A342464 -id:A342464 - cp's OEIS frontend

A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

2, 3, 5, 9, 7, 25, 35, 15, 11, 49, 117649, 625, 717409, 1225, 55, 225, 13, 121, 1771561, 2401, 36226650889, 184877, 1127357, 875, 902613283, 514675673281, 3780549773, 1500625, 83852850675321384784127, 3025, 62004635, 21, 17, 169, 4826809, 14641, 8254129, 143, 2924207, 77, 8223741426987700773289, 59797108943, 546826709

Offset: 0

Antti Karttunen and Michael De Vlieger, Mar 14 2021

nonn

This sequence (which could be viewed as a binary tree, like the underlying A005940 and A329886) is similar to A324289, but unlike its underlying tree A283477 that generates only numbers that are products of distinct primorial numbers (i.e., terms of A129912), here the underlying tree A329886 generates all possible products of primorial numbers, i.e., terms of A025487, but in different order.

Cf. A005940, A025487, A108951, A129912, A276086, A283980, A324886, A342457 [= 2*A246277(a(n))], A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342463 [= A342001(a(n))], A342464 [= A051903(a(n))].

Cf. A324289 (a subset of these terms, in different order).

Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] :=
Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[f@ a[#] &, 43, 0]] (* Michael De Vlieger, Mar 17 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A342456(n) = A276086(A329886(n));

a(n) = A276086(A329886(n)) = A324886(A005940(1+n)).
For all n >= 0, gcd(a(n), A329886(n)) = 1.
For all n >= 1, A055396(a(n))-1 = A061395(A329886(n)) = A290251(n) = 1+A080791(n).
For all n >= 0, a(2^n) = A000040(2+n).

A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 6, 4, 6, 4, 2, 4, 1, 2, 6, 4, 10, 6, 6, 4, 8, 12, 10, 8, 22, 4, 8, 2, 1, 2, 6, 4, 6, 2, 6, 2, 18, 10, 8, 6, 18, 12, 16, 4, 26, 16, 24, 8, 20, 14, 4, 6, 26, 16, 14, 8, 30, 6, 8, 4, 1, 2, 6, 4, 14, 12, 12, 8, 18, 12, 24, 4, 8, 12, 14, 4, 24, 20, 28, 20, 26, 16, 16, 12, 32, 26, 24, 14, 28, 16

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

From David A. Corneth's Feb 27 2019 comment in A276150 follows that the only odd terms in this sequence are 1's occurring at 0 and at two's powers.

Subsequences starting at each n = 2^k are slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, etc.. Compare also to the behaviors of A324342 and A342463.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001222, A005940, A108951, A276150, A329886, A342456, A342457, A342461, A342463, A342464.

Cf. also A324342, A324383, A324387, A324888.

A342462(n) = bigomega(A342456(n)); \\ Other code as in A342456.

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A342462(n) = A276150(A329886(n));

a(n) = A001222(A342456(n)) = A001222(A342457(n)).
a(n) = A276150(A329886(n)) = A324888(A005940(1+n)).
a(n) >= A342461(n).
For n >= 0, a(2^n) = 1.

A342457 Terms of A342456 prime-shifted so far towards lower primes that they become even: a(n) = 2*A246277(A342456(n)).

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 6, 6, 2, 4, 64, 16, 324, 36, 10, 36, 2, 4, 64, 16, 2304, 96, 486, 24, 7290, 104976, 21600, 1296, 1708593750000, 100, 93750, 10, 2, 4, 64, 16, 144, 6, 216, 6, 172186884, 7776, 2160, 216, 216000000, 236196, 10497600, 54, 10935000000000, 53144100, 1476225000000, 7290, 122500000000, 10935000, 140, 360

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

These terms have the same prime signature as the corresponding terms in A342456, thus applying omega and bigomega to these gives the same derived sequences A342461 and A342462.

Cf. A246277, A276086, A283980, A329038, A329886, A342456, A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342464 [= A051903(a(n))].

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A342457(n) = 2*A246277(A342456(n)); \\ Uses also code from A342456.

a(n) = 2*A246277(A342456(n)) = 2*A329038(A329886(n)).

cp's OEIS Frontend

A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

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A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A342457 Terms of A342456 prime-shifted so far towards lower primes that they become even: a(n) = 2*A246277(A342456(n)).

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