A081770 -id:A081770 - cp's OEIS frontend

A017113 a(n) = 8*n + 4.

4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420, 428, 436, 444, 452, 460, 468

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(65).
n such that 16 is the largest power of 2 dividing A003629(k)^n - 1 for any k. - Benoit Cloitre, Mar 23 2002
Continued fraction expansion of tanh(1/4). - Benoit Cloitre, Dec 17 2002
Consider all primitive Pythagorean triples (a,b,c) with c - a = 8, sequence gives values for b. (Corresponding values for a are A078371(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004
Also numbers of the form a^2 + b^2 + c^2 + d^2, where a,b,c,d are odd integers. - Alexander Adamchuk, Dec 01 2006
If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-5) is equal to the number of 4-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Aug 26 2007
A007814(a(n)) = 2; A037227(a(n)) = 5. - Reinhard Zumkeller, Jun 30 2012
Numbers k such that 3^k + 1 is divisible by 41. - Bruno Berselli, Aug 22 2018
Lexicographically smallest arithmetic progression of positive integers avoiding Fibonacci numbers. - Paolo Xausa, May 08 2023
From Martin Renner, May 24 2024: (Start)
Also number of points in a grid cross with equally long arms and a width of two points, e.g.:
                                         * *
                        * *              * *
           * *          * *              * *
  * *    * * * *    * * * * * *    * * * * * * * *
  * *    * * * *    * * * * * *    * * * * * * * *
           * *          * *              * *
                        * *              * *
                                         * *
etc. (End)

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1100 from Vincenzo Librandi)
E. Catalan, Extrait d'une lettre, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206. [If N is a prime number of the form 4*m+1, then 8*N+4 is the sum of four odd squares.]
Cody Clifton, Commutativity in non-Abelian Groups, May 06 2010.
Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.2.
Dr Barker, How to Avoid the Fibonacci Numbers, YouTube video, 2023.
Meimei Gu and Rongxia Hao, 3-extra connectivity of 3-ary n-cube networks, arXiv:1309.5083 [cs.DM], Sep 19, 2013.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

First differences of A016742 (even squares).
Cf. A078370, A078371, A081770 (subsequence).
Cf. A003629, A005408, A007814, A019683, A037227, A051062, A118413.
Cf. A008586, A016825.

a017113 = (+ 4) . (* 8)
a017113_list = [4, 12 ..]  -- Reinhard Zumkeller, Jul 13 2013

[8*n+4: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

LinearRecurrence[{2,-1}, {4,12}, 50] (* G. C. Greubel, Apr 26 2018 *)

a(n)=8*n+4 \\ Charles R Greathouse IV, Sep 23 2013

a(n) = A118413(n+1,3) for n > 2. - Reinhard Zumkeller, Apr 27 2006
a(n) = Sum_{k=0..4*n} (i^k+1)*(i^(4*n-k)+1), where i = sqrt(-1). - Bruno Berselli, Mar 19 2012
a(n) = 4*A005408(n). - Omar E. Pol, Apr 17 2016
E.g.f.: (8*x + 4)*exp(x). - G. C. Greubel, Apr 26 2018
G.f.: 4*(1+x)/(1-x)^2. - Wolfdieter Lang, Oct 27 2020
Sum_{n>=0} (-1)^n/a(n) = Pi/16 (A019683). - Amiram Eldar, Dec 11 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(3*Pi/16).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(3*Pi/16). (End)
a(n) = 2*A016825(n) = A008586(2*n+1). - Elmo R. Oliveira, Apr 10 2025

A280292 a(n) = sopfr(n) - sopf(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 2, 0, 0, 0, 6, 0, 3, 0, 2, 0, 0, 0, 4, 5, 0, 6, 2, 0, 0, 0, 8, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 3, 0, 0, 6, 7, 5, 0, 2, 0, 6, 0, 4, 0, 0, 0, 2, 0, 0, 3, 10, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 5, 2, 0, 0, 0, 6, 9, 0, 0, 2, 0, 0, 0, 4, 0, 3, 0, 2, 0, 0, 0, 8, 0, 7, 3, 7, 0, 0, 0, 4, 0

