A341885 -id:A341885 - cp's OEIS frontend

A340834 Fixed points of A341885.

12, 1222, 1437286, 3441373, 1032893366969

Offset: 1

J. M. Bergot and Robert Israel, Feb 22 2021

Numbers n such that A341885(n) = n.
Includes 2*p*q if p and q are primes such that p^2-4*p*q+q^2+p+q+6 = 0.  This includes 12 for p=2, q=3, 1222 for p=13,q=47, 1437286 for p=439, q=1637, and 76498942675946443126 for p=3201392659, q=11947760057.
Another term: 6538810199342921107066977217325653068509 = 13 * 4401624135264074597*114272683103433355069. - Chai Wah Wu, Feb 25 2021

			a(2) = 1222 is a term because 1222 = 2*13*47 and A341885(1222) = 2*3/2 + 13*14/2 + 47*48/2 = 1222.

Cf. A341885.

f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[1]*(t[1]+1)/2*t[2],t=F)
end proc:
select(t -> f(t)=t, [$1..4000000]);

Block[{a = {}}, Monitor[Do[If[# == i, AppendTo[a, i]] &@ Total[PolygonalNumber@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[i]]], {i, 2, 4*10^6}], i]; a] (* Michael De Vlieger, Feb 22 2021 *)

from sympy import factorint
A340834_list = [n for n in range(2,10**4) if n == sum(k*m*(m+1)//2 for m,k in factorint(n).items())] # Chai Wah Wu, Feb 25 2021

A341885(a(n)) = a(n).

a(5) from Martin Ehrenstein, Mar 07 2021

A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12, 18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18

Offset: 1

N. J. A. Sloane

MacMahon calls this the potency of n.
Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
Consider all ways of writing n as a product of zero, one, or more factors; sequence gives smallest sum of terms. - Amarnath Murthy, Jul 07 2001
a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
Look at the graph of this sequence. At the lower edge of the logarithmic scatterplot there is a set of fuzzy but unmistakable diagonal fringes, sloping toward the southeast. Their spacing gradually increases, and their slopes gradually decrease; they are more distinct toward the lower edge of the range. Is any explanation known? - Allan C. Wechsler, Oct 11 2015
For n >= 2, the glb and lub are: 3 * log(n) / log(3) <= a(n) <= n, where the lub occurs when n = 3^k, k >= 1. (Jakimczuk 2012) - Daniel Forgues, Oct 12 2015
Except for the initial term, row sums of A027746. - M. F. Hasler, Feb 08 2016
Atanassov proves that a(n) <= A065387(n) - n. - Charles R Greathouse IV, Dec 06 2016
From Robert G. Wilson v, Aug 15 2022: (Start)
Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., .
The number of terms which equal k is A000607(k).
The first occurrence of k>1 is A056240(k).
The last occurrence of k>1 is A000792(k).
The Amarnath Murthy comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic.
(End)

			a(24) = 2+2+2+3 = 9.
a(30) = 10: 30 can be written as 30, 15*2, 10*3, 6*5, 5*3*2. The corresponding sums are 30, 17, 13, 11, 10. Among these 10 is the least.

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 Franklin T. Adams-Watters)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Kevin S. Brown, The Sum of the Prime Factors of N.
Es-said En-naoui, Study of the generalized Von Mangoldt function defined by L-additive function, arXiv:2301.09677 [math.GM], 2023.
Hans Havermann, Log plot of 100000 terms
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Rafael Jakimczuk, Sum of Prime Factors in the Prime Factorization of an Integer, International Mathematical Forum, Vol. 7, No. 53 (2012), pp. 2617-2621.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.
Wikipedia, Table of prime factors.
Steve Witham, Linear-log plot (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).)
Steve Witham, Log-log plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)

Cf. A008472 (sopf(n)), A002217, A056240, A000792, A046343, A120007, A036288.
A000607(n) gives the number of values of k for which A001414(k) = n.
Cf. A036349 (indices of even terms), A356163 (their char. function), A335657 (indices of odd terms), A289142 (of multiples of 3), A373371 (their char. function).
For sum of squares of prime factors see A067666, for cubes see A224787.
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A056239 (with a(p)=primepi(p)), A059975 (with a(p)=p-1), A064097 (with a(p)=1+a(p-1)), A113177 (with a(p)=Fib(p)), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
For other completely additive sequences see the cross-references in A104244.

a001414 1 = 0
a001414 n = sum $ a027746_row n
-- Reinhard Zumkeller, Feb 27 2012, Nov 20 2011

[n eq 1 select 0 else (&+[j[1]*j[2]: j in Factorization(n)]): n in [1..100]]; // G. C. Greubel, Jan 10 2019

