A319627 -id:A319627 - cp's OEIS frontend

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

1, 2, 3, 4, 5, 9, 7, 2, 3, 25, 11, 81, 13, 49, 15625, 4, 17, 9, 19, 625, 117649, 121, 23, 27, 1225, 169, 21, 2401, 29, 3125, 31, 10, 1771561, 289, 5764801, 81, 37, 361, 4826809, 5, 41, 7, 43, 14641, 12005, 529, 47, 75, 1127357, 1500625, 24137569, 28561, 53, 441, 14641, 5764801, 47045881, 841, 59, 125, 61, 961, 343, 100, 302875106592253

Offset: 1

Antti Karttunen, Jul 07 2021

Denominator of ratio A324886(n) / A329044(n).

Cf. A346096 (numerators).
Cf. A006530, A020639, A108951, A276086, A319627, A324886, A329044, A345941 [= gcd(n, a(n))], A346095, A346098, A346107, A346109.
Cf. also A337377.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n)); \\ Rest of program given in A324886.

a(n) = A319627(A324886(n)).
a(n) = A329044(n) / A346095(n) = A329044(n) / gcd(A324886(n), A329044(n)).
A020639(a(n)) = A006530(n).
A108951(a(n)) = A346107(n).
A346105(a(n)) = A346109(n).

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Original entry on oeis.org

1, -2, -3, 0, -5, 3, -7, 0, 0, 10, -11, 0, -13, 14, 5, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -5, -31, 0, 33, 34, 7, 0, -37, 38, 39, 0, -41, -21, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -33, -67, 0, 69, -14, -71, 0, -73, 74, 0, 0, 11, -39, -79, 0, 0, 82

Offset: 1

Antti Karttunen, Jun 07 2022

Because the ratio n / A064989(n) = A319626(n) / A319627(n) is multiplicative, so is also its Dirichlet inverse (which also is a sequence of rational numbers). This sequence gives the numerators when presented in its lowest terms, while A354366 gives the denominators. See the examples.

			The ratio a(n)/A354366(n) for n = 1..22: 1, -2, -3/2, 0, -5/3, 3, -7/5, 0, 0, 10/3, -11/7, 0, -13/11, 14/5, 5/2, 0, -17/13, 0, -19/17, 0, 21/10, 22/7.

Index entries for sequences computed from indices in prime factorization

Cf. A013929 (positions of 0's), A055615, A319626, A319627, A354350.
Cf. A354366 (denominators).
Cf. also A349629, A354351, A354827.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A354365(n) = numerator((moebius(n)*n)/A064989(n));

a(n) = A055615(n) / gcd(A055615(n), A064989(n)).

A354366 Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 1, 3, 7, 1, 11, 5, 2, 1, 13, 1, 17, 1, 10, 7, 19, 1, 1, 11, 1, 1, 23, 1, 29, 1, 14, 13, 3, 1, 31, 17, 22, 1, 37, 5, 41, 1, 1, 19, 43, 1, 1, 1, 26, 1, 47, 1, 21, 1, 34, 23, 53, 1, 59, 29, 1, 1, 33, 7, 61, 1, 38, 3, 67, 1, 71, 31, 1, 1, 5, 11, 73, 1, 1, 37, 79, 1, 39, 41, 46, 1, 83, 1, 55, 1, 58

Offset: 1

Antti Karttunen, Jun 07 2022

Equally, denominators of Dirichlet inverse of fraction n / A064989(n). See also comments in A354365.

Index entries for sequences computed from indices in prime factorization

Cf. A055615, A064989, A319626, A319627, A354360 (positions of 1's).
Cf. A354365 (numerators).
Cf. also A349630.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A354366(n) = denominator((moebius(n)*n)/A064989(n));

a(n) = A064989(n) / gcd(A055615(n), A064989(n)).

A358217 Number of prime factors (with multiplicity) in A319627(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 3, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 1, 1, 3, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 0

Offset: 1

Antti Karttunen, Nov 04 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial numbers

Cf. A001222, A025487 (positions of zeros), A064989, A319627,

Cf. A358219 (positions where differs from A358218).

