A329344 -id:A329344 - 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

A329348 The least significant nonzero digit in the primorial base expansion of primorial inflation of n, A108951(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 3, 2, 2, 1, 4, 1, 5, 1, 1, 6, 2, 8, 4, 1, 2, 6, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 4, 6, 4, 1, 2, 4, 8, 6, 2, 1, 3, 1, 2, 3, 2, 13, 12, 1, 4, 6, 5, 1, 3, 1, 2, 5, 4, 2, 12, 1, 2, 1, 2, 1, 2, 11, 2, 6, 8, 1, 2, 6, 4, 6, 2, 7, 2, 1, 2, 10, 1, 1, 12, 1, 8, 4

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

Number of occurrences of the least primorial present in the greedy sum of primorials adding to A108951(n).

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the least primorial in the sum is 3, we have a(24) = 3.

Cf. A002110, A034386, A067029, A108951, A117366, A276086, A276088, A324886, A324888, A329040, A329345, A329343, A329344, A329349, A331188, A331289, A331290, A331291.

Cf. also A331292 (the next digit to left of this one), A331293.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); };
A329348(n) = A276088(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A067029(n) = if(1==n, 0, factor(n)[1, 2]); \\ From A067029
A329348(n) = A067029(A324886(n));

A002110(n) = prod(i=1, n, prime(i));
A329348(n) = if(1==n, n, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p); \\ Antti Karttunen, Jan 15 2020

a(n) = A067029(A324886(n)) = A276088(A108951(n)).
a(n) <= A324888(n).
From Antti Karttunen, Jan 15-17 2020: (Start)
a(n) = A331188(n) mod A117366(n).
a(n) = A001511(A246277(A324886(n))).
(End)

Name changed by Antti Karttunen, Jan 17 2020

A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 1, 0, 0, 4, -2, 0, -1, 0, 1, 4, 0, 0, -1, 2, 0, -2, 1, 0, 2, 0, -4, 4, 0, 6, 0, 0, 0, 4, -3, 0, -2, 0, 1, 2, 0, 0, -2, 4, 5, 4, 1, 0, -2, 2, 4, 4, 0, 0, -1, 0, 0, 0, -4, 11, 9, 0, 1, 4, 2, 0, -2, 0, 0, 2, 1, 14, 9, 0, -3, 2, 0, 0, -2, 9, 0, 4, 4, 0, 6, 10, 1, 4, 0, 5, 0, 0, 9, 7, 2, 0, 9, 0, 4, 1

Offset: 1

Antti Karttunen, Nov 18 2019

sign

Cf. A001222, A007814, A034386, A108951, A276086, A324886, A328114, A329344, A329618, A329620, A329622.

With[{b = Reverse@ Prime@ Range@ 120}, Array[Max@ IntegerDigits[#, MixedRadix[b]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] - PrimeOmega[#] &, 105] ] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A329344(n) = A328114(A108951(n));
A329619(n) = (A329344(n) - bigomega(n));

a(n) = A329344(n) - A001222(n).
a(n) = A328114(A108951(n)) - A007814(A108951(n)).
a(p) = 0 for all primes p.

A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 40, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 115, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 228, 229, 233, 239, 241, 251, 252, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Antti Karttunen, May 26 2021

nonn

Numbers k such that A108951(k) is in A276156.

			A108951(40) = 240 and 240 is in A276156 because 240 = A002110(4) + A002110(3) = 210 + 30, therefore 40 is included in this sequence.

Positions of ones in A329344, in A344592 and in A344593.
Positions of squarefree terms in A324886.
Union of A008578 and A351959.
Cf. A002110, A108951, A276156, A351957 (characteristic function).
Cf. also A351958.

Name changed by Antti Karttunen, Apr 04 2022

A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 6, 2, 1, 2, 1, 4, 6, 2, 1, 1, 4, 2, 1, 4, 1, 1, 1, 1, 6, 2, 2, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 1, 8, 6, 4, 1, 2, 2, 8, 6, 2, 1, 3, 1, 2, 3, 2, 1, 12, 1, 4, 6, 5, 1, 1, 1, 2, 2, 4, 16, 12, 1, 2, 6, 2, 1, 2, 1, 2, 6, 8, 1, 10, 12, 4, 6, 2, 1, 6, 1, 2, 2, 1, 1, 12, 1, 8, 1

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 21 = 3 * 7, A108951(21) = A034386(3) * A034386(7) = 6 * 210, so the factor of the largest primorial present (210) in the greedy sum is 6 (as 1260 = 210 + 210 + 210 + 210 + 210 + 210), thus a(21) = 6.
For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the largest primorial in the sum is 1, we have a(24) = 1.

