A071178 -id:A071178 - cp's OEIS frontend

A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

1, 2, 3, 4, 5, 9, 7, 8, 6, 25, 11, 27, 13, 49, 15, 16, 17, 18, 19, 125, 35, 121, 23, 81, 10, 169, 12, 343, 29, 75, 31, 32, 77, 289, 21, 54, 37, 361, 143, 625, 41, 245, 43, 1331, 45, 529, 47, 243, 14, 50, 221, 2197, 53, 36, 55, 2401, 323, 841, 59, 375, 61, 961, 175, 64

Offset: 1

Antti Karttunen, May 03 2014

nonn

For other numbers than the powers of 2 (that are fixed), this permutation reverses the sequence of exponents in the prime factorization of n from the exponent of 2 to that of the largest prime factor, except that the exponents of 2 and the greatest prime factor present are adjusted by one. Note that some of the exponents might be zeros.
Self-inverse permutation of natural numbers, composition of A122111 & A241909 in either order: a(n) = A122111(A241909(n)) = A241909(A122111(n)).
This permutation preserves both bigomega and the (index of) largest prime factor: for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
From the above it follows, that this fixes both primes (A000040) and powers of two (A000079), among other numbers.
Even positions from n=4 onward contain only terms of A070003, and the odd positions only the terms of A102750, apart from 1 which is at a(1), and 2 which is at a(2).

Antti Karttunen, Table of n, a(n) for n = 1..8192
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

A241912 gives the fixed points; A241913 their complement.
Cf. A006530, A137502, A070003, A102750, A278525.
{A000027, A122111, A241909, A241916} form a 4-group.
The sum of prime indices of a(n) is A243503(n).
Even bisection of A358195 = Heinz numbers of rows of A358172.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A019565, A161511, A246277, A253565, A326844, A356958, A358137.

nn = 65; f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[If[IntegerQ@ #, n/4, g@ Reverse@(# - Join[{1}, ConstantArray[0, Length@ # - 2], {1}] &@ f@ n)] &@ Log2@ n, {n, 4, 4 nn, 4}] (* Michael De Vlieger, Aug 27 2016 *)

A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2); \\ Antti Karttunen, Jul 02 2018

(define (A241916 n) (A122111 (A241909 n)))

a(1)=1, and for n>1, a(n) = A006530(n) * A137502(n)/2.
a(n) = A122111(A241909(n)) = A241909(A122111(n)).
If 2n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=1..k-1} prime(x_k-x_i+1). The opposite version is A000027, even bisection of A246277. - Gus Wiseman, Dec 28 2022

Description clarified by Antti Karttunen, Jul 02 2018

A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 90, 61, 899, 63, 64, 65

Offset: 1

Antti Karttunen, May 31 2014

nonn

This self-inverse permutation (involution) of positive integers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
It also preserves the exponent of the largest prime factor (A071178), from which follows that the sequence A102750 is closed with respect to this permutation, i.e., for all n in A102750, a(n) is either same n or some other term of A102750.
Considered as an operation on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements a self-inverse bijection which is a composition of the effects of A242419 and A225891. (Or equally: A105119 and A242419). For details, please see the respective Comments sections and/or Example section of this entry.

			For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
First we apply A242419, which reverses the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, so that the first stair from the top (one unit wide and three units high) is moved to the last position, the second one (two units wide and two units high) stays in the middle, and the original bottom step (two units wide and one unit high) will be the new topmost step, thus we get the following Young diagram:
   _ _
  |   |_ _
  |       |
  |       |_
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (2,4,4,5,5,5), encoded in A112798 by p_2 * p_4^2 * p_5^3 = 3 * 7^2 * 11^3 = 195657.
Then we apply A225891, which rotates the exponents of distinct primes in the factorization of n one left, in this context the vertical line segments one step up, with the top-one going to the bottomost, and so we get:
   _ _
  |   |
  |   |_ _
  |       |
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,4,4,4,5), encoded in A112798 by p_2^2 * p_4^3 * p_5 = 3^2 * 7^3 * 11 = 33957, thus a(2200) = 33957.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A243068.

Cf. A112798, A242419, A105119, A225891, A006530, A071178, A102750, A243057, A243059.

