A258851 -id:A258851 - cp's OEIS frontend

A258861 The pi-based antiderivative of n: the least m such that A258851(m) equals n.

0, 2, 3, 5, 4, 11, 13, 6, 19, 23, 29, 10, 8, 41, 43, 14, 53, 59, 61, 15, 12, 22, 79, 83, 89, 26, 21, 103, 107, 109, 25, 34, 16, 18, 139, 38, 151, 33, 163, 167, 173, 35, 181, 191, 28, 197, 199, 211, 223, 58, 229, 233, 24, 30, 27, 51, 49, 269, 55, 277, 281, 74

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=1 of A259016, A259153.

Cf. A000040, A000720, A258851, A258862, A258995.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

A258851[n_] := If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
a[n_] := For[m = 0, True, m++, If[A258851[m] == n, Return[m]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 10 2023 *)

a(n) = min { m >= 0 : A258851(m) = n }.
A258851(a(n)) = n.
a(n) <= A000040(n) for n>0.

A258862 Second pi-based antiderivative of n: the least m such that A258851^2(m) equals n.

Original entry on oeis.org

0, 3, 5, 11, 4, 10, 41, 13, 15, 83, 109, 29, 19, 35, 191, 43, 30, 277, 74, 14, 8, 42, 77, 431, 461, 21, 22, 563, 66, 599, 26, 78, 12, 61, 141, 163, 877, 18, 214, 218, 226, 38, 114, 201, 105, 1201, 215, 1297, 302, 55, 1447, 1471, 89, 25, 103, 170, 58, 291, 51

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=2 of A259016.

Cf. A000040, A000720, A258851, A258852, A258861, A258995.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(t));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

d[n_] := d[n] = If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]];
A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
a[n_] := A[n, 2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

a(n) = min { m >= 0 : A258851^2(m) = n }.
A258851^2(a(n)) = A258852(a(n)) = n.
a(n) <= A000040^2(n) for n>0.
a(n) <= A258861^2(n); a(21) = 42 < A258861^2(21) = A258861(22) = 79; A258851^2(42) = A258851^2(79) = 21.

A258995 Third pi-based antiderivative of n: the least m such that A258851^3(m) equals n.

Original entry on oeis.org

0, 5, 11, 10, 4, 29, 35, 41, 14, 431, 599, 78, 15, 38, 201, 191, 25, 382, 186, 43, 19, 65, 94, 3001, 535, 22, 42, 633, 317, 4397, 21, 141, 8, 74, 574, 214, 1286, 61, 253, 247, 1417, 163, 115, 217, 66, 546, 138, 10631, 1997, 51, 12097, 12301, 362, 26, 563, 1013

Offset: 0

Alois P. Heinz, Jun 16 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=3 of A259016.

Cf. A000040, A000720, A258851, A258853, A258861, A258862.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= d(d(d(t)));
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

d[n_] := d[n] = If[n == 0, 0, n*Total[Last[#]*PrimePi[First[#]]/First[#] & /@ FactorInteger[n]]];
A[n_, k_] := For[m = 0, True, m++, If[Nest[d, m, k] == n, Return[m]]];
a[n_] := A[n, 3];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 17 2024 *)

a(n) = min { m >= 0 : A258851^3(m) = n }.
A258851^3(a(n)) = A258853(a(n)) = n.
a(n) <= A000040^3(n) for n>0.
a(n) <= A258861^3(n).

A278510 a(n) = A258851(n) - A056239(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 9, 8, 7, 0, 16, 0, 10, 14, 28, 0, 28, 0, 27, 20, 15, 0, 47, 24, 18, 48, 38, 0, 47, 0, 75, 30, 23, 34, 78, 0, 26, 36, 78, 0, 66, 0, 57, 80, 31, 0, 122, 48, 78, 46, 68, 0, 128, 50, 109, 52, 38, 0, 129, 0, 41, 112, 186, 60, 99, 0, 87, 62, 109, 0, 197, 0, 48, 132, 98, 70, 118, 0, 201, 208, 53, 0, 180, 76, 56, 76, 164, 0, 211, 84, 117, 82, 61

Offset: 1

Antti Karttunen, Dec 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A056239, A258851.

Cf. A008578 (positions of zeros).

a(n) = A258851(n) - A056239(n).

A353575 Primepi based arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A258851(A181819(A276086(n))).