Offset: 1

Michel Marcus, Dec 31 2016

nonn

Alladi and Erdős (1977) proved that for all numbers m>=0, m!=1, the sequence of numbers k such that a(k) = m has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956). For example, when m=0, the sequence is the squarefree numbers (A005117), whose density is 6/Pi^2, and when m=2 the sequence is A081770, whose density is 1/Pi^2. - Amiram Eldar, Nov 02 2020

Sum of prime factors minus sum of distinct prime factors. Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916. - Gus Wiseman, Feb 21 2025

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 164-166.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, Reciprocals of certain large additive functions, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.
A multiplicative version is A003557, firsts A064549 (sorted A001694).
For length instead of sum we have A046660.
For product instead of sum we have A066503, firsts A381076.
Positions of first appearances are A280286 (sorted A381075).
For indices instead of factors we have A380955, firsts A380956 (sorted A380957).
For exponents instead of factors we have A380958, firsts A380989.
A000040 lists the primes, differences A001223.
A001222 counts prime factors (distinct A001221).
A003963 gives product of prime indices, distinct A156061, excess A380986.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A027746 lists prime factors, distinct A027748.
A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).
Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.

Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* Michael De Vlieger, Feb 25 2019 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]);
a(n) = sopfr(n) - sopf(n);

a(n) = A001414(n) - A008472(n).
a(A005117(n)) = 0.
a(n) = A001414(A003557(n)). - Antti Karttunen, Oct 07 2017
Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - Werner Schulte, Feb 24 2019
From Amiram Eldar, Nov 02 2020: (Start)
a(n) = a(A057521(n)).
Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).
Sum_{n<=x, n nonsquarefree} 1/a(n) ~ c*x + O(sqrt(x)*log(x)), where c = Integral_{t=0..1} (F(t)-6/Pi^2)/t dt, and F(t) = Product_{p prime} (1-1/p)*(1-1/(t^p - p)) (De Koninck et al., 1981; Finch, 2018), or, equivalently c = Sum_{k>=2} d(k)/k = 0.1039..., where d(k) = (6/Pi^2)*A338559(k)/A338560(k) is the asymptotic density of the numbers m with a(m) = k (Alladi and Erdős, 1977; Ivić, 2003). (End)

More terms from Antti Karttunen, Oct 07 2017

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 11 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

Positions of 0's are A005117, complement A013929.
For length instead of sum we have A046660.
Positions of 1's are A081770.
For factors instead of indices we have A280292, firsts A280286 (sorted A381075).
A multiplicative version is A290106.
Counting partitions by this statistic gives A364916.
Dominates A374248.
Positions of first appearances are A380956, sorted A380957.
For prime multiplicities instead of prime indices we have A380958.
For product instead of sum we have A380986.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Total[Union[prix[n]]],{n,100}]

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 12, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 14 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 300 are {1,1,2,3,3}, so a(300) = 18 - 6 = 12.

Positions of nonzeros are A038838.
For length instead of product we have A046660.
For factors instead of indices we have A066503, see A007947 (squarefree kernel).
For sum of factors instead of product of indices we have A280292, see A280286, A381075.
For quotient instead of difference we have A290106, for factors A003557.
For sum instead of product we have A380955 (firsts A380956, sorted A380957).
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@prix[n]-Times@@Union[prix[n]],{n,100}]

a(n) = A003963(n) - A156061(n).

A380958 Number of prime factors of n (with multiplicity) minus sum of distinct prime exponents of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 13 2025

nonn

			The prime factors of 2100 are {2,2,3,5,5,7}, with distinct multiplicities {1,2}, so a(2100) = 6 - (1+2) = 3.

Positions of 0's are A130091, complement A130092.
The RHS (sum of distinct prime exponents) is A136565.
For prime factors instead of exponents see A280292, firsts A280286, sorted A381075.
For prime indices instead of exponents see A380955, firsts A380956, sorted A380957.
Position of first appearance of n is A380989(n).
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A005361 gives product of prime signature.
A055396 gives least prime index, greatest A061395.
A056239 (reverse A296150) adds up prime indices, row sums of A112798, counted by A001222.
A124010 lists prime exponents (signature); see A001222, A001221, A051903, A051904.
Cf. A000720, A046660, A071625, A075254, A075255, A076694, A081770, A116861, A178503, A290106, A380986.