(define (A242420 n) (* (expt (A006530 n) (- (A071178 n) 1)) (A243057 n)))

a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).
For all k in A102750, a(k) = A243057(k) = A243059(k).
By composing related permutations:
a(n) = A225891(A242419(n)) = A242419(A105119(n)).

A346087 a(n) = min(A071178(n), A329348(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Each term of A345941 is a power of the greatest prime factor of n, a(n) is the exponent of that term: a(n) = A100995(A345941(n)).

Values 0..5 occur for the first time at n = 1, 2, 4, 108, 324, 9375.

Cf. A006530, A100995, A071178, A324886, A329044, A329348, A345941, A346100.

A345941(n) = gcd(n,A329044(n));
A346087(n) = isprimepower(A345941(n)); \\ Rest of program given in A329044.

a(n) = min(A071178(n), A329348(n)).
a(n) = A100995(A345941(n)).
For all n >= 1, A006530(n)^a(n) = A345941(n).

A253783 a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

Original entry on oeis.org

0, 2, 3, 3, 4, 4, 5, 3, 4, 5, 6, 5, 7, 6, 5, 4, 8, 5, 9, 6, 6, 7, 10, 5, 5, 8, 4, 7, 11, 6, 12, 4, 7, 9, 6, 6, 13, 10, 8, 6, 14, 7, 15, 8, 6, 11, 16, 6, 6, 6, 9, 9, 17, 5, 7, 7, 10, 12, 18, 7, 19, 13, 7, 5, 8, 8, 20, 10, 11, 7, 21, 6, 22, 14, 6, 11, 7, 9, 23, 7, 5, 15, 24, 8, 9, 16, 12, 8, 25, 7, 8, 12, 13, 17, 10, 6, 26, 7, 8, 7, 27, 10, 28

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

An auxiliary recurrence for computing A075167.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A051119, A061395, A075167, A071178.

Cf. also A106490.

a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

A370492 Numbers k such that A006530(k) = A071178(k).

Original entry on oeis.org

4, 27, 54, 108, 216, 432, 864, 1728, 3125, 3456, 6250, 6912, 9375, 12500, 13824, 18750, 25000, 27648, 28125, 37500, 50000, 55296, 56250, 75000, 84375, 100000, 110592, 112500, 150000, 168750, 200000, 221184, 225000, 253125, 300000, 337500, 400000, 442368, 450000

Offset: 1

Amiram Eldar, Feb 20 2024

nonn

Union of {4}, numbers of the form 3^3 * 2^k (k >= 0), and the sequence of sets of numbers of the form p^p * k, where k is a p-smooth number (number not divisible by any prime > p), for primes p >= 3.

			54 = 2 * 3^3 is a term since A006530(54) = A071178(54)= 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A071178.
Subsequence of A370493.
Subsequences: A051674, A072694, A076265.

Select[Range[2, 10^6], Equal @@ FactorInteger[#][[-1, ;;]] &]

is(n)={my(f = factor(n), m = #f~); n > 1 && f[m,1] == f[m,2];}

Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(prime(k)^prime(k) * Product_{i=1..k-1} (1 - 1/prime(i))) = 0.32503862758571995411... .

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
G. Back and M. Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
A. E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 4375, 8750, 13125, 26250, 39375, 78750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450

Offset: 0

Antti Karttunen, Aug 21 2016

Prime product form of primorial base expansion of n.
Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019
From Antti Karttunen, Feb 18 2022: (Start)
The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.
Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.
This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)
From Bill McEachen, Oct 15 2022: (Start)
From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.
Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)
The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

			For n = 24, which has primorial base representation (see A049345) "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*6 + 0*2 + 0*1, thus a(24) = prime(3)^4 * prime(2)^0 * prime(1)^0 = 5^4 = 625.
For n = 35 = "1021" as 35 = 1*A002110(3) + 0*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*30 + 0*6 + 2*2 + 1*1, thus a(35) = prime(4)^1 * prime(2)^2 * prime(1) = 7 * 3*3 * 2 = 126.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Program in LODA-assembly
Antti Karttunen, Program in LODA-assembly [Cached copy]
Index entries for sequences related to primorial base