Original entry on oeis.org

0, 1, 1, 4, 2, 7, 1, 4, 4, 12, 7, 20, 2, 7, 7, 20, 12, 33, 3, 11, 11, 32, 19, 53, 4, 15, 15, 44, 26, 73, 1, 4, 4, 12, 7, 20, 4, 12, 12, 32, 20, 52, 7, 20, 20, 52, 33, 84, 11, 32, 32, 84, 53, 136, 15, 44, 44, 116, 73, 188, 2, 7, 7, 20, 12, 33, 7, 20, 20, 52, 33, 84, 12, 33, 33, 84, 54, 135, 19, 53, 53, 136, 87, 219

Offset: 0

Antti Karttunen, Apr 29 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A181819, A258851, A276086, A328835, A353379.

Cf. also A342002, A353574 and A353576.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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
A353379(n) = A258851(A181819(n));
A353575(n) = A353379(A276086(n));

a(n) = A353379(A276086(n)) = A258851(A328835(n)).

A354871 a(n) = gcd(A056239(n), A258851(n)).

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 4, 6, 5, 1, 4, 7, 1, 8, 1, 2, 3, 9, 1, 6, 1, 6, 2, 10, 1, 11, 5, 1, 1, 1, 6, 12, 1, 4, 6, 13, 1, 14, 1, 1, 1, 15, 2, 8, 1, 1, 4, 16, 1, 2, 1, 2, 1, 17, 1, 18, 1, 8, 6, 3, 1, 19, 3, 1, 1, 20, 1, 21, 1, 4, 2, 1, 1, 22, 1, 8, 1, 23, 4, 2, 1, 4, 4, 24, 1, 2, 1, 1, 1, 1, 1, 25, 1, 3, 4

Offset: 1

Antti Karttunen, Jun 11 2022

nonn

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

Cf. A056239, A258851, A278510, A354872, A354873.

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354871(n) = gcd(A056239(n), A258851(n));

a(n) = gcd(A056239(n), A258851(n)).
a(n) = gcd(A056239(n), A278510(n)) = gcd(A258851(n), A278510(n)).
For n > 1, a(n) = A056239(n) / A354872(n) = A258851(n) / A354873(n).

A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

1, 4, 2, 7, 12, 3, 12, 19, 30, 4, 11, 54, 41, 56, 5, 20, 26, 225, 79, 110, 6, 15, 87, 58, 588, 131, 156, 7, 32, 37, 310, 94, 1815, 193, 238, 8, 33, 216, 69, 861, 162, 3042, 269, 304, 9, 32, 140, 1500, 117, 2156, 218, 6069, 355, 414, 10, 21, 120, 427, 5488, 183, 3835, 314, 8664, 491, 580, 11, 52, 44, 455, 1254, 26620, 255, 6834, 422, 14283, 629, 682, 12

Offset: 1

Antti Karttunen, Jul 29 2022

Each column is strictly monotonic.

			The top left corner of the array:
   k =  1    2    3      4    5      6    7       8      9     10   11       12
  2k =  2    4    6      8   10     12   14      16     18     20   22       24
-----+--------------------------------------------------------------------------
n= 1 |  1,   4,   7,    12,  11,    20,  15,     32,    33,    32,  21,      52,
   2 |  2,  12,  19,    54,  26,    87,  37,    216,   140,   120,  44,     351,
   3 |  3,  30,  41,   225,  58,   310,  69,   1500,   427,   455,  86,    2075,
   4 |  4,  56,  79,   588,  94,   861, 117,   5488,  1254,  1022, 132,    8183,
   5 |  5, 110, 131,  1815, 162,  2156, 183,  26620,  2561,  2717, 214,   31581,
   6 |  6, 156, 193,  3042, 218,  3835, 255,  52728,  4828,  4316, 304,   67093,
   7 |  7, 238, 269,  6069, 314,  6834, 373, 137564,  7695,  8075, 404,  154615,
   8 |  8, 304, 355,  8664, 422, 10241, 457, 219488, 12098, 12426, 524,  261003,
   9 |  9, 414, 491, 14283, 532, 17296, 609, 438012, 20909, 18653, 668,  535877,
  10 | 10, 580, 629, 25230, 718, 27231, 787, 975560, 29388, 31552, 836, 1050409,

Cf. A000027 (column 1), A097240 (column 3), A246278, A258851.

