A329614 -id:A329614 - cp's OEIS frontend

A329613 Numbers k such that A071187(k) <> A329614(k).

324, 1296, 2500, 5625, 9604, 10000, 11664, 20736, 21609, 38416, 50625, 58564, 60025, 82944, 90000, 114244, 131769, 160000, 194481, 234256, 236196, 250000, 257049, 334084, 345744, 360000, 366025, 456976, 521284, 614656, 640000, 714025, 717409, 751689, 810000, 944784, 960400, 1119364, 1172889, 1185921, 1265625, 1327104, 1336336

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

All terms are squares; their square roots are in A329611.

David A. Corneth, Table of n, a(n) for n = 1..15072 (first 661 terms from Antti Karttunen, terms <= 10^14)
David A. Corneth, PARI program

Subsequence of A000290.

Cf. A055932, A071187, A329611, A329614.

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
A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1]));
A329614(n) = A071187(A108951(n));
k=0; n=0; while(n<2^16, n++; u = n*n; if(A071187(u)!=A329614(u), k++; write("b329613.txt", k, " ", u); print1(u,", ")));

PARI
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A329611 Numbers n for which A071187(n^2) <> A329614(n^2).

Original entry on oeis.org

18, 36, 50, 75, 98, 100, 108, 144, 147, 196, 225, 242, 245, 288, 300, 338, 363, 400, 441, 484, 486, 500, 507, 578, 588, 600, 605, 676, 722, 784, 800, 845, 847, 867, 900, 972, 980, 1058, 1083, 1089, 1125, 1152, 1156, 1176, 1183, 1225, 1350, 1372, 1444, 1445, 1452, 1521, 1568, 1587, 1682, 1764, 1800, 1805, 1859, 1922, 1936, 1944

Offset: 1

Antti Karttunen, Nov 23 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..661

Cf. A000196, A071187, A329614, A329613.

a(n) = A000196(A329613(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

A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 4, 9, 12, 32, 8, 64, 24, 18, 5, 128, 12, 256, 16, 36, 48, 512, 10, 27, 96, 16, 32, 1024, 24, 2048, 6, 72, 192, 54, 15, 4096, 384, 144, 20, 8192, 48, 16384, 64, 32, 768, 32768, 12, 81, 36, 288, 128, 65536, 20, 108, 40, 576, 1536, 131072, 30, 262144, 3072, 64, 7, 216, 96, 524288, 256, 1152, 72, 1048576, 18, 2097152, 6144, 48

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).
Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).
Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).

Block[{a}, 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; Array[DivisorSigma[0, a@ #] &, 75]] (* Michael De Vlieger, Jan 24 2020, after Jean-François Alcover at A108951 *)

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
A329605(n) = numdiv(A108951(n));

A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

A329605(n) = if(1==n,1,my(f=factor(n),e=0,d); forstep(i=#f~,1,-1, e += f[i,2]; d = (primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1]))); f[i,1] = (e+1); f[i,2] = d); factorback(f)); \\ Antti Karttunen, Jan 14 2020

a(n) = A000005(A108951(n)).
a(n) >= A329382(n) >= A329617(n) >= A329378(n).
A020639(a(n)) = A329614(n).
From Antti Karttunen, Jan 14 2020: (Start)
a(A052126(n)) = A329382(n).
a(A002110(n)) = A000142(1+n), for all n >= 0.
a(n) > A056239(n).
a(A329902(n)) = A002183(n).
A000265(a(n)) = A331286(n).
gcd(n,a(n)) = A331283(n).
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.
A000035(a(n)) = A000035(A000005(n)) = A010052(n).
(End)

A071187 Smallest prime factor of number of divisors of n; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, May 15 2002

a(n) = 2 for nonsquare n. - David A. Corneth, Jul 24 2017

			324 = 18^2 = 2^2 * 3^4 has (2+1)*(4+1) = 3 * 5 = 15 divisors, thus a(324) = A020639(15) = 3. - _Antti Karttunen_, Nov 18 2019

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A020639, A071188, A071189, A108951.

Differs from A329614 for the first time at n=324, where a(324) = 3, while A329614(324) = 5. A329613 gives the positions of differences.

a[n_] := FactorInteger[DivisorSigma[0, n]][[1, 1]]; Array[a, 90] (* Jean-François Alcover, Oct 01 2016 *)

A071187(n) = if(1==n, n, my(f = factor(numdiv(n))); vecmin(f[, 1])); \\ Antti Karttunen, Jul 24 2017

first(n) = my(v = vector(n, i, 2)); for(i=1,sqrtint(n), v[i^2] = numdiv(i^2)); v

a(n) = A020639(A000005(n)).
a(A108951(n)) = A329614(n). - Antti Karttunen, Nov 17 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Jan 15 2024

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A329606 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329605(i) = A329605(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 3, 8, 9, 10, 5, 11, 12, 13, 14, 15, 9, 16, 7, 17, 18, 19, 20, 21, 22, 7, 10, 23, 12, 24, 6, 25, 26, 27, 28, 29, 30, 31, 32, 33, 18, 34, 11, 10, 35, 36, 9, 37, 17, 38, 15, 39, 32, 40, 41, 42, 43, 44, 45, 46, 47, 11, 48, 49, 22, 50, 16, 51, 25, 52, 13, 53, 54, 18, 19, 55, 26, 56, 12, 57, 58, 59, 60, 61, 62, 63, 64, 65, 41, 66, 23, 67, 68, 69, 70, 71, 40, 15, 72, 73, 30, 74, 75, 22

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Restricted growth sequence transform of A329605(n) = A000005(A108951(n)).
For all i, j:
  a(i) = a(j) => A329614(i) = A329614(j).

Cf. A000005, A108951, A329605, A329608, A329614.

up_to = 100000;
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
A329605(n) = numdiv(A108951(n));
v329606 = rgs_transform(vector(up_to, n, A329605(n)));
A329606(n) = v329606[n];

A329608 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A329605(n) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j) => A329614(i) = A329614(j).

Cf. A000005, A108951, A305801, A329605, A329606, A329614.

up_to = 100000;
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
A329605(n) = numdiv(A108951(n));
Aux329608(n) = if((n%2)&&isprime(n),0,A329605(n));
v329608 = rgs_transform(vector(up_to, n, Aux329608(n)));
A329608(n) = v329608[n];

cp's OEIS Frontend

A329613 Numbers k such that A071187(k) <> A329614(k).

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A329611 Numbers n for which A071187(n^2) <> A329614(n^2).

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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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A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A071187 Smallest prime factor of number of divisors of n; a(1) = 1.

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A329606 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329605(i) = A329605(j) for all i, j.

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A329608 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A329605(n) for all other n, except for odd primes p, f(p) = 0.

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A329612 a(n) = gcd(d(n), d(A108951(n))), where d(n) gives the number of divisors of n, A000005(n), and A108951 is fully multiplicative with a(prime(i)) = prime(i)# = prime(1) * ... * prime(i).

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