A078899 -id:A078899 - cp's OEIS frontend

A303759 Number of times the largest prime power factor of n (A034699) is largest prime power factor for numbers <= n; a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 4, 1, 5, 1, 1, 3, 2, 5, 3, 1, 2, 3, 3, 1, 6, 1, 4, 4, 2, 1, 2, 1, 2, 3, 4, 1, 2, 5, 4, 3, 2, 1, 6, 1, 2, 5, 1, 5, 6, 1, 4, 3, 7, 1, 6, 1, 2, 3, 4, 7, 6, 1, 3, 1, 2, 1, 8, 5, 2, 3, 8, 1, 7, 7, 4, 3, 2, 5, 2, 1, 2, 9, 4, 1, 6, 1, 8, 9

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of A034699.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961 (positions of ones), A034699.

Cf. also A078899, A284600, A302789.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= max(1, seq(i[1]^i[2], i=ifactors(n)[2]));
      b(t):= b(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018

f[n_] := Max[Power @@@ FactorInteger[n]];
b[_] = 0;
a[n_] := With[{t = f[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Jan 03 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d))));
v303759 = ordinal_transform(vector(up_to,n,A034699(n)));
A303759(n) = v303759[n];

A304734 Ordinal transform of the largest prime factor of the n-th term of EKG-sequence.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 18 2018

Ordinal transform of A304733(n) = A006530(A064413(n)).

Antti Karttunen, Table of n, a(n) for n = 1..20090
Index entries for sequences related to EKG sequence

Cf. A006530, A064413, A064741, A304733.

Cf. also A078899.

terms = 105;
ekGrapher[s_List] := Block[{m = s[[-1]], k = 3}, While[MemberQ[s, k] || GCD[m, k] == 1, k++]; Append[s, k]];
A304733 = FactorInteger[#][[-1, 1]]& /@ Nest[ekGrapher, {1, 2}, terms];
b[_] = 0;
a[n_] := a[n] = With[{t = A304733[[n]]}, b[t] = b[t]+1];
Array[a, terms] (* Jean-François Alcover, Dec 20 2021, after Robert G. Wilson v in A064413 *)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A304733(n) = A006530(A064413(n)); \\ Needs also code for A064413.
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v304734 = ordinal_transform(vector(65539,n,A304733(n)));
A304734(n) = v304734[n];

A331296 Number of values of k, 1 <= k <= n, with A263297(k) = A263297(n), where A263297(n) = max(A001222(n), A061395(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 18 2020

nonn

Ordinal transform of A263297.

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

Cf. A001222, A061395, A263297.

Cf. also A078899.

A263297[n_] := If[n == 1, 0, With[{f = FactorInteger[n]}, Max[PrimePi[Max[f[[All, 1]]]], Total[f[[All, 2]]]]]];
b[_] = 0;
a[n_] := With[{t = A263297[n]}, b[t] = b[t] + 1];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A263297(n) = if(n<2, 0,  my(f=factor(n)); max(vecsum(f[, 2]), primepi(f[#f~, 1]))); \\ From A263297
v331296 = ordinal_transform(vector(up_to, n, A263297(n)));
A331296(n) = v331296[n];

A385653 Least k such that A385652(k) = n.

Original entry on oeis.org

2, 4, 8, 12, 18, 24, 27, 36, 48, 54, 72, 80, 90, 100, 120, 125, 135, 150, 160, 180, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980

Offset: 1

Pontus von Brömssen, Jul 06 2025

nonn

A385654(n) is uniquely popular on the interval [2,a(n)]; see A289662.

Equivalently, a(n) is the least k >= 2 such that A078899(k) = n.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A006530, A078899, A289662, A385503, A385652, A385654.

gpf(n) = if (n==1,1, vecmax(factor(n)[,1])); \\ A006530
f(n) = my(v=vector(n, k, gpf(k)), s=Set(v)); vecmax(apply(x->#x, vector(#s, i, select(x->(x==s[i]), v)))); \\ A385652
a(n) = my(k=2); while (f(k) !=n, k++); k; \\ Michel Marcus, Jul 06 2025

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
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See Links section.
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a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

A328638 a(n) is the sum of m such that 2 <= m <= n and gpf(m) = gpf(n), where gpf(i) is the greatest prime factor of i (A006530), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 9, 7, 14, 18, 15, 11, 30, 13, 21, 30, 30, 17, 48, 19, 50, 42, 33, 23, 72, 75, 39, 99, 70, 29, 105, 31, 62, 66, 51, 105, 135, 37, 57, 78, 145, 41, 147, 43, 110, 190, 69, 47, 183, 196, 240, 102, 130, 53, 237, 165, 252, 114, 87, 59, 300

Offset: 1

J. Stauduhar, Oct 22 2019

For n >= 2, a(n) is the sum of the terms that precede n on the row, of the A125624 array, that contains n.

