cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A344594 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342920(i) = A342920(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 5, 1, 2, 6, 7, 1, 8, 1, 5, 6, 2, 1, 9, 10, 2, 7, 5, 1, 11, 1, 12, 6, 2, 13, 14, 1, 2, 6, 15, 1, 8, 1, 5, 16, 2, 1, 17, 18, 19, 6, 5, 1, 20, 21, 3, 6, 2, 1, 22, 1, 2, 23, 24, 25, 4, 1, 5, 6, 26, 1, 27, 1, 2, 28, 5, 29, 4, 1, 30, 31, 2, 1, 14, 32, 2, 6, 3, 1, 33, 34, 5, 6, 2, 35, 36, 1, 37, 38, 39, 1, 4, 1, 3, 40
Offset: 1

Views

Author

Antti Karttunen, May 26 2021

Keywords

Comments

Restricted growth sequence transform of A342920, where A342920(n) = A342002(A108951(n)) = A329047(n) / A344592(n).

Links

Crossrefs

Cf. A108951, A324886, A329047, A342002, A342920, A344592.
Cf. also A344593.

Programs

  • PARI
    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
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342002(n) = { my(u=A276086(n)); (A003415(u) / A003557(u)); };
A342920(n) = A342002(A108951(n));
v344594 = rgs_transform(vector(up_to, n, A342920(n)));
A344594(n) = v344594[n];

A346103 Numerators of sequence whose Dirichlet convolution with itself yields A342920.

Original entry on oeis.org

1, 1, 1, 7, 1, 3, 1, 57, 47, 3, 1, 19, 1, 3, 11, 747, 1, 139, 1, 19, 11, 3, 1, 319, 199, 3, 81, 19, 1, 231, 1, -265, 11, 3, 251, 873, 1, 3, 11, 191, 1, 79, 1, 19, 299, 3, 1, -157, 5943, 595, 11, 19, 1, 151, 187, 31, 11, 3, 1, 269, 1, 3, 507, -957, 527, 31, 1, 19, 11, 223, 1, 18787, 1, 3, 8915, 19, 483, 31, 1, 2147, 19355
Offset: 1

Views

Author

Antti Karttunen, Jul 09 2021

Keywords

Links

Crossrefs

Cf. A046644 (gives the denominators).
Cf. A108951, A324886, A342001, A342002, A342920, A346104.

Programs

  • PARI
    up_to = 2310;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
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
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A342920(n) = A342002(A108951(n));
vA346103aux = DirSqrt(vector(up_to, n, A342920(n)));
A346103(n) = numerator(vA346103aux[n]);

A346104 Dirichlet inverse of A342920.

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, -5, -11, 0, -1, 1, -1, 0, -4, -1, -1, 1, -1, 1, -4, 0, -1, -8, -49, 0, 7, 1, -1, -48, -1, 25, -4, 0, -124, 12, -1, 0, -4, 0, -1, -10, -1, 1, -13, 0, -1, 41, -1485, 1, -4, 1, -1, 22, -92, 10, -4, 0, -1, 76, -1, 0, -39, 23, -262, 2, -1, 1, -4, 134, -1, -138, -1, 0, -1033, 1, -240, 2, -1, 11, -201, 0, -1, 6
Offset: 1

Views

Author

Antti Karttunen, Jul 09 2021

Keywords

Comments

The lowermost scatter plot gives an appearance of an elliptic galaxy seen from its side.

Links

Crossrefs

Cf. A108951, A324886, A342001, A342002, A342920, A346103.
Cf. also A342417.

Programs

  • PARI
    up_to = 2310;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA034386(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
A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A342920(n) = A342002(A108951(n));
v346104 = DirInverseCorrect(vector(up_to,n,A342920(n)));
A346104(n) = v346104[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

Views

Author

Paul Boddington, Jul 21 2005

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Python
    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
  • Sage
    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

Formula

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

Extensions

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

A324886 a(n) = A276086(A108951(n)).

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 11, 15, 35, 49, 13, 625, 17, 121, 117649, 225, 19, 1225, 23, 2401, 1771561, 169, 29, 875, 717409, 289, 55, 14641, 31, 184877, 37, 21, 4826809, 361, 36226650889, 1500625, 41, 529, 24137569, 77, 43, 143, 47, 28561, 1127357, 841, 53, 1715, 902613283, 514675673281, 47045881, 83521, 59, 3025, 8254129, 214358881, 148035889, 961, 61
Offset: 1

Views

Author

Antti Karttunen, Mar 30 2019

Keywords

Links

Crossrefs

Cf. A034386, A108951, A117366, A276086, A324887, A324888, A324896, A329047, A342920, A344592, A346106, A346107.
Permutation of A324576.
Cf. also A346105.

Programs

  • Mathematica
    With[{b = MixedRadix[Reverse@ Prime@ Range@ 120]}, Array[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], b] &, 58]] (* Michael De Vlieger, Nov 18 2019 *)
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
(* b is A108951 *)
b[n_] := b[n] = Module[{pe = FactorInteger[n], p, e}, If[Length[pe] > 1, Times @@ b /@ Power @@@ pe, {{p, e}} = pe; Times @@ (Prime[Range[ PrimePi[p]]]^e)]]; b[1] = 1;
a[n_] := A276086[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021, after _Antti Karttunen in A296086 *)
  • PARI
    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(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));

Formula

a(n) = A276086(A108951(n)).
a(n) = A117366(n) * A324896(n).
A001222(a(n)) = A324888(n).
A020639(a(n)) = A117366(n).
A032742(a(n)) = A324896(n).
a(A000040(n)) = A000040(1+n).
From Antti Karttunen, Jul 09 2021: (Start)
For n > 1, a(n) = A003961(A329044(n)).
a(n) = A346091(n) * A344592(n).
a(n) = A346106(n) / A346107(n).
A003415(a(n)) = A329047(n).
A003557(a(n)) = A344592(n).
A342001(a(n)) = A342920(n) = A329047(n) / A344592(n).
(End)

A342002 Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 2, 9, 11, 37, 16, 47, 3, 11, 14, 43, 19, 53, 4, 13, 17, 49, 22, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 33, 101, 134, 373, 169, 443, 2, 11, 13, 47, 20, 61, 17, 69, 86, 277, 121, 347, 24, 83, 107, 319, 142, 389, 31
Offset: 0

Views

Author

Antti Karttunen, Feb 28 2021

Keywords

Comments

The scatter plot shows an interesting structure.
The terms are essentially the "wild" or "unherited" part of the arithmetic derivative (A003415) of those natural numbers (A048103) that are not immediately beyond all hope of reaching zero by iteration (as the terms of A100716 are), ordered by the primorial base expansion of n as in A276086. Sequence A342018 shows the positions of the terms here that have just moved to the "no hope" region, while A342019 shows how many prime powers in any term have breached the p^p limit. Note that the results are same as for A327860(n), as the division by "regular part", A328572(n) does not affect the "wild part" of the arithmetic derivative of A276086(n). - Antti Karttunen, Mar 12 2021
I decided to name this sequence in honor of Lithuanian artist Mikalojus Čiurlionis, 1875 - 1911, as the scatter plot of this sequence reminds me thematically of his work "Pyramid sonata", with similar elements: fractal repetition in different scales and high tension present, discharging as lightning. Like Čiurlionis's paintings, this sequence has many variations, see the Formula and Crossrefs sections. - Antti Karttunen, Apr 30 2022

Links

Crossrefs

Cf. A002110 (positions of 1's), A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001, A342005, A342006, A342016, A342022 (rgs-transform), A342417, A342419.
Cf. A342463 [= a(A329886(n))], A342920 [= a(A108951(n))], A342921 [= a(A276156(n))], A342017 [= A342007(a(n))], A342019 [= A129251(a(n))].
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860), A353640 (a(n) mod 4).
Cf. A344760, A344761, A344762, A346252, A346253 and A345930, A353572, A353574 for permuted and other variants.
Cf. A351952 (similar definition, but using factorial base, with quite a different look).