Cf. also A344027.

up_to = 78;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A356155sq(row,col) = A258851(A246278sq(row,col));
A356155list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A356155sq(col,(a-(col-1))))); (v); };
v356155 = A356155list(up_to);
A356155(n) = v356155[n];

A354872 a(n) = A056239(n) / gcd(A056239(n), A258851(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 1, 5, 1, 5, 3, 2, 1, 5, 1, 7, 1, 3, 1, 6, 1, 1, 7, 8, 7, 1, 1, 9, 2, 1, 1, 7, 1, 7, 7, 10, 1, 3, 1, 7, 9, 2, 1, 7, 4, 7, 5, 11, 1, 7, 1, 12, 1, 1, 3, 8, 1, 3, 11, 8, 1, 7, 1, 13, 2, 5, 9, 9, 1, 7, 1, 14, 1, 2, 5, 15, 3, 2, 1, 8, 5, 11, 13, 16, 11, 7, 1, 9, 3, 2, 1, 10

Offset: 2

Antti Karttunen, Jun 11 2022

Numerator of fraction A056239(n) / A258851(n).

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A258851, A278510, A354871, A354873 (denominators).

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354872(n) = { my(u=A056239(n)); (u/gcd(u, A258851(n))); };
\\ Or:
A354872(n) = numerator(A056239(n)/A258851(n));

a(n) = A056239(n) / A354871(n) = A056239(n) / gcd(A056239(n), A258851(n)).

A354873 a(n) = A258851(n) / gcd(A056239(n), A258851(n)).

Original entry on oeis.org

1, 1, 2, 1, 7, 1, 4, 3, 11, 1, 5, 1, 3, 19, 8, 1, 33, 1, 32, 13, 7, 1, 52, 5, 25, 9, 22, 1, 53, 1, 16, 37, 31, 41, 14, 1, 35, 11, 14, 1, 73, 1, 64, 87, 41, 1, 64, 7, 85, 55, 19, 1, 135, 29, 116, 31, 49, 1, 136, 1, 53, 15, 32, 23, 107, 1, 32, 73, 117, 1, 204, 1, 61, 35, 54, 79, 127, 1, 208, 27, 67, 1, 47, 43, 71, 22

Offset: 2

Antti Karttunen, Jun 11 2022

Denominator of fraction A056239(n) / A258851(n).

Antti Karttunen, Table of n, a(n) for n = 2..16384
Index entries for sequences computed from indices in prime factorization

Cf. A056239, A258851, A278510, A354871, A354872 (numerators).

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A354873(n) = { my(u=A258851(n)); (u/gcd(u, A056239(n))); };
\\ Or:
A354873(n) = denominator(A056239(n)/A258851(n));

a(n) = A258851(n) / A354871(n) = A258851(n) / gcd(A056239(n), A258851(n)).

A258863 First differences of the pi-based arithmetic derivative sequence A258851.

Original entry on oeis.org

0, 1, 1, 2, -1, 4, -3, 8, 0, -1, -6, 15, -14, 9, 4, 13, -25, 26, -25, 24, -6, -5, -12, 43, -22, -5, 29, -10, -34, 43, -42, 69, -43, -6, 10, 43, -72, 23, 9, 40, -71, 60, -59, 50, 23, -46, -26, 113, -72, 29, -30, 21, -60, 119, -77, 58, -54, -13, -32, 119, -118

Offset: 0

Alois P. Heinz, Jun 12 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000720, A258851.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
a:= n-> d(n+1)-d(n):
seq(a(n), n=0..100);

a(n) = A258851(n+1) - A258851(n).

cp's OEIS Frontend

A258861 The pi-based antiderivative of n: the least m such that A258851(m) equals n.

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A258862 Second pi-based antiderivative of n: the least m such that A258851^2(m) equals n.

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A258995 Third pi-based antiderivative of n: the least m such that A258851^3(m) equals n.

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A278510 a(n) = A258851(n) - A056239(n).

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A353575 Primepi based arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A258851(A181819(A276086(n))).

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A354871 a(n) = gcd(A056239(n), A258851(n)).

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A356155 The pi-based arithmetic derivative applied to prime shift array: Square array A(n,k) = A258851(A246278(n,k)), read by falling antidiagonals.

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A354872 a(n) = A056239(n) / gcd(A056239(n), A258851(n)).

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A354873 a(n) = A258851(n) / gcd(A056239(n), A258851(n)).

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