Cf. A276085 (a left inverse) and also A276087, A328403.
Cf. A000040, A001221, A001222, A002110, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156, A283477, A324198 (= gcd(n, a(n))), A328584 (= lcm(n, a(n))), A324646, A324289, A328386, A328403, A328475, A328571, A328572, A328578, A328612, A328613, A328620, A328624, A328627, A328763, A328766, A328828, A328835, A328841, A328842, A328843, A328844, A329041, A324580 [= n*a(n)], A324895 (largest proper divisor of a(n)), A351252, A353486 (reduced modulo 4), A358840 (modulo 6), A353489, A353516.
Cf. A048103 (terms sorted into ascending order), A100716 (natural numbers not present in this sequence).
Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version), A328477.
Cf. A328316 (iterates started from zero).
Cf. A327858, A327859, A327860, A327963, A328097, A328098, A328099, A328110, A328112, A328382 for various combinations with arithmetic derivative (A003415).
Cf. also A327167, A329037.
Cf. A019565 and A054842 for base-2 and base-10 analogs and A276076 for the analogous "factorial base exp-function", from which this differs for the first time at n=24, where a(24)=625 while A276076(24)=7.
Cf. A327969, A351088, A351458 for sequences with conjectures involving this sequence.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 51}] (* Michael De Vlieger, Aug 23 2016, Version 10.2 *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ f@ n], {n, 0, 73}] (* Michael De Vlieger, Aug 30 2016, Pre-Version 10 *)
a[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];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen's Sage code *)

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; }; \\ Antti Karttunen, May 12 2017

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; \\ (Better than above one, avoids unnecessary construction of primorials). - Antti Karttunen, Oct 14 2019

from sympy import prime
def a(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m # Indranil Ghosh, May 12 2017, after Antti Karttunen's PARI code

from sympy import nextprime
def a(n):
    m, p = 1, 2
    while n > 0:
        n, r = divmod(n, p)
        m *= p**r
        p = nextprime(p)
    return m
print([a(n) for n in range(74)])  # Peter Luschny, Apr 20 2024

def A276086(n):
    m=1
    i=1
    while n>0:
        p = sloane.A000040(i)
        m *= (p**(n%p))
        n = floor(n/p)
        i += 1
    return (m)
# Antti Karttunen, Oct 14 2019, after Indranil Ghosh's Python code above, and my own leaner PARI code from Oct 14 2019. This avoids unnecessary construction of primorials.

(define (A276086 n) (let loop ((n n) (t 1) (i 1)) (if (zero? n) t (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (* t (expt p d)) (+ 1 i))))))

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n))))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

