x 31, 22 x 39, 20 x 25, 19 x 48.
In contrast to A246271, here it holds that a(u) <= a(u*v) >= a(v) for any u, v. This follows because a number n has reached the "state of 4k+1 purity" (meaning that A065338(n) = 1) only if all its possible divisors have reached it as well.
This explains why for example 23 seems to be so common value. The reason is, that it occurs for the first time as a(30), and 30, being the third primorial is a product of three commonest primes: 30 = 2*3*5, thus for any n which is a multiple of 30, a(n) >= 23. Similarly a(42) = 49, a(84) = 49, a(126) = 49, so any number k which has 2, 3 and 7 as its prime factors must have a(k) >= 49.

			Consider n = 6 = 2*3 = p_1 * p_2. Five is the least number of iterations of A003961(n) (which increments by one the prime indices of prime factorization of n), before both primes are of the form 4k+1:
  p_2 = 3, p_3 = 5 (4k+3 & 4k+1),
  p_3 = 5, p_4 = 7 (4k+1 & 4k+3),
  p_4 = 7, p_5 = 11 (4k+3 & 4k+3),
  p_5 = 11, p_6 = 13 (4k+3 & 4k+1),
  p_6 = 13, p_7 = 17 (4k+1 & 4k+1),
thus a(6) = 5.

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

A004613 gives the positions of zeros.
A246349 gives the positions of records and A246350 the corresponding values.
Cf. A003961, A065338, A246269, A246270, A246271, A246277.

default(primelimit, 2^22)
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
A065338(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = (f[i, 1]%4)); factorback(f);
A246272(n) = {my(i); i=0; while((A065338(n)!=1), i++; n = A003961(n)); i};
for(n=1, 10001, write("b246272.txt", n, " ", A246272(n)));

(define (A246272 n) (let loop ((i 0) (n n)) (if (= 1 (A065338 n)) i (loop (+ i 1) (A003961 n)))))

;; Requires memoizing definec-macro.
(definec (A246272 n) (if (= 1 (A065338 n)) 0 (+ 1 (A246272 (A003961 n)))))

If A065338(n) = 1, a(n) = 0, otherwise 1 + a(A003961(n)).
Other identities:
a(n) = a(A007947(n)) for all n. [Duplicate prime factors have no effect on the result].

A246270 Number of prime factors of the form 4k+3 (counted with multiplicity) in A003961(n): a(n) = A065339(A003961(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A001222, A003961, A007949, A065339, A246269, A246272.

default(primelimit, 2^22)
A246269(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = (nextprime(f[i, 1]+1)%4)); factorback(f);
A246270(n) = bigomega(A246269(n));
for(n=1, 10080, write("b246270.txt", n, " ", A246270(n)));

(define (A246270 n) (A065339 (A003961 n)))

(define (A246270 n) (A007949 (A246269 n)))

a(n) = A065339(A003961(n)).
a(n) = A001222(A246269(n)).
a(n) = A007949(A246269(n)).
Other identities.
If n = u*v, a(n) = a(u)+a(v).
For all n >= 0, a(2^n) = n.

cp's OEIS Frontend

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

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A246272 Starting from n, the number of iterations of A003961 needed before the result has only prime factors of the form 4k+1 (a(1) = 0). [Where A003961(n) shifts the prime factorization of n one step towards larger primes].

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A246270 Number of prime factors of the form 4k+3 (counted with multiplicity) in A003961(n): a(n) = A065339(A003961(n)).

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