Cf. A002110, A034386, A071178, A108951, A276086, A276153, A324886, A324888, A329040, A329343, A329344, A329345, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276153(n) = { my(e=0, p=2); while(n, e=n%p; n = n\p; p = nextprime(1+p)); (e); };
A329349(n) = A276153(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A071178(n) = if(1==n,0,my(es=factor(n)[,2]); es[#es]);
A329349(n) = A071178(A324886(n));

a(n) = A276153(A108951(n)) = A071178(A324886(n)).

a(n) <= A324888(n).

A342464 Largest digit value when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 6, 4, 4, 2, 1, 2, 1, 2, 6, 4, 8, 5, 5, 3, 6, 8, 5, 4, 11, 2, 6, 1, 1, 2, 6, 4, 4, 1, 3, 1, 16, 5, 4, 3, 9, 10, 8, 3, 10, 12, 10, 6, 10, 7, 2, 3, 18, 10, 5, 4, 12, 2, 4, 2, 1, 2, 6, 4, 13, 12, 10, 8, 12, 8, 13, 2, 4, 6, 7, 2, 15, 15, 12, 10, 9, 8, 7, 6, 10, 12, 10, 9, 11, 6, 9, 6, 18, 15

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

Cf. A005940, A051903, A108951, A276086, A328114, A329344, A329886, A342456, A342461, A342462.

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A342464(n) = A328114(A329886(n));

a(n) = A328114(A329886(n)) = A051903(A342456(n)) = A329344(A005940(1+n)).

A344593 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344592(i) = A344592(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 1, 4, 1, 5, 1, 6, 7, 8, 1, 9, 1, 10, 11, 12, 1, 13, 14, 15, 1, 16, 1, 17, 1, 1, 18, 19, 20, 21, 1, 22, 23, 1, 1, 1, 1, 24, 25, 26, 1, 27, 18, 28, 29, 30, 1, 31, 32, 33, 34, 35, 1, 36, 1, 37, 38, 39, 40, 41, 1, 42, 43, 44, 1, 45, 1, 46, 47, 48, 49, 50, 1, 51, 11, 52, 1, 53, 54, 55, 56, 57, 1, 58, 59, 60, 61, 62, 63, 64, 1, 65, 66, 11, 1

Offset: 1

Antti Karttunen, May 26 2021

nonn

Restricted growth sequence transform of A344592, where A344592(n) = A003557(A276086(A108951(n))).

For all i, j: a(i) = a(j) => A329344(i) = A329344(j).

			Both a(14) = 6 and a(32768) = 6, because A344592(14) = 11 is the sixth distinct value occurring in A344592, and A344592(32768) = A003557(A276086(A108951(32768))) = A003557(A276086(32768)) = A003557(401115) = A003557(3 * 5 * 11^2 * 13 * 17) = 11 also, which is the second time 11 occurs in A344592.

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

Cf. A003557, A108951, A276086, A324886, A329344, A344591 (positions of ones), A344592.

Cf. also A329045, A329345, A344594.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));
v344593 = rgs_transform(vector(up_to, n, A344592(n)));
A344593(n) = v344593[n];

A329346 a(n) = A322356(A324886(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 19, 1, 1, 1, 1, 13, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 7, 1, 13, 1, 13, 1, 1, 1, 1, 13

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

			For n = 128 = 2^7, A108951(128) = A034386(2)^7 = 128. As 128 = 4 * 30 + 1*6 + 1* 2, A276086(128) = 36015 = 7^4 * 5^1 * 3^1, and there are two such primes that both p and p-2 divide n, and p-2 is also prime, namely, 7 and 5, thus a(128) = 7*5 = 35. This is also the first occurrence of composite number in this sequence.

Cf. A002110, A034386, A108951, A276086, A324886, A329040, A322356, A329343, , A329344, A329347, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };
A329346(n) = A322356(A324886(n));

a(n) = A322356(A324886(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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A329348 The least significant nonzero digit in the primorial base expansion of primorial inflation of n, A108951(n).

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A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

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A344591 Numbers k such that the primorial inflation of k is a sum of distinct primorial numbers.

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A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

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A342464 Largest digit value when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A344593 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344592(i) = A344592(j), for all i, j >= 1.

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