A068346 -id:A068346 - cp's OEIS frontend

A368702 Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

14, 186, 258, 322, 338, 3318, 3962, 5334, 6106, 7674, 8970, 9186, 9978, 10930, 11994, 12154, 12614, 12970, 13218, 13234, 14626, 15226, 15914, 16378, 17122, 18226, 18658, 19058, 19874, 20194, 20962, 21082, 21106, 21218, 44718, 49358, 57346, 58354, 75442, 76162, 81802, 87814, 95114, 102794, 113505, 114918, 130802

Offset: 1

Antti Karttunen, Jan 09 2024

nonn

Numbers k for which A003415(k) is one of the terms of A327978.

Cf. A002110, A003415, A068346, A327978.

Subsequence of A046100 and of A328244.

A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A068346(n) = A003415(A003415(n));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA368702flat(n) = { my(u=A068346(n)); ((u>1)&&(1==A276150(u))); };
k=0; for(n=1, A002620(A002110(7)), if(isA368702flat(n), k++; write("b368702.txt", k, " ", n)));

A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, 0, 1, 5, 10, 0, 1, 12, 1, 16, 44, 7, 39, 16, 42, 608, 391, 55, 135, 365, 455, 1120, 2990, 800, 1100, 1400, 5425, 12575, 21025, 0, 6, 7, 1, 20, 103, 16, 1, 1, 32, 271, 320, 24, 78, 572, 459, 1031, 2887, 635, 1670, 1155, 3335, 19540, 22130, 4225, 7700, 18675, 28100, 68900, 155425, 9, 18, 20, 54, 704, 631, 24, 251

Offset: 0

Antti Karttunen, Feb 10 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Index entries for sequences related to primorial base

Cf. A003415, A068346, A276086, A327860, A370130.

Cf. A002110 (positions of 0's after the initial zero), A328233 (positions of 1's), A328240 (positions of primes), A369651 (= a(A143293(n-1)), for n >= 1).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A370131(n) = A003415(A327860(n));

a(n) = A068346(A276086(n)) = A003415(A327860(n)).

A348304 a(n) = Sum_{d|n} d'', where d'' is the second arithmetic derivative of d (A068346).

Original entry on oeis.org

0, 0, 0, 4, 0, 1, 0, 20, 5, 1, 0, 37, 0, 6, 12, 100, 0, 16, 0, 49, 7, 1, 0, 101, 7, 8, 32, 90, 0, 15, 0, 276, 9, 1, 16, 144, 0, 10, 32, 137, 0, 15, 0, 117, 33, 10, 0, 421, 9, 47, 24, 104, 0, 151, 32, 202, 13, 1, 0, 191, 0, 14, 32, 916, 21, 12, 0, 161, 15, 24, 0, 428, 0, 16, 35

Offset: 1

Wesley Ivan Hurt, Oct 10 2021

Sum of the 2nd arithmetic derivatives of the divisors of n.

			a(8) = 20; a(8) = 1'' + 2'' + 4'' + 8'' = 0 + 0 + 4 + 16 = 20.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Inverse Möbius transform of A068346 (2nd arithmetic derivative).

Cf. A319684, A348279.

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(ad(d))); \\ Michel Marcus, Oct 11 2021

a(p) = 0 for primes p, since we have 1'' + p'' = 0' + 1' = 0 + 0 = 0.

a(n) = Sum_{d|n} A068346(d). - Antti Karttunen, Dec 07 2021

A348426 Numbers k for which sigma(k) = k + k'', where k'' is the second derivative of k (A068346).

Original entry on oeis.org

1, 161, 209, 221, 4265, 12690, 15941, 22217, 24041, 25637, 30377, 38117, 39077, 48617, 49097, 55877, 68441, 73817, 76457, 80357, 88457, 95237, 98117, 99941, 105641, 110057, 115397, 122537, 130217, 131141, 136517, 143237, 147941, 148697, 152357, 154457, 159077

Offset: 1

Marius A. Burtea, Oct 18 2021

nonn

If p and q are different prime numbers and p + q is in A007850 (Giuga numbers) then m = p*q is a term because sigma(m) = sigma(p*q) = p*q + p + q + 1 and m + m'' = p*q + (p + q)' = p*q + p + q + 1 and sigma(m) = m + m''.

			sigma(1) = 1 and 1 + 1'' = 1 so 1 is a term.
sigma(161) = 1 + 7 + 23 + 161 = 192 and 161 + 161'' = 161 + 30' = 161 + 31 = 192 so 161 is a term.
sigma(12690) = sigma(2*3^3*5*47) = 34560 and 12690 + 12690'' = 12690 + A068346(12690) = 12690 + 21870 = 34560 so 12690 is a term.

