A076694 -id:A076694 - cp's OEIS frontend

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

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).

A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

Original entry on oeis.org

1, 6, 30, 210, 900, 7776, 27000, 279936, 810000, 9261000, 24300000, 362797056, 729000000, 13060694016, 21870000000, 408410100000, 656100000000, 16926659444736, 19683000000000, 609359740010496, 590490000000000, 18010885410000000, 17714700000000000

Offset: 0

Gus Wiseman, Feb 18 2025

nonn

Is this sequence strictly increasing?
From David Consiglio, Jr., Feb 20 2025: (Start)
The answer to the question above is: no, a(21) < a(20). And all subsequent odd indexed terms are lower than their even predecessors.
All terms must be a product of x primes (with multiplicity) to the y power where x-y = n and x mod y = 0. There are very few combinations of numbers that meet these criteria, so checking all of them to find the minimum outcome is quite fast.
Example --> n=5
6 primes to the 1 power --> 6 distinct primes
  2*3*5*7*11*13 = 30030
7 primes to the 2 power -- disallowed (5 mod 2 = 1)
8 primes to the 3 power -- disallowed (4 mod 3 = 1)
9 primes to the 4 power -- disallowed (9 mod 4 = 1)
10 primes to the 5 power --> 2 distinct primes
  2*2*2*2*2*3*3*3*3*3 = 7776
The minimum value is 7776 and thus a(5) = 7776. (End)

			The terms together with their prime indices begin:
        1: {}
        6: {1,2}
       30: {1,2,3}
      210: {1,2,3,4}
      900: {1,1,2,2,3,3}
     7776: {1,1,1,1,1,2,2,2,2,2}
    27000: {1,1,1,2,2,2,3,3,3}
   279936: {1,1,1,1,1,1,1,2,2,2,2,2,2,2}
   810000: {1,1,1,1,2,2,2,2,3,3,3,3}
  9261000: {1,1,1,2,2,2,3,3,3,4,4,4}

David Consiglio, Jr., Table of n, a(n) for n = 0..100
David Consiglio, Jr., Python program

Position of first appearance of n in A001222 - A136565.
For factors instead of exponents we have A280286 (sorted A381075), firsts of A280292.
For indices instead of exponents we have A380956 (sorted A380957), firsts of A380955.
A000040 lists the primes, differences A001223.
A005361 gives product of prime exponents.
A055396 gives least prime index, greatest A061395.
A056239 (reverse A296150) adds up prime indices, row sums of A112798.
A124010 lists prime exponents (signature); A001221, A051903, A051904.
Cf. A046660, A071625, A075254, A075255, A076694, A130091, A130092, A290106, A380986.

prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
q=Table[Total[prisig[n]]-Total[Union[prisig[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

a(10)-a(11) from Michel Marcus, Feb 20 2025

a(12) and beyond from David Consiglio, Jr., Feb 20 2025

A075653 a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).

Original entry on oeis.org

1, 4, 6, 6, 10, 11, 14, 10, 12, 17, 22, 17, 26, 23, 23, 18, 34, 23, 38, 27, 31, 35, 46, 29, 30, 41, 30, 37, 58, 40, 62, 34, 47, 53, 47, 41, 74, 59, 55, 47, 82, 54, 86, 57, 53, 71, 94, 53, 56, 57, 71, 67, 106, 59, 71, 65, 79, 89, 118, 70, 122, 95, 73, 66, 83, 82, 134, 87, 95

Offset: 1

Joseph L. Pe, Oct 11 2002

For 1 <= k <= n, add sigma(k) if k is a prime factor of n, otherwise add 1. For example, a(6) = 11 since for k = 1,2,.. we have 1 + sigma(2) + sigma(3) + 1 + 1 + 1 = 1 + (1+2) + (1+3) + 1 + 1 + 1 = 11. - Wesley Ivan Hurt, Oct 18 2021

			6 + sum of prime factors of 6 = 6 + 2 + 3 = 11, so a(6) = 11.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008472, A075254, A075255, A076694.

