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Index values for new maxima in A065925.
+20
3
1, 3, 5, 21, 25, 35, 55, 185, 265, 563, 569, 733, 3350, 3469, 6010, 6779, 10689, 11143, 14505, 18016, 30561, 31588, 37757, 59611, 65816, 67106, 75904, 118361, 214379, 241564, 255196, 414221, 648152, 703812, 881848, 1153107, 1277048, 1407769, 1719552, 1775846, 2722607, 3177728
REFERENCES
J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 99-100. ASIN: B002ACVZ6O [From Jason Earls, Nov 26 2009]
PROG
(PARI) sopf(n) = local(fac, i); fac=factor(n); sum(i=1, matsize(fac)[1], fac[i, 1])
A065926(m)= {local(a, n, k); a=0; for(k=1, m, n=1; while(sopf(n)!=sopf(n+k), n++); if(n>a, a=n; print1(k, ", ")))} \\ Klaus Brockhaus
5, 7, 114, 209, 493, 516, 855, 1194, 1274, 2815, 2834, 3180, 3186, 5225, 6010, 8056, 8357, 8954, 11439, 13684, 14599, 15748, 17298, 17384, 17784, 20940, 25886, 36223, 36938, 41796, 53725, 64855, 67942, 69167, 72468, 79143, 79516, 86232, 96845, 120708, 125709
PROG
(PARI) sopf(n) = local(fac, i); fac=factor(n); sum(i=1, matsize(fac)[1], fac[i, 1])
A065927(m)= {local(a, n, k); a=0; for(k=1, m, n=1; while(sopf(n)!=sopf(n+k), n++); if(n>a, a=n; print1(n, ", ")))} // Klaus Brockhaus
Smallest k such that omega(n+k) = omega(k).
+10
1
2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 2, 3, 2, 5, 4, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 14, 5, 4, 3, 2, 5, 4, 3, 2, 3, 2, 5, 4, 3, 2, 3, 2, 5, 4, 3, 2, 6, 8, 5, 4, 3, 2, 3, 2, 4, 3, 2, 4, 3, 2, 5, 4, 3, 2, 3, 2, 7, 8, 5, 4, 3, 2, 3, 2, 3, 2, 6, 10, 5, 4, 3, 2, 6, 6, 6, 15, 5, 4, 3, 2, 5, 4, 3, 2, 3, 2, 5, 4
PROG
(PARI): ome(m)= {local(k, n); for(k=1, m, n=1; while(omega(n)!=omega(n+k), n++); print1(n, ", "))} ome(200)
(PARI) { for (n=1, 1000, k=1; while(omega(n + k) != omega(k), k++); write("b065569.txt", n, " ", k) ) } \\ Harry J. Smith, Oct 23 2009
Least number k such that sopfr(k) = sopfr(k + n), where sopfr(k) is the integer log of k.
+10
1
5, 10, 7, 20, 7, 14, 20, 40, 13, 14, 21, 28, 14, 40, 19, 33, 11, 26, 56, 28, 49, 42, 115, 56, 35, 28, 31, 57, 11, 38, 50, 66, 63, 11, 17, 52, 11, 112, 42, 51, 22, 98, 11, 84, 57, 35, 52, 95, 138, 13, 33, 56, 22, 62, 77, 114, 61, 22, 39, 76, 44, 13, 91, 57, 70
EXAMPLE
a(1) = 5 because 5 is the least number such that sopfr(5) = sopfr(5 + 1) = 5 .
a(2) = 10 because 10 is the least number such that sopfr(10) = sopfr(10 + 2) = 7 .
MAPLE
with(numtheory):P:=proc(q) local a, b, k, n; for n from 1 to q do for k from 1 to q do
a:=ifactors(k)[2]; b:=ifactors(k+n)[2];
if add(a[k][1]*a[k][2], k=1..nops(a))=add(b[k][1]*b[k][2], k=1..nops(b))
then print(k); break; fi; od; od; end: P(10^9);
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