Cf. A000203, A003415, A068346, A007850.

f:=func; [n:n in [1..160000]| DivisorSigma(1,n) eq n+Floor(f(Floor(f(n))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[160000], DivisorSigma[1, #] == # + d[d[#]] &] (* Amiram Eldar, Oct 18 2021 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = sigma(k) == k+ad(ad(k)); \\ Michel Marcus, Oct 18 2021

A351131 Triangular numbers (A000217) whose second arithmetic derivative (A068346) is also a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 10, 66, 78, 105, 231, 325, 465, 561, 595, 861, 1378, 2211, 2278, 2485, 3081, 3570, 3655, 4278, 4465, 5253, 6441, 6670, 8515, 8778, 9453, 9870, 10011, 10153, 12561, 13530, 15051, 18145, 21115, 21945, 22578, 23005, 25878, 27966, 28441, 40470, 45753

Offset: 1

Marius A. Burtea, Feb 07 2022

nonn

			6 = A000217(3), 6'' = 5' = 1 = A000217(1), so 6 is a term.
66 = A000217(11), 66'' = 61' = 1 = A000217(1), so 66 is a term.
325 = A000217(25), 325'' = 155' = 36 = A000217(8), so 325 is a term.

Cf. A000217, A003415, A068346.

tr:=func;  f:=func; [n:n in [d*(d+1) div 2:d in [0..310]]| tr(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Table[n*(n + 1)/2, {n, 0, 300}], IntegerQ[Sqrt[8*d[d[#]] + 1]] &] (* Amiram Eldar, Feb 07 2022 *)

der(n) = my(f=factor(n)); vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = ispolygonal(m, 3) && ispolygonal(der(der(m)), 3); \\ Michel Marcus, Feb 16 2022

from itertools import count, islice
from sympy import factorint, integer_nthroot, isprime, nextprime
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def ad(n):
    return 0 if n < 2 else sum(n*e//p for p, e in factorint(n).items())
def agen(): # generator of terms
    for i in count(0):
        t = i*(i+1)//2
        if istri(ad(ad(t))):
            yield t
print(list(islice(agen(), 45))) # Michael S. Branicky, Feb 16 2022

A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

Original entry on oeis.org

4, 27, 104, 260, 296, 405, 525, 740, 910, 945, 1460, 1806, 1818, 2504, 3125, 3140, 3176, 3656, 3860, 4563, 5540, 6056, 6930, 7016, 8420, 8636, 9224, 10820, 12573, 13256, 14024, 15140, 15464, 15944, 16136, 19940, 20456, 21690, 21860, 22856, 23336, 24020, 24260

Offset: 1

Marius A. Burtea, Apr 09 2022

nonn

If m is a term in A051674, then m'' = m, phi(m'') = phi(m) so the sequence is infinite.
If p > 3 is at the intersection of A023208 and A005383 then m = 8*p is a term. Indeed, m'' = (8*p)'' = (12*p + 8)' = (4*(3*p + 2))' = 12*(p + 1) and phi(m'') = phi(12*(p + 1)) = phi(24*(p + 1)/2) = 8*(p - 1)/2 = 4*(p - 1) and phi(m) = phi(8*p) = 4*(p - 1).
If p > 5 is at the intersection of A023221 and A005383 then m = 20*p is a term. Indeed, m'' = (20*p)'' = (24*p + 20)' = (4*(6*p + 5))' = 4*(6*p + 6)  = 24*(p + 1) and phi(m'') = phi(24*(p + 1)) = phi(48*(p + 1)/2) = 16*(p - 1)/2 = 8*(p - 1) and phi(m) = phi(20*p) = 8*(p - 1).

			phi(4'') = phi(4) because 4'' = 4, so 4 is a term.
phi (27'') = phi(27) because 27'' = 27, so 27 is a term.
phi(104'') = phi(164') = phi(168) = phi (8*3*7) = 4*2*6 = 48 and phi(104) = phi(8*13) = 4*12 = 48, so 104 is a term.

Cf. A000010, A005383, A051674, A068346, A023208, A023221, A190402,

f:=func; [n:n in [2..24300]| not IsPrime(n) and EulerPhi(n) eq EulerPhi(Floor(f(Floor(f(n))))) ];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[25000], EulerPhi[#] == EulerPhi[d[d[#]]] &] (* Amiram Eldar, Apr 10 2022 *)

A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).

