A380955 -id:A380955 - cp's OEIS frontend

A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

1, 4, 8, 16, 27, 64, 81, 256, 243, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 0

Gus Wiseman, Feb 12 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.

Also the position of first appearance of n in A374248.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   529: {9,9}
   729: {2,2,2,2,2,2}
   961: {11,11}
  1369: {12,12}
  1681: {13,13}
  1849: {14,14}
  2209: {15,15}

For length instead of sum we have A151821.
For factors instead of indices we have A280286 (sorted A381075), firsts of A280292.
Counting partitions by this statistic gives A364916.
Positions of first appearances in A380955.
The sorted version is A380957.
For product instead of sum we have firsts of A380986.
A multiplicative version is A380987 (sorted A380988), firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
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, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

After a(12) = 961, this appears to converge to prime(n)^2.

A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 243, 256, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

Gus Wiseman, Feb 13 2025

nonn

Also appears to be sorted firsts of A374248.

For length instead of sum we have A151821.
Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916.
Sorted positions of first appearances in A380955.
The unsorted version is A380956.
For product instead of sum we have sorted firsts of A380986.
The multiplicative version is A380988, unsorted A380987, firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
For factors instead of indices we have A381075, see A280286, A280292.
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.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A381431 Heinz number of the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 10, 13, 11, 11, 16, 17, 15, 19, 14, 13, 13, 23, 20, 25, 17, 27, 22, 29, 13, 31, 32, 17, 19, 17, 25, 37, 23, 19, 28, 41, 17, 43, 26, 33, 29, 47, 40, 49, 35, 23, 34, 53, 45, 19, 44, 29, 31, 59, 26, 61, 37, 39, 64, 23, 19, 67, 38

Offset: 1

Gus Wiseman, Feb 26 2025

nonn

The image first differs from A320340, A364347, A350838 in containing a(150) = 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
   7: {4}
  11: {5}
  10: {1,3}
  13: {6}
  11: {5}
  11: {5}
  16: {1,1,1,1}

The conjugate is A048767, union A351294, complement A351295, fix A048768 (count A217605).
Taking length instead of sum in the definition gives A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
The union is A381432, complement A381433.
Values appearing only once are A381434, more than once A381435.
These are the Heinz numbers of rows of A381436, conjugate A381440.
Greatest prime index of each term is A381437, counted by A381438.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A005117, A047966, A051903, A066328, A091602, A116861, A130091, A212166, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Times@@Prime/@egs[prix[n]],{n,100}]

A122111(a(n)) = A048767(n).

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

A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 4, 1, 1, 1, 2, 2, 4, 5, 3, 1, 6, 5, 5, 1, 1, 1, 1, 7, 3, 2, 8, 4, 1, 6, 6, 9, 3, 1, 1, 3, 3, 7, 2, 2, 2, 5, 1, 10, 6, 11, 1, 1, 1, 1, 1, 7, 8, 7, 3, 3, 12, 9, 8, 4, 1, 1, 13, 7, 14, 6, 1, 5, 2, 10, 15, 3, 1, 1, 1, 4, 4, 4, 3, 9, 7, 1, 16, 3, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2025

Row-lengths are A051903.
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 section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
   1: (empty)
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 3
   7: 4
   8: 1 1 1
   9: 2 2
  10: 4
  11: 5
  12: 3 1
  13: 6
  14: 5
  15: 5
  16: 1 1 1 1

Row-lengths are A051903.
Row sums are A056239.
First part in each row is A066328.
Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).
Rows appearing only once have Heinz numbers A381434, more than once A381435.
Last part in each row is A381437, counted by A381438.
The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[egs[prix[n]],{n,100}]

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).

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
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 Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
Triangle begins:
   1: (empty)
   2: 1
   3: 1 1
   4: 2
   5: 1 1 1
   6: 1 1 1
   7: 1 1 1 1
   8: 3
   9: 2 2
  10: 1 1 1 1
  11: 1 1 1 1 1
  12: 2 1 1
  13: 1 1 1 1 1 1
  14: 1 1 1 1 1
  15: 1 1 1 1 1
  16: 4
  17: 1 1 1 1 1 1 1
  18: 2 2 1
  19: 1 1 1 1 1 1 1 1

Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
First part in each row is A051903, conjugate A066328.
Last part in each row is A051904, conjugate A381437 (counted by A381438).
Row sums are A056239.
Row lengths are A066328.
Partitions of this type are counted by A239455, complement A351293.
The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
Rows appearing only once have Heinz numbers A381540, more than once A381541.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.

Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 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 section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

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.

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).

cp's OEIS Frontend

A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

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A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

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A381431 Heinz number of the section-sum partition of the prime indices of n.

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

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A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

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

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A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

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A381437 Last part of the section-sum partition of the prime indices of n.

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