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

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

Author

Gus Wiseman, Feb 20 2025

Keywords

• nonn

Comments

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.

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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[[#]]]&]