A342462 -id:A342462 - 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).

A342463 a(n) = A342001(A342456(n)); "wild part" of the arithmetic derivative of A342456(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 12, 8, 1, 2, 6, 4, 50, 24, 16, 16, 1, 2, 6, 4, 126, 62, 46, 26, 1486, 100, 1142, 48, 2056, 32, 342, 10, 1, 2, 6, 4, 94, 24, 72, 18, 242, 120, 1588, 54, 3408, 92, 1740, 22, 6846, 2972, 4340, 766, 5048, 1374, 652, 376, 71156, 22710, 20390, 64, 738580, 4272, 568, 20, 1, 2, 6, 4, 264, 12, 196, 8, 318

Offset: 0

Antti Karttunen, Mar 15 2021

Like in A342462, also here the subsequences starting at each n = 2^k seem to be slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, ...

Cf. A003415, A003557, A005940, A108951, A329886, A342001, A342002, A342456, A342462, A342920.

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[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &@ f@ a[#] &, 73, 0]] (* Michael De Vlieger, Mar 17 2021 *)

\\ Needs also code from A342456.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A342463(n) = A342001(A342456(n));

a(n) = A342001(A342456(n)) = A342002(A329886(n)) = A342920(A005940(1+n)).

A342461 Number of nonzero 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, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 2, 5, 4, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 3, 3, 2, 4, 3, 4, 3, 4, 3, 4, 2, 3, 4, 4, 3, 4, 3, 4, 3, 3, 4

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

Cf. A001221, A005940, A108951, A267263, A329886, A342456, A342457, A342462.

Cf. also A324341, A329040.

A342461(n) = omega(A342456(n)); \\ Other code as in A342456.

a(n) = A001221(A342456(n)) = A001221(A342457(n)).
a(n) = A267263(A329886(n)) = A329040(A005940(1+n)).
a(n) <= A342462(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)).

A342464 Largest digit value 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, 1, 1, 1, 2, 6, 4, 4, 2, 1, 2, 1, 2, 6, 4, 8, 5, 5, 3, 6, 8, 5, 4, 11, 2, 6, 1, 1, 2, 6, 4, 4, 1, 3, 1, 16, 5, 4, 3, 9, 10, 8, 3, 10, 12, 10, 6, 10, 7, 2, 3, 18, 10, 5, 4, 12, 2, 4, 2, 1, 2, 6, 4, 13, 12, 10, 8, 12, 8, 13, 2, 4, 6, 7, 2, 15, 15, 12, 10, 9, 8, 7, 6, 10, 12, 10, 9, 11, 6, 9, 6, 18, 15

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

Cf. A005940, A051903, A108951, A276086, A328114, A329344, A329886, A342456, A342461, A342462.

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)}; \\ From A283980
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A342464(n) = A328114(A329886(n));

a(n) = A328114(A329886(n)) = A051903(A342456(n)) = A329344(A005940(1+n)).

A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).

Original entry on oeis.org

0, 2, 7, 18, 35, 62, 98, 148, 210, 290, 385, 502, 637, 798, 980, 1192, 1428, 1698, 1995, 2330, 2695, 3102, 3542, 4028, 4550, 5122, 5733, 6398, 7105, 7870, 8680, 9552, 10472, 11458, 12495, 13602, 14763, 15998, 17290, 18660, 20090, 21602, 23177, 24838, 26565, 28382, 30268, 32248, 34300, 36450

Offset: 0

Petros Hadjicostas, Mar 10 2021

nonn

One-half of the antidiagonal sums of array A220508.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A004126, A169607, A220508, A342462.

CoefficientList[Series[(2x^2+3x+2) x/((x+1)^2 (x-1)^4),{x,0,70}],x] (* or *) LinearRecurrence[{2,1,-4,1,2,-1},{0,2,7,18,35,62},70] (* Harvey P. Dale, Jul 08 2023 *)

/* First program */
seq1(n)={my(x='x+O('x^n)); Vec((2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4), -n)}
/* Second program (array M is A220508) */
seq2(nn) = {my(M=matrix(nn+1, nn+1)); my(a=vector(nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, n^2-n, if(k < n, n^2-2*n+k, k^2-n)))); for(n=1, nn+1, a[n] = sum(k=1, n, M[n-k+1,k])/2); a}

a(n) = (n+1)*(1 - (-1)^n)/16 + (7/4)*(binomial(n+3, 3) - binomial(n+2, 2)).
a(n) = (A342362(n) - (n + 1))/4.
a(2*n) = A169607(n) and a(2*n + 1) = 2*A004126(n + 1).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5. - Chai Wah Wu, Mar 11 2021

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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A342463 a(n) = A342001(A342456(n)); "wild part" of the arithmetic derivative of A342456(n).

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A342461 Number of nonzero 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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A342464 Largest digit value when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A342397 Expansion of the o.g.f. (2*x^2 + 3*x + 2)*x/((x + 1)^2*(x - 1)^4).

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A342397 Expansion of the o.g.f. (2x^2 + 3x + 2)x/((x + 1)^2(x - 1)^4).