A001414 := proc(n) add( op(1,i)*op(2,i),i=ifactors(n)[2]) ; end proc:
seq(A001414(n), n=1..100); # Peter Luschny, Jan 17 2011

a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)

a(n)=local(f); if(n<1,0,f=factor(n); sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))

A001414(n) = (n=factor(n))[,1]~*n[,2] \\ M. F. Hasler, Feb 07 2009

from sympy import factorint
def A001414(n):
    return sum(p*e for p,e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016

[sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0,len(factor(n)))) for n in range(1,79)] # Giuseppe Coppoletta, Jan 19 2015

If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
Dirichlet g.f. f(s)*zeta(s), where f(s) = Sum_{p prime} p/(p^s-1) = Sum_{k>0} primezeta(k*s-1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e. - Franklin T. Adams-Watters, Jun 02 2006
For n > 1: a(n) = Sum_{k=1..A001222(n)} A027746(n,k). - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} (-1)^a(n)/n^s = ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s), if Re(s)>1 and 0 if s=1 (Alladi and Erdős, 1977). - Amiram Eldar, Nov 02 2020
a(n) >= k*log(n), where k = 3/log(3). This bound is sharp. - Charles R Greathouse IV, Jul 28 2025

A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset: 1

Antti Karttunen, Aug 21 2016

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)-1).
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

A left inverse of A276086.
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
Other identities.
For all n >= 0:
a(A276086(n)) = n.
a(A000040(1+n)) = A002110(n).
a(A002110(1+n)) = A143293(n).
From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)
a(A283477(n)) = A283985(n).
a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:
a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:
a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
The sum or difference of the rhs-sequences is A108951:
a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
Here the two sequences are inverse permutations of each other:
a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
a(A346101(n)) = A289234(n). [Self-inverse]
Other correspondences:
a(A324350(x,y)) = A324351(x,y).
a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
(End)

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

A059975 For n > 1, a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors; fully additive with a(p) = p-1.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41

Offset: 1

Yong Kong (ykong(AT)curagen.com), Mar 05 2001

nonn

n*a(n) is the number of complex multiplications needed for the fast Fourier transform of n numbers, writing n = r1 * r2 where r1 is a prime.

This sequence with offset 1 and a(1) = 0 is completely additive with a(p^e) = e*(p-1) for prime p and e >= 0. - Werner Schulte, Feb 23 2019

			a(18) = 5 since 18 = 2*3^2, a(18) = 1*(2-1) + 2*(3-1) = 5.

Herbert S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 2..1000 from Vincenzo Librandi, terms 1001..10000 from Amiram Eldar)
K. V. Lever, Problem 89-11: The complexity of the standard form of an integer, SIAM Rev. 31 (3) (1989) 493-498.
Herbert S. Wilf, Algorithms and complexity, Internet Edition, 1994, p. 56.

Essentially same as A087656 apart from offset.
Cf. A000005, A138618, A309155, A309239, A327250, A341865, A373368 [= gcd(n, a(n))], A373369 [= gcd(A001414(n), a(n))].
Cf. A003159 (positions of even terms), A096268 (with offset 1, parity of terms), A373385 (positions of multiples of 3).
Leftmost column of irregular table A355029.
Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).

A059975 := proc(n)
        local a,pf,p,e ;
        a := 0 ;
        for pf in ifactors(n)[2] do
                p := op(1,pf) ;
                e := op(2,pf) ;
                a := a+e*(p-1) ;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 17 2011

Table[Total[(First /@ FactorInteger[n] - 1) Last /@ FactorInteger[n]], {n, 1, 100}] (* Danny Marmer, Nov 13 2014 *)
f[p_, e_] := e*(p - 1); a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 27 2023 *)

a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1));} \\ Amiram Eldar, Mar 27 2023

a(n) = Sum ( e_i * (p_i - 1) ) where n = Product ( p_i^e_i ) is the canonical factorization of n.
a(n) = min(A001222(x) : A000005(x)=n).
a(n) = row sums of A138618 - row products of A138618. - Mats Granvik, May 23 2013
a(n) = A001414(n) - A001222(n). - David James Sycamore, Jul 17 2019
a(n) = n - A341865(n). - Antti Karttunen, Jun 05 2024

Definition revised by Hugo van der Sanden, May 21 2010

Term a(1)=0 prepended and Werner Schulte's comment adopted as an alternative definition - Antti Karttunen, Jun 05 2024

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A340834 Fixed points of A341885.

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A001414 Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).

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A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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A059975 For n > 1, a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors; fully additive with a(p) = p-1.

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