a(0) = 1; for n >= 1, a(n) = A053669(n) * a(A276151(n)) = A053669(n) * a(n-A002110(A276084(n))).
a(0) = 1; for n >= 1, a(n) = A053669(n)^A276088(n) * a(A276093(n)).
a(n) = A328841(a(n)) + A328842(a(n)) = A328843(n) + A328844(n).
a(n) = a(A328841(n)) * a(A328842(n)) = A328571(n) * A328572(n).
a(n) = A328475(n) * A328580(n) = A328476(n) + A328580(n).
a(A002110(n)) = A000040(n+1). [Maps primorials to primes]
a(A143293(n)) = A002110(n+1). [Maps partial sums of primorials to primorials]
a(A057588(n)) = A276092(n).
a(A276156(n)) = A019565(n).
a(A283477(n)) = A324289(n).
a(A003415(n)) = A327859(n).
Here the text in brackets shows how the right hand side sequence is a function of the primorial base expansion of n:
A001221(a(n)) = A267263(n). [Number of nonzero digits]
A001222(a(n)) = A276150(n). [Sum of digits]
A067029(a(n)) = A276088(n). [The least significant nonzero digit]
A071178(a(n)) = A276153(n). [The most significant digit]
A061395(a(n)) = A235224(n). [Number of significant digits]
A051903(a(n)) = A328114(n). [Largest digit]
A055396(a(n)) = A257993(n). [Number of trailing zeros + 1]
A257993(a(n)) = A328570(n). [Index of the least significant zero digit]
A079067(a(n)) = A328620(n). [Number of nonleading zeros]
A056169(a(n)) = A328614(n). [Number of 1-digits]
A056170(a(n)) = A328615(n). [Number of digits larger than 1]
A277885(a(n)) = A328828(n). [Index of the least significant digit > 1]
A134193(a(n)) = A329028(n). [The least missing nonzero digit]
A005361(a(n)) = A328581(n). [Product of nonzero digits]
A072411(a(n)) = A328582(n). [LCM of nonzero digits]
A001055(a(n)) = A317836(n). [Number of carry-free partitions of n in primorial base]
Various number theoretical functions applied:
A000005(a(n)) = A324655(n). [Number of divisors of a(n)]
A000203(a(n)) = A324653(n). [Sum of divisors of a(n)]
A000010(a(n)) = A324650(n). [Euler phi applied to a(n)]
A023900(a(n)) = A328583(n). [Dirichlet inverse of Euler phi applied to a(n)]
A069359(a(n)) = A329029(n). [Sum a(n)/p over primes p dividing a(n)]
A003415(a(n)) = A327860(n). [Arithmetic derivative of a(n)]
Other identities:
A276085(a(n)) = n.          [A276085 is a left inverse]
A020639(a(n)) = A053669(n). [The smallest prime not dividing n -> the smallest prime dividing n]
A046523(a(n)) = A278226(n). [Least number with the same prime signature as a(n)]
A246277(a(n)) = A329038(n).
A181819(a(n)) = A328835(n).
A053669(a(n)) = A326810(n), A326810(a(n)) = A328579(n).
A257993(a(n)) = A328570(n), A328570(a(n)) = A328578(n).
A328613(a(n)) = A328763(n), A328620(a(n)) = A328766(n).
A328828(a(n)) = A328829(n).
A053589(a(n)) = A328580(n). [Greatest primorial number which divides a(n)]
A276151(a(n)) = A328476(n). [... and that primorial subtracted from a(n)]
A111701(a(n)) = A328475(n).
A328114(a(n)) = A328389(n). [Greatest digit of primorial base expansion of a(n)]
A328389(a(n)) = A328394(n), A328394(a(n)) = A328398(n).
A235224(a(n)) = A328404(n), A328405(a(n)) = A328406(n).
a(A328625(n)) = A328624(n), a(A328626(n)) = A328627(n). ["Twisted" variants]
a(A108951(n)) = A324886(n).
a(n) mod n = A328386(n).
a(a(n)) = A276087(n), a(a(a(n))) = A328403(n). [2- and 3-fold applications]
a(2n+1) = 2 * a(2n). - Antti Karttunen, Feb 17 2022

Name edited and new link-formulas added by Antti Karttunen, Oct 29 2019

Name changed again by Antti Karttunen, Feb 05 2022

A124010 Triangle in which first row is 0, n-th row (n>1) lists the exponents of distinct prime factors ("ordered prime signature") in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1

Offset: 1

Franklin T. Adams-Watters, Nov 01 2006

A001222(n) = Sum(T(n,k), 1 <= k <= A001221(n)); A005361(n) = Product(T(n,k), 1 <= k <= A001221(n)), n>1; A051903(n) = Max(T(n,k): 1 <= k <= A001221(n)); A051904(n) = Min(T(n,k), 1 <= k <= A001221(n)); A067029(n) = T(n,1); A071178(n) = T(n,A001221(n)); A064372(n)=Sum(A064372(T(n,k)), 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011
Any finite sequence of natural numbers appears as consecutive terms. - Paul Tek, Apr 27 2013
For n > 1: n-th row = n-th row of A067255 without zeros. - Reinhard Zumkeller, Jun 11 2013
Most often the prime signature is given as a sorted representative of the multiset of the nonzero exponents, either in increasing order, which yields A118914, or, most commonly, in decreasing order, which yields A212171. - M. F. Hasler, Oct 12 2018

			Initial values of exponents are:
1, [0]
2, [1]
3, [1]
4, [2]
5, [1]
6, [1, 1]
7, [1]
8, [3]
9, [2]
10, [1, 1]
11, [1]
12, [2, 1]
13, [1]
14, [1, 1]
15, [1, 1]
16, [4]
17, [1]
18, [1, 2]
19, [1]
20, [2, 1]
...

