A127668 -id:A127668 - cp's OEIS frontend

A127669 Number of numbers mapped to A127668(n) with the map described there.

1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 3, 2, 7, 2, 2, 2, 5, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 3, 3, 11, 2, 3, 1, 3, 2, 3, 1, 7, 2, 2, 3, 3, 2, 3, 2, 7, 5, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 3, 2, 2, 2, 11, 1, 3, 3, 5

Offset: 2

Wolfdieter Lang Jan 23 2007

This is not A008481(n), n>=2, which starts similarly, but differs, beginning with n=24.

			a(4)=2 because two numbers are mapped to 11= A127668(4), namely n=p(1)*p(1)=4 and n=p(11)=31. p(n)=A000041(n) (partition numbers).
a(24)=5 but A008481(24)=4.
The five numbers mapped to A127668(24)= 2111 are: 18433, 2594, 2263, 292, 24.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A008481, A127668.

f:= proc(n) local S;
   nops(g(sprintf("%d",n)))
end proc:
g:= proc(s) option remember;
    local S,m,k1;
    if s[1] = "0" then return {} fi;
    S:= {[parse(s)]};
    for m from 1 to length(s)-1 do
      k1:= parse(s[1..m]);
      S:= S union map(t -> [k1,op(t)], select(r -> r[1] <= k1, procname(s[m+1..-1])));
    od;
    S;
end proc:
h:= proc(n) local F;
  F:= map(t -> numtheory:-pi(t[1])$t[2], sort(ifactors(n)[2],(a,b) -> a[1] > b[1]));
  parse(cat(op(F)))
end proc:
seq(f(h(i)),i=2..100); # Robert Israel, Dec 08 2024

a(n) <= pa(Length( A127668(n))), n>=2. Length gives the number of digits and pa(k):=A000041(k) (partition numbers). (It was originally claimed that this is equality, but that is not correct. - Franklin T. Adams-Watters, May 21 2014)

Edited by Franklin T. Adams-Watters, May 21 2014

Corrected by Robert Israel, Dec 08 2024

A056239 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} k*c_k.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 8, 6, 13, 7, 14, 7, 7, 10, 15, 6, 8, 7, 9, 8, 16, 7, 8, 7, 10, 11, 17, 7, 18, 12, 8, 6, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 7, 8, 14, 23, 8, 10, 15, 12, 8, 24, 8, 10

Offset: 1

Leroy Quet, Aug 19 2000

A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. [Cf. A248692 for example.] Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2, etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)-th prime and last with the a(n)-th power of 2. Produces triangular numbers from primorials. - Henry Bottomley, Nov 22 2001
Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.
All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012
a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

			a(12) = 1*2 + 2*1 = 4, since 12 = 2^2 *3^1 = (p_1)^2 *(p_2)^1.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Row sums of A112798.
Cf. A003963 (gives the corresponding products).
Cf. also A000041, A000217, A000720, A001221, A001222, A002110, A005940, A008475, A027748, A049084, A060437, A081401, A082090, A088314, A088318, A088850, A088880, A088881, A088887, A088902, A108951, A122111, A124010, A127668, A141128, A153734, A154351, A156552, A163517, A178503, A215366, A215369, A215501, A242422, A242423, A242424, A243070, A243353, A243354, A243503, A248692, A249336, A249337, A329605, A329902.

a056239 n = sum $ zipWith (*) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013

