A337478 -id:A337478 - cp's OEIS frontend

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

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

A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

Original entry on oeis.org

1, 6, 120, 19399380, 195534950863140268380, 538938984694949877040715541221415046162838700, 216487559804430601784907786655491617909711008142914104790481010259258659171900

Offset: 0

Antti Karttunen, Aug 07 2020

For n > 0, the least k such that for at least n-1 iterations of map x -> A003961(x), starting from x=k, x stays nondeficient. In other words, from each a(n) starts a chain of at least n nondeficient numbers (A023196) obtained by successive prime shifts, e.g, for a(3) we have: 19399380 -> 334639305 -> 5391411025, where -> stands for applying A003961, the prime shift towards larger primes.
After 1 all other terms here are even, because if an odd number k is nondeficient, then A064989(k) is nondeficient also, where A064989 is the prime shift towards smaller primes. Moreover, because A047802 is defined for every n >= 0, also this sequence is.
From Peter Munn, Aug 13 2020 (Start)
Upper bounds for a(4) and a(5) are:
a(4) <= 195534950863140268380 = A064989(A064989(A064989(20169691981106018776756331))) = A337202(3).
a(5) <= 538938984694949877040715541221415046162838700 = A064989^4((A047802(4)*17*19)/137).
(End)
From David A. Corneth, Aug 21 2020: (Start)
Subsequence of A025487.
Let prime(n)# be the n-th primorial number, A002110(n) = A034386(prime(n)). Then:
a(6) <= 191# * 7#;
a(7) <= 311# * 5#;
a(8) <= 457# * 5#.
(End)
That each term occurs in A025487 follows because (1), the abundancy index of prime(i)^e is larger than that of prime(i+1)^e, that is, sigma(prime(i)^e)/prime(i)^e > sigma(prime(i+1)^e)/prime(i+1)^e, and (2) because the abundancy index of p^(e+d) * q^e is larger than that of p^e * q^(e+d), where p and q are distinct primes, p < q, and e, d > 0. Thus, for any n, we can first find a "prime-factorization compressed version" of it, A071364(n), and then sort the exponents to the non-ascending order with A046523 (and actually, A046523(A071364(n)) = A046523(n), so we need to apply just A046523), to get a term x of A025487, that certainly have the abundancy index >= n [and this inequivalence stays same for their successive prime shifts as well, the abundancy index of A003961(x) being at least that of A003961(n), etc.], and as A046523(n) <= n for all n, it is guaranteed that the least k for which A336835(k) >= n are found from A025487, which is the range of A046523.

Cf. A003961, A005100, A023196, A025487, A047802, A108951, A246277, A329900, A336834, A336835, A337202, A337205, A337477, A337478.

From term a(2) = 120 onward a subsequence of A337386.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
A336389(n) = for(i=1,oo,if(A336835(i)>=n,return(i)));

For n >= 0, A336835(a(n)) >= n.
For all n >= 1, a(n) <= A337202(n-1) [= 2*A246277(A047802(n-1))].
a(n) = A025487(A337477(n)).
a(n) = A108951(A337478(n)).

a(4) - a(6) from combined work of David A. Corneth and Peter Munn Aug 13-26 2020

A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 28 2020

nonn

a(n) is the least k for which A337473(k, n) = 1.

Cf. A003961, A108951, A181815, A336835, A337473, A337475.

Cf. A337476 (position of the first occurrence of each n), A337478.

A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337474(n) = for(i=0,oo,if(1==A337473sq(i,n),return(i)));

\\ This version uses binary search, which is faster in certain cases:
isA337473sq1(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(s<2)); prevpid = pid; e += f[i,2]); (s<2));
A337474(n) = if(!bitand(n,n-1),0,my(imin=0,imax=n,imid); for(i=0,oo, imid=(imax+imin)\2; if(1!=isA337473sq1(imid,n), imin = imid+1, if(1!=isA337473sq1(imid-1,n),return(imid), imax = imid-1))));

a(n) = A336835(A108951(n)).
a(A181815(n)) = A337475(n).
For all n >= 0, a(A337476(n)) = n.
For all n >= 0, a(A337478(n)) >= n.

cp's OEIS Frontend

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

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A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

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A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

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A337476 Position of the first occurrence of n in A337474.

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A337477 Position of the first term >= n in A337475.

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