Flatten[Append[{1}, Table[n + Apply[Plus, Transpose[FactorInteger[n]][[1]]], {n, 2, 100}]]]
Join[{1},Table[n+Total[FactorInteger[n][[All,1]]],{n,2,70}]] (* Harvey P. Dale, Sep 29 2016 *)

a(n) = n + vecsum(factor(n)[,1]); \\ Michel Marcus, Feb 22 2017

a(n) = n + A008472(n).

a(p) = 2p for primes p. - Wesley Ivan Hurt, Oct 18 2021

A208614 Number of partitions of n into distinct primes where all of the prime factors of n are represented in the partition.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, 4, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 4, 2, 1, 5, 9, 6, 5, 4, 1, 6, 6, 7, 4, 3, 1, 4, 1, 4, 9, 20, 7, 3, 1, 7, 8, 6, 1, 15, 1, 5, 19, 11, 13, 9, 1, 21, 52, 7, 1

Offset: 0

Richard Penner, Feb 29 2012

nonn

Inspired by web-based discussion started by Rajesh Bhowmick.

			a(p) = 1 for any prime p.
a(n) = 0 for 1, 4, 6, 8, 9, 22.
a(25) = 3 because 25 = 3 + 5 + 17 = 5 + 7 + 13 = 2 + 5 + 7 + 11.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Rajesh Bhowmick, A bit different from Goldbach's conjecture (February 27-28, 2012)

Cf. A000586 (upper bound). A000586(A076694(n)) is a stricter upper bound.

with(numtheory):
a:= proc(n) local b, l, f;
      b:= proc(h, j) option remember;
            `if`(h=0, 1, `if`(j<1, 0,
            `if`(l[j]>h, 0, b(h-l[j], j-1)) +b(h, j-1)))
          end; forget(b);
      f:= factorset(n);
      l:= sort([({seq(ithprime(i), i=1..pi(n))} minus f)[]]);
      b(n-add(i, i=f), nops(l))
    end:
seq(a(n), n=0..300);  # Alois P. Heinz, Mar 20 2012

restrictedIntegerPartition[ n_Integer, list_List ] := 1 /; n == 0
restrictedIntegerPartition[ n_Integer, list_List ] := 0 /; n < 0 || Total[list] < n || n < Min[list]
restrictedIntegerPartition[ n_Integer, list_List ] := restrictedIntegerPartition[n - First[list], Rest[list]] + restrictedIntegerPartition[n, Rest[list]]
distinctPrimeFactors[ n_Integer ] := distinctPrimeFactors[n] = Map[First, FactorInteger[n]]
oeisA076694[ n_Integer ] := oeisA076694[n] = n - Total[distinctPrimeFactors[n]]
oeisA208614[ n_Integer ] := restrictedIntegerPartition[oeisA076694[n], Sort[Complement[Prime @ Range @ PrimePi @ oeisA076694 @ n, distinctPrimeFactors[n]] , Greater ]]
Table[oeisA208614[n], {n,1,100}]

countRestrictedIntegerPartitions(n, L) := if ( n = 0 ) then 1 else if ( ( n < 0 ) or ( lsum(k, k, L) < n ) or ( n < lmin( L ) ) ) then 0 else block( [ m, R ], m : first(L), R : rest(L), countRestrictedIntegerPartitions(n, R) + countRestrictedIntegerPartitions(n - m, R));
distinctPrimeFactors(n) := map(first,ifactors(n));
oeisA076694(n) := n - lsum(k,k,distinctPrimeFactors(n));
listOfPrimesLessThanOrEqualTo (n) := block( [ list : [] , i], for i : 2 step 0 while i <= n do ( list : cons(i, list) , i : next_prime(i) ) , list );
oeisA208614(n) := block([ m, list ], m : oeisA076694(n), list : sort(listify(setdifference(setify(listOfPrimesLessThanOrEqualTo(m)), setify(distinctPrimeFactors(n)))), ordergreatp), countRestrictedIntegerPartitions(m, list));
makelist(oeisA208614(j), j, 1, 100);

cp's OEIS Frontend

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

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A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

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A075653 a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).

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A208614 Number of partitions of n into distinct primes where all of the prime factors of n are represented in the partition.

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