A381076 -id:A381076 - cp's OEIS frontend

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

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

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

Original entry on oeis.org

4, 9, 8, 25, 16, 49, 32, 81, 64, 121, 128, 169, 256, 625, 512, 289, 1024, 361, 2048, 1444, 1331, 529, 5324, 2116, 2197, 4232, 8788, 841, 17576, 961, 7569, 3844, 4913, 7688, 19652, 1369, 6859, 5476, 12321, 1681, 34225, 1849, 15129, 7396, 12167, 2209, 46225, 8836, 19881

Offset: 2

Michel Marcus, Dec 31 2016

nonn

Michel Marcus, Table of n, a(n) for n = 2..500

Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.
A multiplicative version is A064549 (sorted A001694), firsts of A003557.
For length instead of sum we have A151821.
These are the positions of first appearances in A280292 = A001414 - A008472.
For indices instead of factors we have A380956 (sorted A380957), firsts of A380955.
A multiplicative version for indices is A380987 (sorted A380988), firsts of A290106.
For prime exponents instead of factors we have A380989, firsts of A380958.
The sorted version is A381075.
For product instead of sum see A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* Gus Wiseman, Feb 20 2025 *)

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) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}

For p prime, a(p) = p^2 (see A001248).

A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 32, 49, 64, 81, 121, 128, 169, 256, 289, 361, 512, 529, 625, 841, 961, 1024, 1331, 1369, 1444, 1681, 1849, 2048, 2116, 2197, 2209, 2809, 3481, 3721, 3844, 4232, 4489, 4913, 5041, 5324, 5329, 5476, 6241, 6859, 6889, 7396, 7569, 7688, 7921

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

			The initial terms of A280292 are (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,...), wherein a value appears for the first time at positions 1, 4, 8, 9, 16, 25, ...

Michel Marcus, Table of n, a(n) for n = 1..391

For length instead of sum we have A151821.
The unsorted version is A280286, firsts of A280292.
For indices instead of factors we have A380957 (unsorted A380956), firsts of A380955.
A multiplicative version is A380988 (unsorted A380987), firsts of A290106.
For prime multiplicities instead of factors see A380989, firsts of A380958.
For product instead of sum we have A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A364916 counts partitions by (sum minus sum of distinct parts).
Cf. A000720, A001414, A008472, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

f(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2] - f[j, 1]); \\ A280292
lista(nn) = my(v=Set(vector(nn, i, f(i))), list=List()); for (i=1, #v, my(k=1); while(f(k) != v[i], k++); listput(list, k)); vecsort(Vec(list)); \\ Michel Marcus, Apr 15 2025

Sorted positions of first appearances in A001414 - A008472.

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 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. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
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. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

cp's OEIS Frontend

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

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A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

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A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

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A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

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