Original entry on oeis.org

4, 260, 294, 740, 1460, 3140, 3860, 5540, 8420, 10820, 15140, 19940, 21860, 24020, 24260, 27620, 37460, 40340, 46820, 49460, 55940, 61220, 70340, 85460, 101540, 114020, 124340, 132740, 133220, 144260, 148340, 149540, 155060, 162020, 164420, 167060, 170420, 173540

Offset: 1

Marius A. Burtea, Apr 09 2022

nonn

If p > 3 is at the intersection of A023221 and A005383, then m = 20*p is a term. Indeed, m' = (20*p)' = 24*p + 20 = 4*(6*p + 5), m'' = (4*(6*p + 5))' = 4*(6*p + 6) = 24*(p + 1), phi(m') + phi(m'') = phi(4*(6*p + 5)) + phi(24*(p + 1)) = 2*(6*p + 4) + phi(48*(p + 1)/2) = 2*(6*p + 4) + 16*(p - 1)/2 = 12*p + 8 + 8*p - 8 = 20*p = m.

			phi(4') + phi(4'') = phi(4) + phi(4) = 2 + 2 = 4, so 4 is a term.
phi(260') + phi(260'') = phi(332) + phi(336) = 164 + 96 = 260, so 260 is a term.

Cf. A000010, A023221, A003415, A005383, A068346.

f:=func; [n:n in [2..174000]|not IsPrime(n) and n-EulerPhi(Floor(f(n))) eq EulerPhi(Floor(f(Floor(f(n)))))];

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200000], CompositeQ[#] && EulerPhi[d[#]] + EulerPhi[d[d[#]]] == # &] (* Amiram Eldar, Apr 10 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(adk=ad(k)); !isprime(k) && (k == eulerphi(adk) + eulerphi(ad(adk))); \\ Michel Marcus, Apr 30 2022

A361078 Numbers k for which k = gcd(k', k"), where k' is the arithmetic derivative of k (A003415) and k" is the second derivative of k (A068346).

Original entry on oeis.org

4, 16, 27, 64, 108, 432, 729, 1024, 2916, 3125, 4096, 6912, 12500, 16384, 19683, 27648, 46656, 50000, 84375, 110592, 186624, 314928, 337500, 746496, 800000, 823543, 1048576, 1259712, 2125764, 2278125, 3200000, 3294172, 4194304, 5038848, 5400000, 7077888, 8503056

Offset: 1

Marius A. Burtea, Mar 01 2023

nonn

The sequence is infinite because for p prime, m = p^p (A051674) is a term.
For the prime number p, the number m = 4^p is a term. Indeed: (4^p)' = p*4^p, (4^p)" = (1 + p^2)*4^p and gcd((4^p)', 4^p) = gcd(p*4^p, (1 + p^2)*4^p) = 4^p*gcd(p, 1 + p^2) = 4^p.
Numbers of the form a*b with a, b in A051674 are terms. Indeed, if m = a*b then m' = a'*b + a*b' = a*a + b*b = 2*a*b = 2*m, m" = a*b + 2*a'b + 2*a*b' = a*b + 2*a*b + 2*a*b = 5*a*b = 5*m and gcd(m', m") = (2*m, 5*m) = m.

			4' = 4, 4" = 4 and gcd(4', 4") = gcd(4, 4) = 4, so 4 is a term.
16' = 32, 16" = 32' = 80 and gcd(16', 16") = gcd(32, 80) = 16, so 16 is a term.

Cf. A003415 (n'), A068346 (n"), A051674, A072873, A348284.

f:=func; [n:n in [2..100000]|not IsPrime(n) and  Gcd(Floor(f(n)),Floor(f(Floor(f(n))))) eq n];

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[10^6], GCD[d[#], d[d[#]]] == # &] (* Amiram Eldar, Mar 03 2023 *)

ader(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = gcd(ader(k), ader(ader(k))) == k; \\ Michel Marcus, Mar 03 2023