Table[PrimeOmega[n]-Total[Union[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

a(n) = A001222(n) - A136565(n).

A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.

Original entry on oeis.org

12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 404, 412, 420, 428, 436, 444, 452, 460, 476, 492, 508, 516, 524, 532, 548

Offset: 1

Michael De Vlieger, Aug 16 2023

nonn

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.
Apparently this is A081770 without the leading 4. - R. J. Mathar, Sep 05 2023
From Peter Munn, Mar 05 2024: (Start)
The preceding observation is true for the whole sequence, for reasons outlined below.
To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).
For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.
(End)

			Let b(n) = A126706(n), S = A360767, and T = A363082.
b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.

Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.

Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Intersection of A363082 and A360767.
From Peter Munn, Feb 21 2024: (Start)
a(n) = 2*A039956(n+1).
Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)
From Michael De Vlieger, Mar 08 2024: (Start)
{a(n)} = A366825 \ A366460, i.e., even terms in A366825.
A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)

A091428 Numbers k such that abs(A092673(k)) = 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 12, 13, 15, 17, 19, 20, 21, 23, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 51, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 116, 119, 123, 124, 127, 129

Offset: 1

Jon Perry, Mar 02 2004

nonn

The asymptotic density of this sequence is 5/Pi^2 = 0.506605... . - Amiram Eldar, Jan 11 2023

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

Disjoint union of A056911 and A081770.
Cf. A092673, A359592 (characteristic function).
Distinct from A020486.

Select[Range[130], Abs[MoebiusMu[#] - If[EvenQ[#], MoebiusMu[#/2], 0]] == 1 &] (* Amiram Eldar, Jan 11 2023 *)

A338559 Numerators of the fractions f(n) such that (6/Pi^2)*f(n) is the asymptotic density of the numbers k with A280292(k) = sopfr(k) - sopf(k) = n.

Original entry on oeis.org

1, 0, 1, 1, 1, 17, 5, 17, 61, 559, 269, 5851, 5279, 954913, 1693849, 6394159, 1430687, 33257690117, 393069739, 330504317141, 146861034421, 3447587278559, 13150098373, 17185160160637123, 68404253084009, 219367146802450039, 527431007100952693, 2089195405327981487

Offset: 0

Amiram Eldar, Nov 02 2020

Alladi and Erdős (1977) proved that for all numbers n>=0, n!=1, the sequence of numbers k such that A280292(k) = n has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956).

			1/1, 0/1, 1/6, 1/12, 1/12, 17/360, 5/72, 17/560, 61/2160, 559/30240, 269/12600, 5851/399168, ...
For n=0, the sequence of numbers k such that A280292(k) = 0 are the squarefree numbers (A005117), whose density is 6/Pi^2. Thus f(0) = a(0)/A338560(0) = 1 and a(0) = 1.
For n=1, there are no numbers k with A280292(k) = 1, thus a(1) = 0.
For n=2, the sequence of numbers k with A280292(k) = 2 is A081770 whose density is 1/Pi^2. Thus f(2) = a(2)/A338560(2) = 1/6 and a(2) = 1.
For n=7, there are A000607(7) = 3 partitions of 7 into prime parts: {{p_i, b_i}} = {{2, 2}, {3, 1}}, {{2, 1}, {5, 1}}, and {{7, 1}}. The powerful numbers associated with these partitions are 2^(2+1)*3^(1+1) = 72, 2^(1+1)*5^(1+1) = 100, and 7^(1+1) = 49. Thus, f(7) = a(7)/A338560(7) = 1/psi(72) + 1/psi(100) + 1/psi(49) = 1/144 + 1/180 + 1/56 = 17/560, and a(7) = 17.

