A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

2, 1, 21, 25, 39, 35, 57, 55, 49, 65, 75, 77, 129, 95, 91, 105, 183, 119, 125, 143, 133, 185, 147, 161, 169, 195, 175, 209, 339, 217, 255, 253, 259, 305, 247, 285, 273, 245, 301, 299, 345, 323, 325, 357, 371, 435, 669, 391, 361, 403, 399, 473, 343, 469, 481

Offset: 0

Author

Gus Wiseman, Jan 21 2025

Keywords

• nonn
• look

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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The least number whose prime indices satisfy (product) - (sum) = 3 is 25 (prime indices {3,3}), so a(3) = 25.
		

Crossrefs

Position of first appearance of n in A325036.

For sum instead of difference we have A379682, firsts of A379681.

A000040 lists the primes, differences A001223.

A003963 multiplies together prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

The subtraction A325036 takes the following values:

- zero: A301987, counted by A001055 (strict A045778).

- negative: A325037, counted by A114324, see A318029

- positive: A325038, counted by A096276 shifted right

- negative one: A325041, counted by A028422

- one: A325042, counted by A001055 shifted right

- nonnegative: A325044, counted by A096276

- nonpositive: A379721, counted by A319005

Cf. A000720, A178503, A175508, A318950, A319000, A325034, A326151, A326153/A326154, A379319, A379722.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pp=Table[Total[prix[n]]-Times@@prix[n],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[pp,-i][[1,1]],{i,0,mnrm[-DeleteCases[pp,0|_?Positive]]}]

Formula

Satisfies A003963(a(n)) - A056239(a(n)) = n.