			5, 10, 15 and 20 have same gpf as 20, so a(20) = 5 + 10 + 15 + 20 = 50.

Cf. A006530 (gpf), A125624, A078899, A328637.

gpf(n)={my(f=factor(n)[,1]); f[#f]}
a(n)={if(n<=1, n==1, my(t=gpf(n)); sum(i=2, n, if(gpf(i)==t, i, 0)))} \\ Andrew Howroyd, Oct 28 2019

A331295 Number of values of k, 1 <= k <= n, with f(k) = f(n), where f(n) = [A001222(n), A061395(n)].

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 3, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 4, 2, 1, 1, 3, 3, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 4, 2, 1, 1, 4, 1, 1, 2, 3, 1, 2, 1, 1, 3, 4, 1, 2, 1, 3, 1, 1, 5, 3, 4, 1, 2, 1, 1, 6

Offset: 1

Antti Karttunen, Jan 19 2020

nonn

Ordinal transform of A331298, or equally, of the ordered pair [A001222(n), A061395(n)].

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001222, A061395, A078899, A331296, A331297, A331298.

f[n_] := f[n] = {PrimeOmega[n], PrimePi[FactorInteger[n]][[-1, 1]]};
a[n_] := Count[Array[f, n], f[n]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)

up_to = 1001;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
Aux331298(n) = [bigomega(n), A061395(n)];
v331295 = ordinal_transform(vector(up_to, n, Aux331298(n)));
A331295(n) = v331295[n];

A377730 Number of integers less than n that have the same greatest prime factor as n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 3, 0, 1, 2, 3, 0, 4, 0, 3, 2, 1, 0, 5, 4, 1, 6, 3, 0, 5, 0, 4, 2, 1, 4, 7, 0, 1, 2, 6, 0, 5, 0, 3, 7, 1, 0, 8, 6, 8, 2, 3, 0, 9, 4, 7, 2, 1, 0, 9, 0, 1, 8, 5, 4, 5, 0, 3, 2, 9, 0, 10, 0, 1, 10, 3, 6, 5, 0, 11, 11, 1, 0, 10, 4, 1, 2, 7, 0, 12

Offset: 1

Ilya Gutkovskiy, Nov 05 2024

nonn

Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A047983, A078899, A334655, A335097, A377734.

R:= NULL:
for n from 1 to 100 do  p:= max(numtheory:-factorset(n));  if assigned(C[p]) then C[p]:= C[p]+1 else C[p]:= 0 fi;
  R:= R, C[p]
od:R; # Robert Israel, Nov 07 2024

Table[Length[Select[Range[n - 1], FactorInteger[#][[-1, 1]] == FactorInteger[n][[-1, 1]] &]], {n, 90}]

a(n) = |{j < n : gpf(j) = gpf(n)}|.

a(n) = A078899(n) - 1.

cp's OEIS Frontend

A303759 Number of times the largest prime power factor of n (A034699) is largest prime power factor for numbers <= n; a(1) = 1.

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A304734 Ordinal transform of the largest prime factor of the n-th term of EKG-sequence.

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A331296 Number of values of k, 1 <= k <= n, with A263297(k) = A263297(n), where A263297(n) = max(A001222(n), A061395(n)).

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A385653 Least k such that A385652(k) = n.

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A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

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A328638 a(n) is the sum of m such that 2 <= m <= n and gpf(m) = gpf(n), where gpf(i) is the greatest prime factor of i (A006530), with a(1) = 1.

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A331295 Number of values of k, 1 <= k <= n, with f(k) = f(n), where f(n) = [A001222(n), A061395(n)].

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A377730 Number of integers less than n that have the same greatest prime factor as n.

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