# To get 10000 terms. First make prime table: M:=10000; pl:=array(1..M); for i from 1 to M do pl[i]:=0; od: for i from 1 to M do if ithprime(i) > M then break; fi; pl[ithprime(i)]:=i; od:
# Decode Maple's amazing syntax for factoring integers: g:=proc(n) local e,p,t1,t2,t3,i,j,k; global pl; t1:=ifactor(n); t2:=nops(t1); if t2 = 2 and whattype(t1) <> `*` then p:=op(1,op(1,t1)); e:=op(2,t1); t3:=pl[p]*e; else
t3:=0; for i from 1 to t2 do j:=op(i,t1); if nops(j) = 1 then e:=1; p:=op(1,j); else e:=op(2,j); p:=op(1,op(1,j)); fi; t3:=t3+pl[p]*e; od: fi; t3; end; # N. J. A. Sloane, May 10 2006
A056239 := proc(n) add( numtheory[pi](op(1,p))*op(2,p), p = ifactors(n)[2]) ; end proc: # R. J. Mathar, Apr 20 2010
# alternative:
with(numtheory): a := proc (n) local B: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: add(B(n)[i], i = 1 .. nops(B(n))) end proc: seq(a(n), n = 1 .. 130); # Emeric Deutsch, May 19 2015

a[1] = 0; a[2] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p];
a[n_] := a[n] = Total[#[[2]]*a[#[[1]]] & /@ FactorInteger[n]]; a /@ Range[91] (* Jean-François Alcover, May 19 2011 *)
Table[Total[FactorInteger[n] /. {p_, c_} /; p > 0 :> PrimePi[p] c], {n, 91}] (* Michael De Vlieger, Jul 12 2017 *)

A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); \\ Antti Karttunen, Oct 26 2014, edited Jan 13 2020

from sympy import primepi, factorint
def A056239(n): return sum(primepi(p)*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 01 2023

(require 'factor) ;; Uses the function factor available in Aubrey Jaffer's SLIB Scheme library.
(define (A056239 n) (apply + (map A049084 (factor n))))
;; Antti Karttunen, Oct 26 2014

Totally additive with a(p) = PrimePi(p), where PrimePi(n) = A000720(n).
a(n) = Sum_{k=1..A001221(n)} A049084(A027748(k))*A124010(k). - Reinhard Zumkeller, Apr 27 2013
From Antti Karttunen, Oct 11 2014: (Start)
a(n) = n - A178503(n).
a(n) = A161511(A156552(n)).
a(n) = A227183(A243354(n)).
For all n >= 0:
a(A002110(n)) = A000217(n). [Cf. Henry Bottomley's comment above.]
a(A005940(n+1)) = A161511(n).
a(A243353(n)) = A227183(n).
Also, for all n >= 1:
a(A241909(n)) = A243503(n).
a(A122111(n)) = a(n).
a(A242424(n)) = a(n).
A248692(n) = 2^a(n). (End)
a(n) < A329605(n), a(n) = A001222(A108951(n)), a(A329902(n)) = A112778(n). - Antti Karttunen, Jan 14 2020

A329025 If n = Product (p_j^k_j) then a(n) = concatenation (pi(p_j)), where pi = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 12, 4, 1, 2, 13, 5, 12, 6, 14, 23, 1, 7, 12, 8, 13, 24, 15, 9, 12, 3, 16, 2, 14, 10, 123, 11, 1, 25, 17, 34, 12, 12, 18, 26, 13, 13, 124, 14, 15, 23, 19, 15, 12, 4, 13, 27, 16, 16, 12, 35, 14, 28, 110, 17, 123, 18, 111, 24, 1, 36, 125, 19, 17, 29, 134

Offset: 1

Ilya Gutkovskiy, Nov 02 2019

Concatenate of indices of distinct prime factors of n, in increasing order.

			a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 123.

Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A037916, A080695, A084317, A127668, A304038, A329026.

a[n_] := FromDigits[Flatten@IntegerDigits@(PrimePi[#[[1]]] & /@ FactorInteger[n])]; Table[a[n], {n, 1, 70}]

a(prime(n)^k) = n for k > 0.

cp's OEIS Frontend

A127669 Number of numbers mapped to A127668(n) with the map described there.

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A056239 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} k*c_k.

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A329025 If n = Product (p_j^k_j) then a(n) = concatenation (pi(p_j)), where pi = A000720.

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