A156061 -id:A156061 - cp's OEIS frontend

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

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[[#]]]&]

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).

A290107 a(1) = 1; for n > 1, a(n) = product of distinct exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 13 2017

nonn

			For n = 36 = 2^2 * 3^2, the only distinct exponent that occurs is 2, thus a(36) = 2.
For n = 144 = 2^4 * 3^2, the distinct exponents are 2 and 4, thus a(144) = 2*4 = 8.
For n = 4500 = 2^2 * 3^2 * 5^3, the distinct exponents are 2 and 3, thus a(4500) = 2*3 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A156061, A181819.
Differs from A005361 for the first time at n=36.
Differs from A072411 for the first time at n=144, and also from A157754 for the second time (after the initial term).

Table[If[n == 1, 1, Apply[Times, Union[FactorInteger[n][[All, -1]] ]]], {n, 120}] (* Michael De Vlieger, Aug 14 2017 *)

A290107(n) = factorback(vecsort((factor(n)[, 2]), ,8));

(define (A290107 n) (A156061 (A181819 n)))

a(n) = A156061(A181819(n)).

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

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.

All terms are odd.

			The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}

For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
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}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[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[[#]]]&]

A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 06 2018

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).

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[[#]]]&]

A225395 Replace each prime number with its rank in the recursive prime factorization of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 8, 5, 9, 2, 3, 6, 4, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 12, 3, 13, 8, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 4, 15, 4, 16, 10, 17, 6, 18, 11, 8, 1, 18, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 12, 22, 3, 2, 13, 23, 8, 21, 14, 20, 5, 24, 6, 24, 9, 22, 15, 24, 2, 25, 4, 10, 3, 26, 14, 27, 6, 24, 16, 28

Offset: 1

Paul Tek, May 06 2013

a(A000040(n)) = n, hence all natural numbers appear in this sequence.
a(2n) = n.
It appears that a(35) = 12 is the only instance where a composite index yields a larger value than any smaller index. Checked to 10^7. - Charles R Greathouse IV, Jul 30 2016

			The number 9967 is the 1228th prime number.
Hence a(9967) = 1228.
The recursive prime factorization of 31250 is 2*5^(2*3).
The numbers 2, 3 and 5 are respectively the 1st, 2nd and 3rd prime numbers.
Hence a(31250) = a(2*5^(2*3)) = 1*3^(1*2) = 9.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence

Cf. A000040, A000026, A156061.

a225395 n = product $ zipWith (^)
    (map a049084 $ a027748_row n) (map a225395 $ a124010_row n)
-- Reinhard Zumkeller, May 10 2013

a[1] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p]; a[n_] := a[n] = Times @@ (PrimePi[#[[1]]]^a[#[[2]]]& /@ FactorInteger[n]); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 07 2013 *)

a(n)=if(n<3, return(1)); my(f=factor(n)); prod(i=1,#f~, primepi(f[i,1])^a(f[i,2])) \\ Charles R Greathouse IV, Jul 30 2016

Perl
```
# See Tek link.
```

Multiplicative, with a(prime(i)^j) = i^a(j).

a(n) = prod(A049084(A027748(k))^a(A124010(k)): k=1..A001221(n)). - Reinhard Zumkeller, May 10 2013

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A328880 If n = Product (p_j^k_j) then a(n) = Product (a(pi(p_j)) + 1), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 3, 2, 3, 8, 5, 6, 7, 6, 12, 2, 4, 6, 3, 8, 9, 10, 4, 6, 4, 14, 3, 6, 9, 24, 6, 2, 15, 8, 12, 6, 7, 6, 21, 8, 8, 18, 7, 10, 12, 8, 13, 6, 3, 8, 12, 14, 3, 6, 20, 6, 9, 18, 5, 24, 7, 12, 9, 2, 28, 30, 4, 8, 12, 24, 9, 6, 10, 14, 12, 6, 15, 42, 11, 8

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (a(1) + 1) * (a(2) + 1) = (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A048250, A156061, A184160, A225395, A282446, A328879.

a[1] = 1; a[n_] := Times @@ (a[PrimePi[#[[1]]]] + 1 & /@ FactorInteger[n]); Table[a[n], {n, 1, 80}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + a(primepi(f[i])))} \\ Andrew Howroyd, Oct 29 2019

a(n) = a(prime(n)) - 1.

cp's OEIS Frontend

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

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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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A290107 a(1) = 1; for n > 1, a(n) = product of distinct exponents in the prime factorization of n.

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A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

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

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A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

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

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A225395 Replace each prime number with its rank in the recursive prime factorization of n.

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