Reinhard Zumkeller, Rows n = 1..10000 of triangle, flattened
Index entries for sequences computed from exponents in factorization of n

Cf. A027748, A001221 (row lengths, n>1), A001222 (row sums), A027746, A020639, A064372, A067029 (first column).

Sorted rows: A118914, A212171.

a124010 n k = a124010_tabf !! (n-1) !! (k-1)
a124010_row 1 = [0]
a124010_row n = f n a000040_list where
   f 1 _      = []
   f u (p:ps) = h u 0 where
     h v e | m == 0 = h v' (e + 1)
           | m /= 0 = if e > 0 then e : f v ps else f v ps
           where (v',m) = divMod v p
a124010_tabf = map a124010_row [1..]
-- Reinhard Zumkeller, Jun 12 2013, Aug 27 2011

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end; # N. J. A. Sloane, Dec 20 2007
PrimeSignature := proc(n) local F, e, k; F := ifactors(n)[2]; [seq(e, e = seq(F[k][2], k = 1..nops(F)))] end:
ListTools:-Flatten([[0], seq(PrimeSignature(n), n = 1..73)]); # Peter Luschny, Jun 15 2025

row[1] = {0}; row[n_] := FactorInteger[n][[All, 2]] // Flatten; Table[row[n], {n, 1, 80}] // Flatten (* Jean-François Alcover, Aug 19 2013 *)

print1(0); for(n=2,50, f=factor(n)[,2]; for(i=1,#f,print1(", "f[i]))) \\ Charles R Greathouse IV, Nov 07 2014

A124010_row(n)=if(n,factor(n)[,2]~,[0]) \\ M. F. Hasler, Oct 12 2018

from sympy import factorint
def a(n):
    f=factorint(n)
    return [0] if n==1 else [f[i] for i in f]
for n in range(1, 21): print(a(n)) # Indranil Ghosh, May 16 2017

n = Product_k A027748(n,k)^a(n,k).

Name edited by M. F. Hasler, Apr 08 2022

A122111 Self-inverse permutation of the positive integers induced by partition enumeration in A112798 and partition conjugation.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 64, 24, 18, 7, 128, 15, 256, 20, 36, 48, 512, 14, 27, 96, 25, 40, 1024, 30, 2048, 11, 72, 192, 54, 21, 4096, 384, 144, 28, 8192, 60, 16384, 80, 50, 768, 32768, 22, 81, 45, 288, 160, 65536, 35, 108, 56, 576, 1536, 131072, 42

Offset: 1

Franklin T. Adams-Watters, Oct 18 2006

Factor n; replace each prime(i) with i, take the conjugate partition, replace parts i with prime(i) and multiply out.
From Antti Karttunen, May 12-19 2014: (Start)
For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).
Because the partition conjugation doesn't change the partition's total sum, this permutation preserves A056239, i.e., A056239(a(n)) = A056239(n) for all n.
(Similarly, for all n, A001221(a(n)) = A001221(n), because the number of steps in the Ferrers/Young-diagram stays invariant under the conjugation. - Note added Apr 29 2022).
Because this permutation commutes with A241909, in other words, as a(A241909(n)) = A241909(a(n)) for all n, from which follows, because both permutations are self-inverse, that a(n) = A241909(a(A241909(n))), it means that this is also induced when partitions are conjugated in the partition enumeration system A241918. (Not only in A112798.)
(End)
From Antti Karttunen, Jul 31 2014: (Start)
Rows in arrays A243060 and A243070 converge towards this sequence, and also, assuming no surprises at the rate of that convergence, this sequence occurs also as the central diagonal of both.
Each even number is mapped to a unique term of A102750 and vice versa.
Conversely, each odd number (larger than 1) is mapped to a unique term of A070003, and vice versa. The permutation pair A243287-A243288 has the same property. This is also used to induce the permutations A244981-A244984.
Taking the odd bisection and dividing out the largest prime factor results in the permutation A243505.
Shares with A245613 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.
Conversely, with A245614 (the inverse of above), shares the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.
(End)
The Maple program follows the steps described in the first comment. The subprogram C yields the conjugate partition of a given partition. - Emeric Deutsch, May 09 2015
The Heinz number of the partition that is conjugate to the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r). Example: a(3) = 4. Indeed, the partition with Heinz number 3 is [2]; its conjugate is [1,1] having Heinz number 4. - Emeric Deutsch, May 19 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1024 terms from Antti Karttunen)
Antti Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