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A358217(n) = bigomega(A319627(n));

a(n) = A001222(A319627(n)).

Apparently, a(n) <= A358218(n) for all n.

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A319626 Primorial deflation of n (numerator): Let f be the completely multiplicative function over the positive rational numbers defined by f(p) = A034386(p) for any prime number p; f constitutes a permutation of the positive rational numbers; let g be the inverse of f; for any n > 0, a(n) is the numerator of g(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 15, 46, 47, 24, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 33, 67, 68, 69

Offset: 1

Rémy Sigrist, Sep 25 2018

See A319627 for the corresponding denominators.
The restriction of f to the natural numbers corresponds to A108951.
The function g is completely multiplicative over the positive rational numbers with g(2) = 2 and g(q) = q/p for any pair (p, q) of consecutive prime numbers.
The ratio A319626(n)/A319627(n) can be viewed as a "primorial deflation" of n (see also A329900), with the inverse operation being n = A108951(A319626(n)) / A108951(A319627(n)), where A319627(k) = 1 for all k in A025487. - Daniel Suteu, Dec 29 2019

			f(21/5) = (2*3) * (2*3*5*7) / (2*3*5) = 42, hence g(42) = 21/5 and a(42) = 21.

Daniel Suteu, Table of n, a(n) for n = 1..10000

A left inverse of A108951. Coincides with A329900 on A025487.

Cf. A006530, A053585, A064989, A181815, A307035, A319627, A319630, A329902, A330749, A330750 (rgs-transform), A330751 (ordinal transform).

Array[#1/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &, 120] (* Michael De Vlieger, Aug 27 2020 *)

a(n) = my (f=factor(n)); numerator(prod(i=1, #f~, my (p=f[i,1]); (p/if (p>2, precprime(p-1), 1))^f[i,2]))

a(n) = n / gcd(n, A064989(n)) = n / A330749(n).
a(n) <= n with equality iff n belongs to A319630.
A006530(a(n)) = A006530(n).
A053585(a(n)) = A053585(n).
From Antti Karttunen, Dec 29 2019: (Start)
a(A108951(n)) = n.
a(A025487(n)) = A329900(A025487(n)) = A181815(n).
Many of the formulas given in A329900 apply here as well:
a(n!) = A307035(n), a(A002182(n)) = A329902(n), and so on.
(End)

"Primorial deflation" prefixed to the name by Antti Karttunen, Dec 29 2019

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

Original entry on oeis.org

1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)*A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.

According to Daniel Suteu, also the ratio (A319626(n) / A319627(n)) can be viewed as a "primorial deflation". That definition coincides with this one when restricted to terms of A025487, as for all k in A025487, A319626(k) = a(k), and A319627(k) = 1. - Antti Karttunen, Dec 29 2019

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

A left inverse of A108951. Coincides with A319626 on A025487.

Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.

Array[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *)
Array[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)

A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); };

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1,oo,if(n%prime(i),return(i-1))); }
A329900(n) = if(n%2,1,prime(A276084(n))*A329900(A111701(n)));

For odd n, a(n) = 1, for even n, a(n) = A000040(A276084(n)) * a(A111701(n)).
For even n, a(n) = A000040(A276084(n)) * a(n/A002110(A276084(n))).
A108951(a(n)) = A328479(n), for n >= 1.
a(A108951(n)) = n, for n >= 1.
a(A328479(n)) = a(n),  for n >= 1.
a(A328478(n)) = 1, for n >= 1.
a(A002110(n)) = A000040(n), for n >= 1.
a(A000142(n)) = A307035(n), for n >= 0.
a(A283477(n)) = A019565(n), for n >= 0.
a(A329886(n)) = A005940(1+n), for n >= 0.
a(A329887(n)) = A163511(n), for n >= 0.
a(A329602(n)) = A329888(n), for n >= 1.
a(A025487(n)) = A181815(n), for n >= 1.
a(A124859(n)) = A181819(n), for n >= 1.
a(A181817(n)) = A025487(n), for n >= 1.
a(A181821(n)) = A122111(n), for n >= 1.
a(A002182(n)) = A329902(n), for n >= 1.
a(A260633(n)) = A329889(n), for n >= 1.
a(A033833(n)) = A330685(n), for n >= 1.
a(A307866(1+n)) = A330686(n), for n >= 1.
a(A330687(n)) = A330689(n), for n >= 1.

