A374248 -id:A374248 - cp's OEIS frontend

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 2, 3, 0, 4, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 4, 3, 0, 1, 0, 4, 0, 2, 0, 0, 0, 1, 0, 0, 2, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 3, 6, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 11 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 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

Positions of 0's are A005117, complement A013929.
For length instead of sum we have A046660.
Positions of 1's are A081770.
For factors instead of indices we have A280292, firsts A280286 (sorted A381075).
A multiplicative version is A290106.
Counting partitions by this statistic gives A364916.
Dominates A374248.
Positions of first appearances are A380956, sorted A380957.
For prime multiplicities instead of prime indices we have A380958.
For product instead of sum we have A380986.
A000040 lists the primes, differences A001223.
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, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Total[Union[prix[n]]],{n,100}]

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

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[[#]]]&]

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).

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[[#]]]&]

A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 7, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 9, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

Contains no ones.

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The prime factors of 96 are {2,2,2,2,2,3}, with sum 13, and we have permutations such as (2,2,2,2,3,2), with run-compression (2,3,2), with sum 7, so a(96) = 13 - 7 = 6.

Positions of first appearances are A280286.
For least instead of greatest sum of run-compression we have A280292.
Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
For prime indices instead of factors we have A374248.
This is an opposite version of A374250, for prime indices A373956.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 is run-compression of prime indices, row-sums A066328.
A373949 counts compositions by sum of run-compression, opposite A373951.
A373957 gives greatest number of runs in a permutation of prime factors.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
A374252 counts permutations of prime factors by number of runs.
Cf. A000040, A026549, A037201, A046660, A124767, A246655, A333755, A373954, A374246, A374247.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Total[prifacs[n]]-Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]

a(n) = A001414(n) - A374250(n).

cp's OEIS Frontend

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

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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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A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.

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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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A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

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