Cf. A088902 (fixed points).
Cf. A112798, A241918 (conjugates the partitions listed in these two tables).
Cf. A243060 and A243070. (Limit of rows in these arrays, and also their central diagonal).
Cf. also A080576, A036036, A001221, A001222, A061395, A243503, A056239, A052126, A003961, A064989, A071178, A241917, A241919, A242378, A242424, A006530, A105560, A070003, A102750, A066829, A028260, A026424, A244990, A244991, A244992, A108951, A181815, A181819, A181820, A238745, A238690, A242421.
Cf. A319988 (parity of this sequence for n > 1), A336124 (a(n) mod 4).
{A000027, A122111, A241909, A241916} form a 4-group.
{A000027, A122111, A153212, A242419} form also a 4-group.
Other related permutations: A048673-A064216, A244981-A244982, A244983-A244984, A243287-A243288, A243505-A243506, A245613-A245614, A075157, A075158, A129594, A069799, A242415, A245451, A245452, A245454, also A336321 & A336322 (composed with A225546).
Cf. also array A350066 [A(i, j) = a(a(i)*a(j))].

with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0; for i to nops(P) do if j <= P[i] then c := c+1 else  end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: seq(c(n), n = 1 .. 59); # Emeric Deutsch, May 09 2015
# second Maple program:
a:= n-> (l-> mul(ithprime(add(`if`(jAlois P. Heinz, Sep 30 2017

A122111[1] = 1; A122111[n_] := Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@Length@l, m = Min@l}, Length@Union@l]] &@Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@n]; Array[A122111, 60] (* JungHwan Min, Aug 22 2016 *)
a[n_] := Function[l, Product[Prime[Sum[If[jJean-François Alcover, Sep 23 2020, after Alois P. Heinz *)

A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m)); \\ Antti Karttunen, Jul 20 2020

from sympy import factorint, prevprime, prime, primefactors
from operator import mul
def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a105560(n): return 1 if n==1 else prime(a001222(n))
def a(n): return 1 if n==1 else a105560(n)*a(a064989(n))
[a(n) for n in range(1, 101)] # Indranil Ghosh, Jun 15 2017

;; Uses Antti Karttunen's IntSeq-library.
(definec (A122111 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A122111 (A064989 n)))))
;; Antti Karttunen, May 12 2014

;; Uses Antti Karttunen's IntSeq-library.
(definec (A122111 n) (if (<= n 1) n (* (A000079 (A241917 n)) (A003961 (A122111 (A052126 n))))))
;; Antti Karttunen, May 12 2014

;; Uses Antti Karttunen's IntSeq-library.
(definec (A122111 n) (if (<= n 1) n (* (expt (A000040 (A071178 n)) (A241919 n)) (A242378bi (A071178 n) (A122111 (A051119 n))))))
;; Antti Karttunen, May 12 2014

From Antti Karttunen, May 12-19 2014: (Start)
a(1) = 1, a(p_i) = 2^i, and for other cases, if n = p_i1 * p_i2 * p_i3 * ... * p_{k-1} * p_k, where p's are primes, not necessarily distinct, sorted into nondescending order so that i1 <= i2 <= i3 <= ... <= i_{k-1} <= ik, then a(n) = 2^(ik-i_{k-1}) * 3^(i_{k-1}-i_{k-2}) * ... * p_{i_{k-1}}^(i2-i1) * p_ik^(i1).
This can be implemented as a recurrence, with base case a(1) = 1,
and then using any of the following three alternative formulas:
  a(n) = A105560(n) * a(A064989(n)) = A000040(A001222(n)) * a(A064989(n)). [Cf. the formula for A242424.]
  a(n) = A000079(A241917(n)) * A003961(a(A052126(n))).
  a(n) = (A000040(A071178(n))^A241919(n)) * A242378(A071178(n), a(A051119(n))). [Here ^ stands for the ordinary exponentiation, and the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n.]
a(n) = 1 + A075157(A129594(A075158(n-1))). [Follows from the commutativity with A241909, please see the comments section.]
(End)
From Antti Karttunen, Jul 31 2014: (Start)
As a composition of related permutations:
a(n) = A153212(A242419(n)) = A242419(A153212(n)).
a(n) = A241909(A241916(n)) = A241916(A241909(n)).
a(n) = A243505(A048673(n)).
a(n) = A064216(A243506(n)).
Other identities. For all n >= 1, the following holds:
A006530(a(n)) = A105560(n). [The latter sequence gives greatest prime factor of the n-th term].
a(2n)/a(n) = A105560(2n)/A105560(n), which is equal to A003961(A105560(n))/A105560(n) when n > 1.
A243505(n) = A052126(a(2n-1)) = A052126(a(4n-2)).
A066829(n) = A244992(a(n)) and vice versa, A244992(n) = A066829(a(n)).
A243503(a(n)) = A243503(n). [Because partition conjugation does not change the partition size.]
A238690(a(n)) = A238690(n). - per Matthew Vandermast's note in that sequence.
A238745(n) = a(A181819(n)) and a(A238745(n)) = A181819(n). - per Matthew Vandermast's note in A238745.
A181815(n) = a(A181820(n)) and a(A181815(n)) = A181820(n). - per Matthew Vandermast's note in A181815.
(End)
a(n) = A181819(A108951(n)). [Prime shadow of the primorial inflation of n] - Antti Karttunen, Apr 29 2022

A067029 Exponent of least prime factor in prime factorization of n, a(1)=0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Feb 17 2002

Even bisection is A001511: a(2n) = A007814(n) + 1. - Ralf Stephan, Jan 31 2004
Number of occurrences of the smallest part in the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(24)=3 because the partition with Heinz number 24 = 3*2*2*2 is [2,1,1,1]. - Emeric Deutsch, Oct 02 2015
Together with A028234 is useful for defining sequences that are multiplicative with a(p^e) = f(e), as recurrences of the form: a(1) = 1 and for n > 1, a(n) = f(A067029(n)) * a(A028234(n)). - Antti Karttunen, May 29 2017

			a(18) = a(2^1 * 3^2) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler, Problem 779: Prime factor and exponent, (2022).

Cf. A051903, A020639, A028233, A034684, A071178, first column of A124010, A247180.

a067029 = head . a124010_row
-- Reinhard Zumkeller, Jul 05 2013, Jun 04 2012

A067029 := proc(n)
    local f,lp,a;
    a := 0 ;
    lp := n+1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        if p < lp then
            a := op(2,f) ;
            lp := p;
        fi;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 08 2015
seq(ifelse(n = 1, 0, ifactors(n)[2][1][2]), n = 1..90); # Peter Luschny, Jun 15 2025

Join[{0},Table[FactorInteger[n][[1,2]],{n,2,100}]] (* Harvey P. Dale, Oct 14 2011 *)

a(n) = if (n==1, 0, factor(n)[1,2]); \\ Michel Marcus, May 15 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)] # Indranil Ghosh, May 15 2017

;; Naive implementation of A020639 is given under that entry. All of these functions could be also defined with definec to make them faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(define (A067029 n) (if (< n 2) 0 (let ((mp (A020639 n))) (let loop ((e 0) (n (/ n mp))) (cond ((integer? n) (loop (+ e 1) (/ n mp))) (else e)))))) ;;  Antti Karttunen, May 29 2017

a(n) = A124010(n,1). - Reinhard Zumkeller, Aug 27 2011
A028233(n) = A020639(n)^a(n). - Reinhard Zumkeller, May 13 2006
a(A247180(n)) = 1. - Reinhard Zumkeller, Nov 23 2014
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (Product_{i=1..k-1} (1 - 1/prime(i)))/(prime(k)-1) = 1/(prime(1)-1) + (1-1/prime(1))*(1/(prime(2)-1) + (1-1/prime(2))*(1/(prime(3)-1) + (1-1/prime(3))*( ... ))) = 1.6125177915... - Amiram Eldar, Oct 26 2021

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A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

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A242420 Self-inverse permutation of positive integers: a(n) = (A006530(n)^(A071178(n)-1)) * A243057(n).

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A346087 a(n) = min(A071178(n), A329348(n)).

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A253783 a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

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A370492 Numbers k such that A006530(k) = A071178(k).

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A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

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