Programs

  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342002(n) = A342001(A276086(n)); \\ Uses also code from A342001.
  • PARI
    A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ Antti Karttunen, Mar 12 2021
  • PARI
    A342002(n) = { my(s=0, p=2, e); while(n, e = (n%p); s += (e/p); n = n\p; p = nextprime(1+p)); numerator(s); }; \\ (Taking denominator instead would give A328571) - Antti Karttunen, Mar 12 2021

Formula

a(n) = A342001(A276086(n)) = A083345(A276086(n)).
a(n) = A327860(n) / A328572(n) = A003415(A276086(n)) / A003557(A276086(n)).
From Antti Karttunen, Jul 18 2021: (Start)
There are several permutations of this sequence. The following formulas show the relations:
a(n) = A344760(A289234(n)).
a(n) = A346252(A328623(n)) = A346253(A328622(n)).
a(n) = A344761(A328626(n)) = A344762(A328625(n)).
(End)

Extensions

Sequence renamed as "Čiurlionis sequence" to honor Lithuanian artist Mikalojus Čiurlionis - Antti Karttunen, Apr 30 2022

A344592 a(n) = A003557(A276086(A108951(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 125, 1, 11, 16807, 15, 1, 35, 1, 343, 161051, 13, 1, 25, 9317, 17, 1, 1331, 1, 2401, 1, 1, 371293, 19, 253333223, 42875, 1, 23, 1419857, 1, 1, 1, 1, 2197, 14641, 29, 1, 49, 371293, 6684099653, 2476099, 4913, 1, 55, 37349, 19487171, 6436343, 31, 1, 5929, 1, 37, 20449, 21, 582622237229761, 1792160394037
Offset: 1

Views

Author

Antti Karttunen, May 26 2021

Keywords

Links

Crossrefs

Cf. A003557, A085731, A108951, A276086, A324886, A328572, A329047, A342002, A342920, A346091, A346092 [= A276085(a(n))].
Cf. A344591 (positions of ones), A344593 (rgs-transform).

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[#/(Times @@ FactorInteger[#][[All, 1]]) &@ Apply[Times, Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &@ Apply[Times, Map[(Times @@ Prime@ Range@ PrimePi@ #1)^#2 & @@ # &, FactorInteger[#]]] &, 66]] (* Michael De Vlieger, Jul 14 2021 *)
  • PARI
    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));
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A003557(A276086(A108951(n)));

Formula

a(n) = A328572(A108951(n)) = A003557(A324886(n)) = A003557(A276086(A108951(n))).
a(n) = A329047(n) / A342920(n).
a(n) = A085731(A324886(n)) = gcd(A324886(n), A329047(n)) = A324886(n) / A346091(n). - Antti Karttunen, Jul 09 2021

A342463 a(n) = A342001(A342456(n)); "wild part" of the arithmetic derivative of A342456(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 12, 8, 1, 2, 6, 4, 50, 24, 16, 16, 1, 2, 6, 4, 126, 62, 46, 26, 1486, 100, 1142, 48, 2056, 32, 342, 10, 1, 2, 6, 4, 94, 24, 72, 18, 242, 120, 1588, 54, 3408, 92, 1740, 22, 6846, 2972, 4340, 766, 5048, 1374, 652, 376, 71156, 22710, 20390, 64, 738580, 4272, 568, 20, 1, 2, 6, 4, 264, 12, 196, 8, 318
Offset: 0

Views

Author

Antti Karttunen, Mar 15 2021

Keywords

Comments

Like in A342462, also here the subsequences starting at each n = 2^k seem to be slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, ...

Links

Crossrefs

Cf. A003415, A003557, A005940, A108951, A329886, A342001, A342002, A342456, A342462, A342920.

Programs

  • Mathematica
    Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] := Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &@ f@ a[#] &, 73, 0]] (* Michael De Vlieger, Mar 17 2021 *)
  • PARI
    \\ Needs also code from A342456.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A342463(n) = A342001(A342456(n));

Formula

a(n) = A342001(A342456(n)) = A342002(A329886(n)) = A342920(A005940(1+n)).
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