A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

N. J. A. Sloane

The term "prime power" is ambiguous. To a mathematician it means any number p^k, p prime, k >= 0, including p^0 = 1.
Any nonzero integer is a product of primes and units, where the units are +1 and -1. This is tied to the Fundamental Theorem of Arithmetic which proves that the factorizations are unique up to order and units. (So, since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power. See A246655 for the sequence without 1.)
These numbers are (apart from 1) the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004
Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ..., p^k. - Amarnath Murthy, Jan 09 2002
These are also precisely the orders of those finite affine planes that are known to exist as of today. (The order of a finite affine plane is the number of points in an arbitrarily chosen line of that plane. This number is unique for all lines comprise the same number of points.) - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006
Except for first term, the index of the second number divisible by n in A002378, if the index equals n. - Mats Granvik, Nov 18 2007
These are precisely the numbers such that lcm(1,...,m-1) < lcm(1,...,m) (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). We have a(n+1)=a(n)+1 if a(n) is a Mersenne prime or a(n)+1 is a Fermat prime; the converse is true except for n=7 (from Catalan's conjecture) and n=1, since 2^1-1 and 2^0+1 are not considered as Mersenne resp. Fermat prime. - M. F. Hasler, Jan 18 2007, Apr 18 2010
The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov, Feb 06 2008
Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500. - M. F. Hasler, Apr 04 2008
Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly the indices for which Phi[k](-1) is prime. - M. F. Hasler, Apr 04 2008
A143201(a(n)) = 1. - Reinhard Zumkeller, Aug 12 2008
Number of distinct primes dividing n=omega(n) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
Numbers n such that Sum_{p-1|p is prime and divisor of n} = Product_{p-1|p is prime and divisor of n}. A055631(n) = A173557(n-1). - Juri-Stepan Gerasimov, Dec 09 2009, Mar 10 2010
Numbers n such that A028236(n) = 1. Klaus Brockhaus, Nov 06 2010
A188666(k) = a(k+1) for k: 2*a(k) <= k < 2*a(k+1), k > 0; notably a(n+1) = A188666(2*a(n)). - Reinhard Zumkeller, Apr 25 2011
A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n) + A192015(n). - Reinhard Zumkeller, Jun 26 2011
A089233(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2013
The positive integers n such that every element of the symmetric group S_n which has order n is an n-cycle. - W. Edwin Clark, Aug 05 2014
Conjecture: these are numbers m such that Sum_{k=0..m-1} k^phi(m) == phi(m) (mod m), where phi(m) = A000010(m). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
Numbers whose (increasingly ordered) divisors are alternatingly squares and nonsquares. - Michel Marcus, Jan 16 2019
Possible numbers of elements in a finite vector space. - Jianing Song, Apr 22 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. 870.
M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993.
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
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].
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018).
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Projective Plane
Index entries for "core" sequences

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018
Cf. A008480, A010055, A065515, A095874, A025473.
Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.
A138929(n) = 2*a(n).
Cf. A000040, A001221, A001477. - Juri-Stepan Gerasimov, Dec 09 2009
A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). - Klaus Brockhaus, Nov 06 2010
A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.
Complementary (in the positive integers) to sequence A024619. - Jason Kimberley, Nov 10 2015

import Data.Set (singleton, deleteFindMin, insert)
a000961 n = a000961_list !! (n-1)
a000961_list = 1 : g (singleton 2) (tail a000040_list) where
g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012, Apr 25 2011

[1] cat [ n : n in [2..250] | IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d,`,n) fi: od:
# second Maple program:
a:= proc(n) option remember; local k; for k from
      1+a(n-1) while nops(ifactors(k)[2])>1 do od; k
    end: a(1):=1: A000961:= a:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2013

Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]
max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; a (* Artur Jasinski *)
Join[{1}, Select[Range[2, 250], PrimePowerQ]] (* Jean-François Alcover, Jul 07 2015 *)

A000961(n,l=-1,k=0)=until(n--<1,until(lA000961(lim=999,l=-1)=for(k=1,lim, l==lcm(l,k) && next; l=lcm(l,k); print1(k,",")) \\ M. F. Hasler, Jan 18 2007

isA000961(n) = (omega(n) == 1 || n == 1) \\ Michael B. Porter, Sep 23 2009

nextA000961(n)=my(m,r,p);m=2*n;for(e=1,ceil(log(n+0.01)/log(2)),r=(n+0.01)^(1/e);p=prime(primepi(r)+1);m=min(m,p^e));m \\ Michael B. Porter, Nov 02 2009

is(n)=isprimepower(n) || n==1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=primes(primepi(lim)),u=List([1])); forprime(p=2,sqrtint(lim\1),for(e=2,log(lim+.5)\log(p),listput(u,p^e))); vecsort(concat(v,Vec(u))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import primerange
def A000961_list(limit): # following Python style, list terms < limit
    L = [1]
    for p in primerange(1, limit):
        pe = p
        while pe < limit:
            L.append(pe)
            pe *= p
    return sorted(L) # Chai Wah Wu, Sep 08 2014, edited by M. F. Hasler, Jun 16 2022

