A258851 -id:A258851 - cp's OEIS frontend

A259155 Sum of all values v whose pi-based arithmetic derivative equals n.

1, 2, 3, 5, 11, 11, 13, 23, 19, 23, 29, 41, 54, 41, 43, 61, 53, 59, 61, 82, 83, 95, 79, 83, 89, 123, 122, 103, 107, 109, 138, 161, 167, 155, 139, 187, 151, 190, 163, 167, 173, 260, 181, 191, 260, 197, 199, 211, 223, 285, 229, 233, 263, 333, 278, 308, 312, 269

Offset: 0

Alois P. Heinz, Jun 19 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Row sums of A259153.

Cf. A000040, A000720, A258851, A259154.

a(n) = Sum_{v>=0, A258851(v)=n} v.
a(n) = Sum_{k=1..A259154(n)} A259153(n,k).
a(n) >= A000040(n) for n>0.

A259169 a(n) = n-th pi-based antiderivative of 8.

Original entry on oeis.org

8, 19, 15, 14, 43, 191, 201, 217, 1113, 1239, 986, 925, 375, 526, 689, 998, 3642, 3966, 5299, 4090, 7363, 20942, 150161, 117915, 218218, 597199, 472182, 494550, 1075362, 796042, 310086, 444985, 1403783, 1578955, 2702706, 10010173

Offset: 0

Alois P. Heinz, Jun 19 2015

nonn

			a(7) = 217 -> 201 -> 191 -> 43 -> 14 -> 15 -> 19 -> 8.
a(8) = 1113 -> 1714 -> 1153 -> 191 -> 43 -> 14 -> 15 -> 19 -> 8.

Row n=8 of A259016.

Cf. A000720, A258848, A258851.

a(n) = min { m >= 0 : A258851^n(m) = 8 }.

A259409 The pi-based arithmetic derivative of the double factorial of n.

Original entry on oeis.org

0, 0, 1, 2, 12, 19, 128, 193, 1600, 2997, 20224, 37692, 319488, 552366, 5164032, 10853055, 103268352, 198691110, 2199453696, 4050806490, 49934499840, 102089892240, 1176592711680, 2471811316695, 32489204613120, 71282307214125, 893769083781120, 2351538388135125

Offset: 0

Alois P. Heinz, Jun 26 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A006882, A258845, A258851.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-2)*n+doublefactorial(n-2)*d(n))
    end:
seq(a(n), n=0..40);

d[n_] := n*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[n_] := a[n] = If[n < 2, 0, a[n-2]*n + (n-2)!!*d[n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

a(n) = A258851(n!!) = A258851(A006882(n)).

A258847 Sum of the k-th pi-based arithmetic derivative of n-k for k=0..n.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 13, 20, 21, 33, 54, 86, 146, 339, 788, 2947, 14870, 94801, 706961, 5566784, 43958933, 317950465, 2406052444, 19645433193, 146175038733, 1479263447899, 16135114175706, 203382520812382, 2606355260220040, 32974597626726301, 406609097787758227

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Antidiagonal sums of A258850.

Cf. A000720, A258851.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= proc(n) option remember; add(A(h, n-h), h=0..n) end:
seq(a(n), n=0..30);

a(n) = Sum_{k=0..n} A258850(n-k,k).

A258849 The n-th pi-based arithmetic derivative of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 4, 4, 12288, 81920, 0, 0, 278380544, 4, 4931584, 278380544, 14768867966976, 4, 128412352512, 14768867966976, 375877192068366336, 14768867966976, 14768867966976, 375877192068366336

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Main diagonal of A258850.

Cf. A000720, A258851.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= n-> A(n$2):
seq(a(n), n=0..23);

a(n) = A258851^n(n) = A258850(n,n).

A259168 a(n) = n-th pi-based antiderivative of 7.

Original entry on oeis.org

7, 6, 13, 41, 35, 38, 163, 138, 253, 346, 1383, 630, 4657, 3210, 5633, 9469, 20838, 22525, 28491, 21035, 16491, 13735, 22114, 54298, 225361, 639070, 479794, 421883, 720634, 461055, 788446, 650762, 688229, 478126, 1465550, 1960533, 2117157

Offset: 0

Alois P. Heinz, Jun 19 2015

nonn

Row n=7 of A259016.

Cf. A000720, A258851.

a(n) = min { m >= 0 : A258851^n(m) = 7 }.

A259170 a(n) = n-th pi-based antiderivative of 9.

Original entry on oeis.org

9, 23, 83, 431, 3001, 27457, 10626, 112087, 87306, 172810, 280217, 390133, 353555, 750243, 1318106, 937023, 367542, 2615090, 3434663, 3281065, 3270774, 4979697, 8021665, 4627825, 5618666, 11169397

Offset: 0

Alois P. Heinz, Jun 19 2015

nonn

Row n=9 of A259016.

Cf. A000720, A258851.

a(n) = min { m >= 0 : A258851^n(m) = 9 }.

A259416 The pi-based arithmetic derivative of the n-th Fibonacci number.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 12, 6, 26, 31, 58, 24, 480, 51, 304, 851, 1537, 251, 6028, 2466, 13789, 12657, 8870, 3121, 222384, 100805, 49405, 290641, 536006, 42613, 2666244, 446497, 3716882, 3111635, 1694821, 10652139, 76062672, 16224451, 27896232, 51359105, 282500593

Offset: 0

Alois P. Heinz, Jun 26 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..58

Cf. A000045, A258851.

a(n) = A258851(A000045(n)).

A353509 a(n) = A353379(n) - A001222(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 2, 2, 0, 0, 4, 0, 4, 2, 2, 0, 7, 0, 2, 0, 4, 0, 9, 0, 0, 2, 2, 2, 8, 0, 2, 2, 7, 0, 9, 0, 4, 4, 2, 0, 10, 0, 4, 2, 4, 0, 7, 2, 7, 2, 2, 0, 16, 0, 2, 4, 0, 2, 9, 0, 4, 2, 9, 0, 14, 0, 2, 4, 4, 2, 9, 0, 10, 0, 2, 0, 16, 2, 2, 2, 7, 0, 16, 2, 4, 2, 2, 2, 15, 0, 4, 4, 8, 0, 9

Offset: 1

Antti Karttunen, Apr 29 2022

nonn

The difference of A258851 (primepi-based arithmetic derivative) and A056239 (sum of prime indices with multiplicity) applied to A181819, the prime shadow of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A056239, A181819, A258851, A278510, A353379.

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A353509(n) = (A258851(A181819(n))-bigomega(n));

a(n) = A278510(A181819(n)) = A353379(n) - A001222(n).

cp's OEIS Frontend

A259155 Sum of all values v whose pi-based arithmetic derivative equals n.

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A259169 a(n) = n-th pi-based antiderivative of 8.

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A259409 The pi-based arithmetic derivative of the double factorial of n.

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Maple

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A258847 Sum of the k-th pi-based arithmetic derivative of n-k for k=0..n.

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Maple

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A258849 The n-th pi-based arithmetic derivative of n.

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Maple

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A259168 a(n) = n-th pi-based antiderivative of 7.

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A259170 a(n) = n-th pi-based antiderivative of 9.

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A259416 The pi-based arithmetic derivative of the n-th Fibonacci number.

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A353509 a(n) = A353379(n) - A001222(n).

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PARI

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