A380987 -id:A380987 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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[[#]]]&]

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Gus Wiseman, Feb 18 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.

All terms are odd.

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
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, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

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,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

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[[#]]]&]

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

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica