A076920 -id:A076920 - cp's OEIS frontend

A076919 a(1) = 1, a(2) = 2, then a(n+1) is the smallest number such that the highest common factor of a(n) and a(n+1) is different from that of a(n) and a(n-1) and is more than 1.

1, 2, 4, 8, 10, 15, 18, 20, 24, 26, 39, 42, 44, 48, 50, 55, 66, 68, 72, 74, 111, 114, 116, 120, 122, 183, 186, 188, 192, 194, 291, 294, 296, 300, 302, 453, 456, 458, 687, 690, 692, 696, 698, 1047, 1050, 1052, 1056, 1058, 1081, 1128, 1130, 1135, 1362, 1364, 1368, 1370, 1375, 1386, 1388, 1392

Offset: 1

Amarnath Murthy, Oct 17 2002

nonn

			15 follows 10 as (8,10) = 2 so 12 and 14 are ruled out.

Cf. A076920.

a[1] = 1; a[2] = 2;
a[n_] := a[n] = Module[{k}, For[k = a[n-1] + 1, True, k++, If[GCD[a[n-1], a[n-2]] != GCD[k, a[n-1]] && GCD[k, a[n-1]] > 1, Return[k]]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 25 2023 *)

More terms from Jean-François Alcover, Oct 25 2023

A227478 Numbers k such that both the sum of the semiprime divisors of k and the sum of the prime divisors of k are squares.

Original entry on oeis.org

1146, 2874, 9870, 33220, 34353, 43140, 50694, 52290, 66440, 86280, 94350, 100804, 101097, 103059, 106140, 121540, 125070, 127897, 132880, 139908, 156870, 172560, 183475, 191140, 193410, 201608, 208692, 212280, 243080, 248378, 265760, 276094, 279816, 303291

Offset: 1

Michel Lagneau, Jul 13 2013

nonn

The sequence is infinite: if a number of the form p(1) * p(2) * ... * p(i)^2 * p(i+1) * ... * p(m) is in the sequence where p(1), ..., p(m) are primes, then the numbers p(1) * p(2) * ... * p(i)^q * p(i+1) * ... * p(m) are also in the sequence for q = 3, 4, ... For example, the infinite subsequence 33220, 66440, 132880, ... contains the numbers of the form 2^q * 5 * 11 * 151 for q = 2, 3, 4, ... where 2+5+11+151 = 169 = 13^2 and 2*2 + 2*5 + 2*11 + 2*151 + 5*11 + 5*151 + 11*151 = 2809 = 53^2.

In this sequence, the corresponding pairs of squares are (961, 196), (2401, 484), (900, 64), (2809, 169), (4900, 361), (7225, 729), (2304, 100), (1521, 100), (2809, 169), (7225, 729), (1225, 64), (3721, 121), (12100, 289), (4900, 361), (2704, 100), (7225, 169), (8100, 400), (2916, 169), (2809, 169), (12769, 225), (1521, 100), (7225, 729), (8464, 225), (13225, 529), (5329, 121), (3721, 121), (1369, 64), (2704, 100), (7225, 169), (13689, 289), (2809, 169), (3364, 100), (12769, 225), (12100, 289), ...

			1146 = 2*3*191 is in the sequence because the divisors are {1, 2, 3, 6, 191, 382, 573, 1146}, so the sum of the semiprime divisors is 6 + 382 + 573 = 961 = 31^2 and the sum of the prime divisors is 2 + 3 + 191 = 196 = 14^2.

Cf. A001358, A008472, A076920, A164722, A227476.

with(numtheory):for n from 2 to 310000 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) then printf(`%d, `,n):else fi:od:

Rest@ Select[Range[3*10^5], AllTrue[{DivisorSum[#, # &, PrimeOmega@ # == 2 &], DivisorSum[#, # &, PrimeQ]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Sep 15 2017 *)

cp's OEIS Frontend

A076919 a(1) = 1, a(2) = 2, then a(n+1) is the smallest number such that the highest common factor of a(n) and a(n+1) is different from that of a(n) and a(n-1) and is more than 1.

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