Amiram Eldar, Table of n, a(n) for n = 0..214
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link. See Theorem 1.7, pp. 286-287.
Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Cf. A000607, A001414, A001615, A005117, A008472, A013661, A059956, A081770, A280292, A338560 (denominators).

psi[1] = 1; psi[n_] := n*Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); delta[ps_] := 1/psi[(Times @@ ps)*(Times @@ Union[ps])]; f[n_] := Total[delta /@ IntegerPartitions[n, Floor[n/2], Select[Range[n], PrimeQ]]]; Numerator @ Array[f, 30, 0]

Sum_{k>=0} a(k)/A338560(k) = zeta(2) = Pi^2/6 (A013661).
Let {P_j} be the set of partitions of n into prime parts, with j = 1..A000607(n). For a partition P_j = {p_i, b_i} of n = Sum_i b_i * p_i, where p_i are distinct primes, and b_i >= 1 are their multiplicities in the partition, let S(P_j) = Product_i p_i^(b_i + 1) be a powerful number associated with the partition. f(n) = a(n)/A338560(n) = Sum_{j=1..r(n)} 1/psi(S(P_j)), where psi is the Dedekind psi function (A001615).
For any r>0, Sum_{n<=x, n nonsquarefree} 1/A280292(n)^r ~ c(r)*x + O(x^(1-r/2)*log(x)) + O(x^(1/2)*log(x)), where c(r) = (6/Pi^2) * Sum_{k>=2} (a(k)/A338560(k)) * (1/k^r) (Ivić, 2003).

A338560 Denominators of the fractions f(n) such that (6/Pi^2)*f(n) is the asymptotic density of the numbers k with A280292(k) = sopfr(k) - sopf(k) = n.

Original entry on oeis.org

1, 1, 6, 12, 12, 360, 72, 560, 2160, 30240, 12600, 399168, 453600, 86486400, 209563200, 1111968000, 363242880, 5557616064000, 163459296000, 70396470144000, 83364240960000, 1773991047628800, 7508350080000, 6120269114319360000, 86090742017280000, 224409867525043200000

Offset: 0

Amiram Eldar, Nov 02 2020

See A338559 for details.

Amiram Eldar, Table of n, a(n) for n = 0..214

Cf. A000607, A001414, A001615, A005117, A008472, A013661, A059956, A081770, A280292, A338559 (numerators).

psi[1] = 1; psi[n_] := n*Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); delta[ps_] := 1/psi[(Times @@ ps)*(Times @@ Union[ps])]; f[n_] := Total[delta /@ IntegerPartitions[n, Floor[n/2], Select[Range[n], PrimeQ]]]; Denominator @ Array[f, 30, 0]

A381076 Sorted positions of first appearances in A066503 (n minus squarefree kernel of n).

Original entry on oeis.org

1, 4, 8, 16, 18, 20, 24, 25, 27, 32, 44, 48, 50, 52, 54, 64, 68, 72, 75, 76, 80, 81, 92, 96, 98, 108, 112, 116, 121, 125, 128, 144, 148, 152, 160, 162, 164, 172, 175, 176, 188, 189, 192, 196, 198, 200, 212, 216, 232, 236, 242, 243, 244, 256, 260, 264, 268, 272

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

In A066503, each value appears for the first time at one of these positions.

For quotient instead of difference we have A001694, sorted firsts of A003557.
Sorted positions of first appearances in A066503.
For indices and sum we have A380957 (unsorted A380956), firsts of A380955.
For indices and quotient we have A380988 (unsorted A380987), firsts of A290106.
For sum instead of product we have A381075, sorted firsts of A280292, see A280286.
For indices instead of factors we have A381077, sorted firsts of A380986.
A000040 lists the primes, differences A001223.
A001414 adds up prime factors (indices A056239), row sums of A027746 (indices A112798).
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
Cf. A001221, A001222, A038838, A046660, A075255, A081770, A116861, A136565, A178503, A175508, A304038, A380989.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Times@@prifacs[n]-Times@@Union[prifacs[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

cp's OEIS Frontend

A017113 a(n) = 8*n + 4.

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A280292 a(n) = sopfr(n) - sopf(n).

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A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

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A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.

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A380958 Number of prime factors of n (with multiplicity) minus sum of distinct prime exponents of n.

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A364999 Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.

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A091428 Numbers k such that abs(A092673(k)) = 1.

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A338559 Numerators of the fractions f(n) such that (6/Pi^2)*f(n) is the asymptotic density of the numbers k with A280292(k) = sopfr(k) - sopf(k) = n.

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A364999 Numbers k neither squarefree nor prime power such that both rad(k)A119288(k) > k and rad(k)A053669(k) > k.