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A000961(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

def A000961_list(n):
    R = [1]
    for i in (2..n):
        if i.is_prime_power(): R.append(i)
    return R
A000961_list(227) # Peter Luschny, Feb 07 2012

a(n) = A025473(n)^A025474(n). - David Wasserman, Feb 16 2006
a(n) = A117331(A117333(n)). - Reinhard Zumkeller, Mar 08 2006
Panaitopol (2001) gives many properties, inequalities and asymptotics, including a(n) ~ prime(n). - N. J. A. Sloane, Oct 31 2014, corrected by M. F. Hasler, Jun 12 2023 [The reference gives pi*(x) = pi(x) + pi(sqrt(x)) + ... where pi*(x) counts the terms up to x, so it is the inverse function to a(n).]
m=a(n) for some n <=> lcm(1,...,m-1) < lcm(1,...,m), where lcm(1...0):=0 as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) for some k (by Catalan's conjecture), except for n=1 and n=7. - M. F. Hasler, Jan 18 2007, Apr 18 2010
A001221(a(n)) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
A008480(a(n)) = 1 for all n >= 1. - Alois P. Heinz, May 26 2018
Sum_{k=1..n} 1/a(k) ~ log(log(a(n))) + 1 + A077761 + A136141. - François Huppé, Jul 31 2024

Description modified by Ralf Stephan, Aug 29 2014

A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 60, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71

Offset: 0

N. J. A. Sloane, R. K. Guy

Can be extended to negative numbers by defining a(-n) = -a(n).
Based on the product rule for differentiation of functions: for functions f(x) and g(x), (fg)' = f'g + fg'. So with numbers, (ab)' = a'b + ab'. This implies 1' = 0. - Kerry Mitchell, Mar 18 2004
The derivative of a number x with respect to a prime number p as being the number "dx/dp" = (x-x^p)/p, which is an integer due to Fermat's little theorem. - Alexandru Buium, Mar 18 2004
The relation (ab)' = a'b + ab' implies 1' = 0, but it does not imply p' = 1 for p a prime. In fact, any function f defined on the primes can be extended uniquely to a function on the integers satisfying this relation: f(Product_i p_i^e_i) = (Product_i p_i^e_i) * (Sum_i e_i*f(p_i)/p_i). - Franklin T. Adams-Watters, Nov 07 2006
See A131116 and A131117 for record values and where they occur. - Reinhard Zumkeller, Jun 17 2007
Let n be the product of a multiset P of k primes. Consider the k-dimensional box whose edges are the elements of P. Then the (k-1)-dimensional surface of this box is 2*a(n). For example, 2*a(25) = 20, the perimeter of a 5 X 5 square. Similarly, 2*a(18) = 42, the surface area of a 2 X 3 X 3 box. - David W. Wilson, Mar 11 2011
The arithmetic derivative n' was introduced, probably for the first time, by the Spanish mathematician José Mingot Shelly in June 1911 with "Una cuestión de la teoría de los números", work presented at the "Tercer Congreso Nacional para el Progreso de las Ciencias, Granada", cf. link to the abstract on Zentralblatt MATH, and L. E. Dickson, History of the Theory of Numbers. - Giorgio Balzarotti, Oct 19 2013
a(A235991(n)) odd; a(A235992(n)) even. - Reinhard Zumkeller, Mar 11 2014
Sequence A157037 lists numbers with prime arithmetic derivative, i.e., indices of primes in this sequence. - M. F. Hasler, Apr 07 2015
Maybe the simplest "natural extension" of the arithmetic derivative, in the spirit of the above remark by Franklin T. Adams-Watters (2006), is the "pi based" version where f(p) = primepi(p), see sequence A258851. When f is chosen to be the identity map (on primes), one gets A066959. - M. F. Hasler, Jul 13 2015
When n is composite, it appears that a(n) has lower bound 2*sqrt(n), with equality when n is the square of a prime, and a(n) has upper bound (n/2)*log_2(n), with equality when n is a power of 2. - Daniel Forgues, Jun 22 2016
If n = p1*p2*p3*... where p1, p2, p3, ... are all the prime factors of n (not necessarily distinct), and h is a real number (we assume h nonnegative and < 1), the arithmetic derivative of n is equivalent to n' = lim_{h->0} ((p1+h)*(p2+h)*(p3+h)*... - (p1*p2*p3*...))/h. It also follows that the arithmetic derivative of a prime is 1. We could assume h = 1/N, where N is an integer; then the limit becomes {N -> oo}. Note that n = 1 is not a prime and plays the role of constant. - Giorgio Balzarotti, May 01 2023

			6' = (2*3)' = 2'*3 + 2*3' = 1*3 + 2*1 = 5.
Note that, for example, 2' + 3' = 1 + 1 = 2, (2+3)' = 5' = 1. So ' is not linear.
G.f. = x^2 + x^3 + 4*x^4 + x^5 + 5*x^6 + x^7 + 12*x^8 + 6*x^9 + 7*x^10 + ...