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 5, 9, 7, 5, 11, 9, 25, 7, 13, 45, 17, 11, 35, 81, 19, 25, 23, 3, 55, 13, 29, 9, 49, 17, 25, 99, 31, 35, 37, 27, 65, 19, 77, 225, 41, 23, 85, 21, 43, 55, 47, 39, 175, 29, 53, 405, 121, 49, 95, 153, 59, 25, 91, 99, 23, 31, 61, 15, 67, 37, 275, 729, 17, 65, 71, 19, 145, 77, 73, 45, 79, 41, 245, 207, 143, 85, 83

Offset: 1

Antti Karttunen, Nov 09 2021

Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
Both ratios are multiplicative because the constituent sequences are.
No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).

Cf. A000203, A003961, A161942, A286385, A319626, A319627, A326042, A336702, A342671, A349163, A349164, A349627, A349628.

Cf. A348992, A349162 (denominators).

Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349161(n) = { my(u=A003961(n)); (u/gcd(u,sigma(n))); };

from math import prod, gcd
from sympy import nextprime, factorint
def A349161(n):
    f = factorint(n).items()
    a = prod(nextprime(p)**e for p, e in f)
    b = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return a//gcd(a,b) # Chai Wah Wu, Mar 17 2023

a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).

a(n) = A003961(A349164(n)).

A337377 Primorial deflation (denominator) of Doudna-tree.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 3, 2, 1, 9, 2, 8, 1, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 11, 7, 14, 5, 21, 5, 20, 3, 5, 3, 2, 1, 9, 2, 8, 1, 49, 25, 50, 9, 15, 3, 4, 1, 125, 27, 18, 2, 81, 8, 32, 1, 13, 11, 22, 7, 33, 7, 28, 5, 55, 21, 14, 5, 63, 10, 40, 3, 7, 5, 10, 3, 3, 1, 4, 1, 25, 9, 6, 1, 27, 4, 16, 1, 121

Offset: 0

Antti Karttunen, Michael De Vlieger and Peter Munn, Aug 25 2020

Like A005940, also this irregular table can be represented as a binary tree:
                                      1
                                      |
                   ...................1...................
                  2                                       1
        3......../ \........1                   4......../ \........1
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    5       3           2       1           9       2           8       1
   7 5    10 3         3 1     4 1        25 9     6 1        27 4    16 1
etc.
A194602 gives the positions of nodes that have value 1. They correspond to terms of A005940 that are products of primorials (A025487). The first 2^k nodes contain A000041(k+1) 1's.
a(n) is even if and only if A005940(1+n) occurs in A277569.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for fraction trees

Cf. A337376 (numerators).
A003961, A005940, A006519, A026741, A064989, A319627 are used in a formula defining this sequence.
Cf. A000041, A025487, A277569.
Positions of 1's: A194602.
Cf. also A329886, A346097.

Array[#2/GCD[#1, #2] & @@ {#, Apply[Times, Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger[#]]]} &@ Function[p, Times @@ Flatten@ Table[Prime[Count[Flatten[#], 0] + 1]^#[[1, 1]] &@ Take[p, -i], {i, Length[p]}]]@ Partition[Split[Join[IntegerDigits[# - 1, 2], {2}]], 2] &[# + 1] &, 96] (* Michael De Vlieger, Aug 27 2020 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A337377(n) = A319627(A005940(1+n));

a(n) = A319627(A005940(1+n)).
For n >= 1, a(2*n) = A003961(a(n)) * A006519(n+1).
a(2*n+1) = A026741(a(n)).

cp's OEIS Frontend

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A354365 Numerators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A354366 Denominators of Dirichlet inverse of primorial deflation fraction A319626(n) / A319627(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A358217 Number of prime factors (with multiplicity) in A319627(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Sage

Formula

Extensions

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula