A121369 -id:A121369 - cp's OEIS frontend

A121368 a(1) = a(2) = 1, a(n) = A007947(a(n-1)) + a(n-2), for n >= 3, i.e., a(n) = a(n-2) plus the largest squarefree divisor of a(n-1).

1, 1, 2, 3, 5, 8, 7, 15, 22, 37, 59, 96, 65, 161, 226, 387, 355, 742, 1097, 1839, 2936, 2573, 5509, 8082, 8203, 16285, 24488, 22407, 46895, 69302, 116197, 185499, 178030, 363529, 541559, 905088, 555701, 1460789, 591330, 2052119, 2643449, 4695568

Offset: 1

Leroy Quet, Jul 23 2006

nonn

			6 is the largest squarefree divisor of a(12) = 96. So a(13) = 6 + a(11) = 65.

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A007947, A121367, A121369.

with(numtheory): A007947 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1], i=1..nops(t1)); end: a:=proc(n) if n=1 or n=2 then 1 else A007947(a(n-1))+a(n-2) fi end: seq(a(n),n=1..25); # Emeric Deutsch, Jul 24 2006

nxt[{a_,b_}]:={b,a+Max[Select[Divisors[b],SquareFreeQ]]}; NestList[nxt,{1,1},50][[All,1]] (* Harvey P. Dale, Jan 21 2017 *)

More terms from Emeric Deutsch, Jul 24 2006

More terms from R. J. Mathar, May 18 2007

A358093 a(n) = n for 1 <= n <= 2; thereafter a(n) is the least unused m such that rad(m) = rad(rad(a(n-1)) + rad(a(n-2))), where rad(m) = A007947(m).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 10, 13, 23, 6, 29, 35, 8, 37, 39, 38, 77, 115, 12, 11, 17, 14, 31, 15, 46, 61, 107, 42, 149, 191, 170, 19, 21, 20, 961, 41, 18, 47, 53, 40, 63, 29791, 26, 57, 83, 70, 51, 121, 62, 73, 45, 22, 1369, 59, 24, 65, 71, 34, 105, 139, 122, 87, 209, 74

Offset: 1

David James Sycamore, Nov 08 2022

nonn

In other words, a(n) is the least m having the same squarefree kernel as the sum of the squarefree kernels of a(n-1) and a(n-2).
Primes do not appear in natural order.
Conjectured to be a permutation of the positive integers.
From Michael De Vlieger, Nov 09 2022: (Start)
Let i = rad(a(n-2)), j = rad(a(n-1)), and s = rad(i+j). If s = p (prime), then a(n) = m = p^e, e >= 1. Because a(n) = m such that rad(m) = p, the j-th occasion of s = p implies a(n) = p^j.
By the same token, generally, the j-th occasion of s implies a(n) = M*s, where M is regular to s, i.e., M is a product restricted to primes p|s, with M appearing in order. For example, if s = 10, then the j-th occasion of s = 10 implies a(n) = 10*A003592(j). These are consequences of conservation of squarefree kernel s, the lexical and greedy nature of the sequence.
It is clear that a(n) >= s, since rad(a(n)) = s, hence s | a(n), and on account of the lexical and greedy nature of the sequence.
Prime a(n) implies s = a(n), while s < a(n) for a(n) that are composite prime powers (i.e., a(n) in A246547) since s is in those cases prime.(End)
From Michael De Vlieger, Nov 13 2022: (Start)
Adjacent terms i and j are coprime as consequence of s mod p = i mod p + j mod p and a(n) = M*s such that rad(M) = rad(s). p | a(n-2) and p | a(n-1) implies p | a(n), and subsequent adjacent terms are likewise divisible by p, contradicting initial a(1) coprime to a(2). Because the sequence starts with dissimilar residues (mod p), p divides only one of {a(n-2), a(n-1), a(n)}.
2 | a(n) for n mod 3 = 2. (End)

			a(5) = 4 because rad(rad(3) + rad(5)) = rad(3 + 5) = rad(8) = 2, and 4 is the least unused number m such that rad(m) = 2.
To find a(36), we have rad(rad(a(34)) + rad(a(35))) = rad(rad(21) + rad(20)) = rad(31) = 31, and since 31 has appeared at a(24), a(36) = 31^2 = 961. [Corrected by _N. J. A. Sloane_, Mar 24 2025]
a(43) = 29791 (31^3) because rad(40) + rad(63) = 31; we already have 31 and 31^2.

Michael De Vlieger, Table of n, a(n) for n = 1..1500
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1500, highlighting primes in red, other prime powers in gold, and other numbers in blue.

Cf. A007947, A005117, A121369, A354184.

nn = 65; c[] = False; p[] = q[] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3]; Array[(q[#]++; p[#]++) &[Times @@ FactorInteger[a[#]][[All, 1]]] &, 2, 2]; u = 4; Set[{i, j}, Map[Times @@ FactorInteger[#][[All, 1]] &, {a[2], a[3]}]]; Do[k = Times @@ FactorInteger[i + j][[All, 1]]; If[PrimeQ[k], m = k^p[k]; p[k]++, m = q[k]; While[Nand[! c[m k], PowerMod[k, k, m] == 0], m++]; m *= k]; q[k]++; Set[{a[n], c[m], i, j}, {m, True, j, k}]; If[m == u, While[c[u], u++]], {n, 4, nn}]; Array[a, nn] (* _Michael De Vlieger, Nov 09 2022 *)

first(n) = { n = max(n, 3); res = vector(n); for(i = 1, 3, res[i] = i); for(i = 4, n, target = rad(rad(res[i-1]) + rad(res[i-2])); resset = Set(res); pr = factor(target)[, 1]; step = target; if(omega(target) >= 1, forstep(j = target, oo, step, if(rad(j) == target, if(#setminus(Set(j), resset) == 1, res[i] = j; next(2) ) ) ) , for(j = 1, oo, if(#setminus(Set(target ^ j), resset) == 1, res[i] = j; next(2) ) ) ) ); res }
rad(n) = factorback(factorint(n)[, 1]); \\ David A. Corneth, Nov 09 2022

nn=300; a=List([1,2]); rad(n)= factorback(factorint(n)[, 1]);
{for(i=3,nn, listput(a,1); j=rad(rad(a[i-1])+rad(a[i-2])); while((rad(j)!=rad(rad(a[i-1])+rad(a[i-2])) || #select(n->n==j, a) > 0),j+= rad(rad(a[i-1])+rad(a[i-2])));  a[i]=j; ); print(a);} \\ Gerry Martens, Nov 10 2022

More terms from David A. Corneth, Nov 09 2022

A121367 a(1) = a(2) = 1, a(n) = a(n-1) + A007947(a(n-2)) for n >= 3, i.e., a(n) = a(n-1) plus the largest squarefree divisor of a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 15, 28, 43, 57, 100, 157, 167, 324, 491, 497, 988, 1485, 1979, 2144, 4123, 4257, 8380, 9799, 13989, 23788, 37777, 49671, 87448, 104005, 125867, 229872, 355739, 384473, 740212, 1124685, 1494791, 1536446, 3031237, 4567683, 7598920

Offset: 1

Leroy Quet, Jul 23 2006

nonn

			14 is the largest squarefree divisor of a(9) = 28. So a(11) = a(10) + 14 = 57.

Cf. A007947, A121368, A121369.

with(numtheory): A007947 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1], i=1..nops(t1)); end: a:=proc(n) if n=1 or n=2 then 1 else a(n-1)+A007947(a(n-2)) fi end: seq(a(n),n=1..20); # Emeric Deutsch, Jul 24 2006

nxt[{a_,b_}]:={b,b+Max[Select[Divisors[a],SquareFreeQ]]}; Transpose[ NestList[ nxt,{1,1},50]][[1]] (* Harvey P. Dale, Jul 23 2015 *)

More terms from Emeric Deutsch, Jul 24 2006

More terms from R. J. Mathar, May 11 2007

A354184 a(1) = a(2) = 1, a(n) = (A007947(31a(n-1)) + A007947(31a(n-2)))/31 for n >= 3, i.e., 31a(n) is the largest squarefree divisor of 31a(n-1) plus the largest squarefree divisor of 31*a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 7, 9, 10, 13, 23, 36, 29, 35, 64, 37, 39, 76, 77, 115, 192, 121, 17, 28, 31, 15, 16, 17, 19, 36, 25, 11, 16, 13, 15, 28, 29, 43, 72, 49, 13, 20, 23, 33, 56, 47, 61, 108, 67, 73, 140, 143, 213, 356, 391, 569, 960, 599, 629, 1228, 1243, 1857, 3100, 1867

Offset: 1

Augusto Santi, May 18 2022

After the first 5 terms, the sequence values repeat periodically with a cycle length of 207. The maximum value of a(n) is 1142300, whose first occurrence appears at n = 111.

			31*2 is the largest squarefree divisor of 31*a(6) = 31*8. 31*7 is the largest squarefree divisor of 31*a(7) = 31*7. So a(8) = (31*2 + 31*7)/31 = 9.

Michael De Vlieger, Table of n, a(n) for n = 1..2075
Augusto Santi, Periodic sequences of integers generated by a(n+1) = rad(a(n)) + rad(a(n-1)), Mathematics Stack Exchange.

Cf. A007947, A121369.

Nest[Append[#, (Times @@ FactorInteger[31 #[[-1]]][[All, 1]] + Times @@ FactorInteger[31 #[[-2]]][[All, 1]])/31] &, {1, 1}, 62] (* Michael De Vlieger, Jul 18 2022 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 1; for (n=3, nn, va[n] = (rad(31*va[n-2]) + rad(31*va[n-1]))/31;); va; \\ Michel Marcus, May 21 2022

from sympy import primefactors
def rad(num):
    primes = primefactors(num)
    value = 1
    for p in primes:
        value *= p
    return value
a = [1, 1]
for n in range(2, 1000):
    a += [(rad(31*a[n-1]) + rad(31*a[n-2])) // 31]

For n >= 6, a(207 + n) = a(n).

cp's OEIS Frontend

A121368 a(1) = a(2) = 1, a(n) = A007947(a(n-1)) + a(n-2), for n >= 3, i.e., a(n) = a(n-2) plus the largest squarefree divisor of a(n-1).

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A358093 a(n) = n for 1 <= n <= 2; thereafter a(n) is the least unused m such that rad(m) = rad(rad(a(n-1)) + rad(a(n-2))), where rad(m) = A007947(m).

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A121367 a(1) = a(2) = 1, a(n) = a(n-1) + A007947(a(n-2)) for n >= 3, i.e., a(n) = a(n-1) plus the largest squarefree divisor of a(n-2).

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Programs

Maple

Mathematica

Extensions

A354184 a(1) = a(2) = 1, a(n) = (A007947(31*a(n-1)) + A007947(31*a(n-2)))/31 for n >= 3, i.e., 31*a(n) is the largest squarefree divisor of 31*a(n-1) plus the largest squarefree divisor of 31*a(n-2).

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A354184 a(1) = a(2) = 1, a(n) = (A007947(31a(n-1)) + A007947(31a(n-2)))/31 for n >= 3, i.e., 31a(n) is the largest squarefree divisor of 31a(n-1) plus the largest squarefree divisor of 31*a(n-2).