G. Balzarotti, P. P. Lava, La derivata aritmetica, Editore U. Hoepli, Milano, 2013.
E. J. Barbeau, Problem, Canad. Math. Congress Notes, 5 (No. 8, April 1973), 6-7.
L. E. Dickson, History of the Theory of Numbers, Vol. 1, Chapter XIX, p. 451, Dover Edition, 2005. (Work originally published in 1919.)
A. M. Gleason et al., The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938-1964, Math. Assoc. America, 1980, p. 295.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, A formula for the n-th prime number, Comptes rendus de l'Académie bulgare des Sciences, Tome 66, No 4, 2013.
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.
A. Buium, Home Page
A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995), no. 2, 309-340.
A. Buium, Geometry of p-jets, Duke Math. J. 82 (1996), no. 2, 349-367.
A. Buium, Arithmetic analogues of derivations, J. Algebra 198 (1997), no. 1, 290-299.
A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000), 95-167.
Brad Emmons and Xiao Xiao, The Arithmetic Partial Derivative, arXiv:2201.12453 [math.NT], 2022.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the Arithmetic Derivative, Journal of Integer Sequences, Vol. 15 (2012), #12.4.1.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
P. Haukkanen, M. Mattila, J. K. Merikoski and T. Tossavainen, Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?, Journal of Integer Sequences, 16 (2013), #13.1.2. - From _N. J. A. Sloane_, Feb 03 2013
P. Haukkanen, J. K. Merikoski and T. Tossavainen, Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative, Mathematical Communications 25 (2020), 107-115.
Antti Karttunen, Program in LODA-assembly
J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8.
Michael Penn, When the derivative of a number is not zero -- The arithmetic derivative., YouTube video, 2022.
Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
D. J. M. Shelly, Una cuestión de la teoria de los numeros, Asociation Esp. Granada 1911, 1-12 S (1911). (Abstract of ref. JFM42.0209.02 on zbMATH.org)
T. Tossavainen, P. Haukkanen, J. K. Merikoski, and M. Mattila, We can differentiate numbers, too, The College Mathematics Journal 55 (2024), no. 2, 100-108.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Linda Westrick, Investigations of the Number Derivative, Siemens Foundation competition 2003 and Intel Science Talent Search 2004.
Wikipedia, Arithmetic derivative

Cf. A086134 (least prime factor of n').
Cf. A086131 (greatest prime factor of n').
Cf. A068719 (derivative of 2n).
Cf. A068720 (derivative of n^2).
Cf. A068721 (derivative of n^3).
Cf. A001787 (derivative of 2^n).
Cf. A027471 (derivative of 3^(n-1)).
Cf. A085708 (derivative of 10^n).
Cf. A068327 (derivative of n^n).
Cf. A024451 (derivative of p#).
Cf. A068237 (numerator of derivative of 1/n).
Cf. A068238 (denominator of derivative of 1/n).
Cf. A068328 (derivative of squarefree numbers).
Cf. A068311 (derivative of n!).
Cf. A168386 (derivative of n!!).
Cf. A260619 (derivative of hyperfactorial(n)).
Cf. A260620 (derivative of superfactorial(n)).
Cf. A068312 (derivative of triangular numbers).
Cf. A068329 (derivative of Fibonacci(n)).
Cf. A096371 (derivative of partition number).
Cf. A099301 (derivative of d(n)).
Cf. A099310 (derivative of phi(n)).
Cf. A342925 (derivative of sigma(n)).
Cf. A349905 (derivative of prime shift).
Cf. A327860 (derivative of primorial base exp-function).
Cf. A369252 (derivative of products of three odd primes), A369251 (same sorted).
Cf. A068346 (second  derivative of n).
Cf. A099306 (third   derivative of n).
Cf. A258644 (fourth  derivative of n).
Cf. A258645 (fifth   derivative of n).
Cf. A258646 (sixth   derivative of n).
Cf. A258647 (seventh derivative of n).
Cf. A258648 (eighth  derivative of n).
Cf. A258649 (ninth   derivative of n).
Cf. A258650 (tenth   derivative of n).
Cf. A185232 (n-th    derivative of n).
Cf. A258651 (A(n,k) = k-th arithmetic derivative of n).
Cf. A085731 (gcd(n,n')), A083345 (n'/gcd(n,n')), A057521 (gcd(n, (n')^k) for k>1).
Cf. A342014 (n' mod n), A369049 (n mod n').
Cf. A341998 (A003557(n')), A342001 (n'/A003557(n)).
Cf. A098699 (least x such that x' = n, antiderivative of n).
Cf. A098700 (n such that x' = n has no integer solution).
Cf. A099302 (number of solutions to x' = n).
Cf. A099303 (greatest x such that x' = n).
Cf. A051674 (n such that n' = n).
Cf. A083347 (n such that n' < n).
Cf. A083348 (n such that n' > n).
Cf. A099304 (least k such that (n+k)' = n' + k').
Cf. A099305 (number of solutions to (n+k)' = n' + k').
Cf. A328235 (least k > 0 such that (n+k)' = u * n' for some natural number u).
Cf. A328236 (least m > 1 such that (m*n)' = u * n' for some natural number u).
Cf. A099307 (least k such that the k-th arithmetic derivative of n is zero).
Cf. A099308 (k-th arithmetic derivative of n is zero for some k).
Cf. A099309 (k-th arithmetic derivative of n is nonzero for all k).
Cf. A129150 (n-th derivative of 2^3).
Cf. A129151 (n-th derivative of 3^4).
Cf. A129152 (n-th derivative of 5^6).
Cf. A189481 (x' = n has a unique solution).
Cf. A190121 (partial sums).
Cf. A258057 (first differences).
Cf. A229501 (n divides the n-th partial sum).
Cf. A165560 (parity).
Cf. A235991 (n' is odd), A235992 (n' is even).
Cf. A327863, A327864, A327865 (n' is a multiple of 3, 4, 5).
Cf. A157037 (n' is prime), A192192 (n'' is prime), A328239 (n''' is prime).
Cf. A328393 (n' is squarefree), A328234 (squarefree and > 1).
Cf. A328244 (n'' is squarefree), A328246 (n''' is squarefree).
Cf. A328303 (n' is not squarefree), A328252 (n' is squarefree, but n is not).
Cf. A328248 (least k such that the (k-1)-th derivative of n is squarefree).
Cf. A328251 (k-th arithmetic derivative is never squarefree for any k >= 0).
Cf. A256750 (least k such that the k-th derivative is either 0 or has a factor p^p).
Cf. A327928 (number of distinct primes p such that p^p divides n').
Cf. A342003 (max. exponent k for any prime power p^k that divides n').
Cf. A327929 (n' has at least one divisor of the form p^p).
Cf. A327978 (n' is primorial number > 1).
Cf. A328243 (n' is a partial sum of primorial numbers and larger than one).
Cf. A328310 (maximal prime exponent of n' minus maximal prime exponent of n).
Cf. A328320 (max. prime exponent of n' is less than that of n).
Cf. A328321 (max. prime exponent of n' is >= that of n).
Cf. A328383 (least k such that the k-th derivative of n is either a multiple or a divisor of n, but not both).
Cf. A263111 (the ordinal transform of a).
Cf. A300251, A319684 (Möbius and inverse Möbius transform).
Cf. A305809 (Dirichlet convolution square).
Cf. A349133, A349173, A349394, A349380, A349618, A349619, A349620, A349621 (for miscellaneous Dirichlet convolutions).
Cf. A069359 (similar formula which agrees on squarefree numbers).
Cf. A258851 (the pi-based arithmetic derivative of n).
Cf. A328768, A328769 (primorial-based arithmetic derivatives of n).
Cf. A328845, A328846 (Fibonacci-based arithmetic derivatives of n).
Cf. A302055, A327963, A327965, A328099 (for other variants and modifications).
Cf. A038554 (another sequence using "derivative" in its name, but involving binary expansion of n).
Cf. A322582, A348507 (lower and upper bounds), also A002620.

A003415:= Concatenation([0,0],List(List([2..10^3],Factors),
i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Aug 31 2017
(APL, Dyalog dialect) A003415 ← { ⍺←(0 1 2) ⋄ ⍵≤1:⊃⍺ ⋄ 0=(3⊃⍺)|⍵:((⊃⍺+(2⊃⍺)×(⍵÷3⊃⍺)) ((2⊃⍺)×(3⊃⍺)) (3⊃⍺)) ∇ ⍵÷3⊃⍺ ⋄ ((⊃⍺) (2⊃⍺) (1+(3⊃⍺))) ∇ ⍵} ⍝ Antti Karttunen, Feb 18 2024

a003415 0 = 0
a003415 n = ad n a000040_list where
  ad 1 _             = 0
  ad n ps'@(p:ps)
     | n < p * p     = 1
     | r > 0         = ad n ps
     | otherwise     = n' + p * ad n' ps' where
       (n',r) = divMod n p
-- Reinhard Zumkeller, May 09 2011

Ad:=func; [n le 1 select 0 else Ad(n): n in [0..80]]; // Bruno Berselli, Oct 22 2013

A003415 := proc(n) local B,m,i,t1,t2,t3; B := 1000000000039; if n<=1 then RETURN(0); fi; if isprime(n) then RETURN(1); fi; t1 := ifactor(B*n); m := nops(t1); t2 := 0; for i from 1 to m do t3 := op(i,t1); if nops(t3) = 1 then t2 := t2+1/op(t3); else t2 := t2+op(2,t3)/op(op(1,t3)); fi od: t2 := t2-1/B; n*t2; end;
A003415 := proc(n)
        local a,f;
        a := 0 ;
        for f in ifactors(n)[2] do
                a := a+ op(2,f)/op(1,f);
        end do;
        n*a ;
end proc: # R. J. Mathar, Apr 05 2012

a[ n_] := If[ Abs @ n < 2, 0, n Total[ #2 / #1 & @@@ FactorInteger[ Abs @ n]]]; (* Michael Somos, Apr 12 2011 *)
dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; Table[dn[n], {n, 0, 100}] (* T. D. Noe, Sep 28 2012 *)

A003415(n) = {local(fac);if(n<1,0,fac=factor(n);sum(i=1,matsize(fac)[1],n*fac[i,2]/fac[i,1]))} /* Michael B. Porter, Nov 25 2009 */

apply( A003415(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]), [0..99]) \\ M. F. Hasler, Sep 25 2013, updated Nov 27 2019

A003415(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= spf); (s); }; \\ Antti Karttunen, Mar 10 2021

a(n) = my(f=factor(n), r=[1/(e+!e)|e<-f[,1]], c=f[,2]); n*r*c; \\ Ruud H.G. van Tol, Sep 03 2023

from sympy import factorint
def A003415(n):
    return sum([int(n*e/p) for p,e in factorint(n).items()]) if n > 1 else 0
# Chai Wah Wu, Aug 21 2014

def A003415(n):
    F = [] if n == 0 else factor(n)
    return n * sum(g / f for f, g in F)
[A003415(n) for n in range(79)] # Peter Luschny, Aug 23 2014

If n = Product p_i^e_i, a(n) = n * Sum (e_i/p_i).
a(m*p^p) = (m + a(m))*p^p, p prime: a(m*A051674(k))=A129283(m)*A051674(k). - Reinhard Zumkeller, Apr 07 2007
For n > 1: a(n) = a(A032742(n)) * A020639(n) + A032742(n). - Reinhard Zumkeller, May 09 2011
a(n) = n * Sum_{p|n} v_p(n)/p, where v_p(n) is the largest power of the prime p dividing n. - Wesley Ivan Hurt, Jul 12 2015
For n >= 2, Sum_{k=2..n} floor(1/a(k)) = pi(n) = A000720(n) (see K. T. Atanassov article). - Ivan N. Ianakiev, Mar 22 2019
From A.H.M. Smeets, Jan 17 2020: (Start)
Limit_{n -> oo} (1/n^2)*Sum_{i=1..n} a(i) = A136141/2.
Limit_{n -> oo} (1/n)*Sum_{i=1..n} a(i)/i = A136141.
a(n) = n if and only if n = p^p, where p is a prime number. (End)
Dirichlet g.f.: zeta(s-1)*Sum_{p prime} 1/(p^s-p), see A136141 (s=2), A369632 (s=3) [Haukkanen, Merikoski and Tossavainen]. - Sebastian Karlsson, Nov 25 2021
From Antti Karttunen, Nov 25 2021: (Start)
a(n) = Sum_{d|n} d * A349394(n/d).
For all n >= 1, A322582(n) <= a(n) <= A348507(n).
If n is not a prime, then a(n) >= 2*sqrt(n), or in other words, for all k >= 1 for which A002620(n)+k is not a prime, we have a(A002620(n)+k) > n. [See Ufnarovski and Åhlander, Theorem 9, point (3).]
(End)

More terms from Michel ten Voorde, Apr 11 2001

cp's OEIS Frontend

A368702 Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

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A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.

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A348304 a(n) = Sum_{d|n} d'', where d'' is the second arithmetic derivative of d (A068346).

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A348426 Numbers k for which sigma(k) = k + k'', where k'' is the second derivative of k (A068346).

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A351131 Triangular numbers (A000217) whose second arithmetic derivative (A068346) is also a triangular number.

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A352331 Numbers k for which phi(k) = phi(k''), where phi is the Euler totient function (A000010) and k'' the second arithmetic derivative of k (A068346).

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A352332 Numbers k for which k = phi(k') + phi(k''), where phi is the Euler totient function (A000010), k' the arithmetic derivative of k (A003415) and k'' the second arithmetic derivative of k (A068346).

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A361078 Numbers k for which k = gcd(k', k"), where k' is the arithmetic derivative of k (A003415) and k" is the second derivative of k (A068346).

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A003415 a(n) = n' = arithmetic derivative of n: a(0) = a(1) = 0, a(prime) = 1, a(mn) = ma(n) + n*a(m).