A352844 Smallest k > 1 such that sopfr(k) - tau(k) = n, or -1 if no such k exists.

2, 3, 8, 5, 15, 7, 21, 25, 35, 11, 33, 13, 39, 117, 65, 17, 51, 19, 57, 121, 95, 23, 69, 169, 115, 483, 161, 29, 87, 31, 93, 279, 155, 651, 217, 37, 111, 333, 185, 41, 123, 43, 129, 387, 215, 47, 141, 423, 235, 954, 318, 53, 159, 477, 265, 841, 354, 59, 177, 61

Offset: 0

Author

Nicolas Bělohoubek, Apr 05 2022

Keywords

• nonn

Comments

Conjecture: There is no -1 in this sequence.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 516 terms from Nicolas Bělohoubek)

Crossrefs

Cf. A000005, A001414.

Programs

PARI

f(m) = my(fp=factor(m)); sum(k=1, #fp~, fp[k,1]*fp[k,2]) - numdiv(fp);
a(n) = my(k=2); while(f(k) != n, k++); k; \\ Michel Marcus, Apr 06 2022

Formula

It appears that the sequence satisfies these rules, for large m:

Rule 1: a(prime(m+1) - 2) = prime(m+1)

Rule 2: a(prime(m+1) - 1) = 3*prime(m+1)

Rule 3: a(prime(m+1) + 1) = 5*prime(m+1)

Rule 4: a(prime(m+1) - 3) = 6*prime(m+1)

Rule 5: a(prime(m+1) + 3) = 7*prime(m+1)

Rule 6: a(prime(m+1)) = 9*prime(m+1)

Rule 7: a(prime(m+1) - 7) = 11*prime(m+1)

Rule 8: a(prime(m+1) + 7) = 12*prime(m+1)

Rule 9: a(prime(m+1) + 9) = 13*prime(m+1)

...

Choose the first rule that applies.