A007947 -id:A007947 - cp's OEIS frontend

A347960 Numbers k for which A348036(k) > A007947(k).

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 900, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1125, 1152, 1156, 1176, 1188, 1200, 1224, 1260, 1296

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Numbers k such that A348039(k) > 1.
Numbers k such that A348037(k) < A003557(k).
Numbers k such that A327564(k) > A348038(k).
Differs from A036785 and A338539 for the first time at n=20, where a(n) = 450, as A036785(20) = A338539(20) = 441 is not included in this sequence.

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038, A348039.

f[p_, e_] := p*(p + 1)^(e - 1); q[n_] := GCD[n, Times @@ f @@@ (fct = FactorInteger[n])] > Times @@ First /@ fct; Select[Range[1300], q] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
isA347960(n) = (A348036(n)>A007947(n));

A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

Original entry on oeis.org

12, 20, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 92, 99, 104, 116, 117, 124, 132, 136, 140, 148, 152, 153, 156, 164, 171, 172, 175, 176, 184, 188, 204, 207, 208, 212, 220, 228, 232, 236, 244, 248, 260, 261, 268, 272, 275, 276, 279, 280, 284, 292, 296, 297, 304, 308, 315, 316, 325, 328, 332, 333

Offset: 1

Michael De Vlieger, Feb 28 2023

Proper subsequence of A126706.

Numbers k such that there does not exist j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

			a(1) = 12, since 12/6 < 3.
a(2) = 20, since 20/10 < 5.
a(3) = 28, since 28/14 < 7.
a(4) = 40, since 40/10 < 5, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A013929, A024619, A119288, A126706, A355432, A360768.

Select[Select[Range[120], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 < #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
f(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[, 1]); f[2]); \\ A119288
isok(k) = !isprimepower(k) && !issquarefree(k) && (k/rad(k) < f(k)); \\ Michel Marcus, Mar 01 2023

This sequence is { k in A126706 : k/A007947(k) < A119288(k) } = A126706 \ A360768.

A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).

Original entry on oeis.org

1, 2, 5, 18, 68, 283, 1161, 4843, 19985, 83074, 349670, 1456458, 6107257, 25547835, 106115655, 440396113, 1833079809, 7642924612, 31705433101, 131711607956, 546283729493, 2257462298234, 9339325821411, 38593708318690, 159600066415313, 661371515924516, 2736805917843710

Offset: 0

Michael De Vlieger, Jun 16 2023

			a(0) = 1 since P(0) = 1 and 1 | 1.
a(1) = 2 since P(1) = 2 and both 1 | 2 and 2 | 2.
a(2) = 5 since P(2) = 6 and rad(m) | 6 for m = {1, 2, 3, 4, 6}.
a(3) = 18 since P(3) = 30 and rad(m) | 30 for m = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30}, etc.
Regarding a(3), we see that there are 18 terms in the tensor product of prime power ranges of 2, 3, and 5 that do not exceed 30:
5^0X | 2^0 2^1 2^2 2^3 2^4    5^1X | 2^0 2^1 2^2    5^2X | 2^0
--------------------------    ------------------    ----------
3^0  |   1   2   4   8  16    3^0  |   5  10  20    3^0  |  25
3^1  |   3   6  12  24        3^1  |  15  30
3^2  |   9  18
3^3  |  27
Hence, a(3) = 18. This approach proves handy for larger n.

Bert Dobbelaere, Python program

Cf. A002110, A007947, A010846.

f[1] = 1; f[n_] := Function[w,
ToExpression@ StringJoin["Block[{n = ", ToString@ n,
    ", k = 0}, Flatten@ Table[k++, ",
    Most@ Flatten@ Map[{#, ", "} &, #], "]; k]"] &@
      MapIndexed[
        Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[",
          ToString@ First@ p, ", n/(",
          ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]],
          ")]}"] ]@ w[[First@ #2]] &, w]]@
   Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &,
     FactorInteger[n][[All, 1]]];
   Map[f, FoldList[Times, 1, Prime@ Range@ 9] ]

a(n) = A010846(A002110(n)).

a(n) >= 2^n.

Corrected a(15) and added a(16)-a(23) from Bert Dobbelaere, Jun 27 2023

a(24)-a(26) from Martin Ehrenstein, Jul 08 2023

A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 4, 0, 1, 3, 0, 1, 2, 4, 5, 6, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 0, 1, 0, 1, 2, 4, 7, 8, 0, 1, 3, 5, 6, 9, 10, 12, 0, 1, 2, 4, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 0, 1, 0, 1, 2, 4, 5, 8, 10, 12, 16

Offset: 1

Michael De Vlieger, Mar 07 2025

Define S(p,n) to be the set of residues r (mod n) taken by the power range of prime divisor p, i.e., {p^m, m >= 1}.
Define T(n) to be the union of the tensor product of distinct terms in S(p,n) for all p|n, where the products are expressed mod n.
Row n of this triangle is T(n), a superset of row n of A381799.
For n > 1, the intersection of row n of this triangle and row n of A038566 is {1}.

			Table of c(n) = A381800(n) and T(n) for select n:
 n  c(n)  T(n)
-----------------------------------------------------------------------------
 1    1   {0}
 2    2   {0, 1}
 3    2   {0, 1}
 4    3   {0, 1, 2}
 5    2   {0, 1}
 6    5   {0, 1, 2, 3, 4}
 8    4   {0, 1, 2, 4}
 9    3   {0, 1, 3}
10    7   {0, 1, 2, 4, 5, 6, 8}
11    2   {0, 1}
12    8   {0, 1, 2, 3, 4, 6, 8, 9}
14    6   {0, 1, 2, 4, 7, 8}
15    8   {0, 1, 3, 5, 6, 9, 10, 12}
16    5   {0, 1, 2, 4, 8}
18   12   {0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16}
20    9   {0, 1, 2, 4, 5, 8, 10, 12, 16}
24   11   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
28    9   {0, 1, 2, 4, 7, 8, 14, 16, 21}
30   19   {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}
36   16   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32}
For n = 10, we have S(2,10) = {1, 2, 4, 6, 8} and S(5,10) = {1, 5}. Therefore we have the following distinct products:
   1  2  4  8  6
   5  0
Hence T(10) = {0, 1, 2, 4, 5, 6, 8}; terms in A003592 belong to these residues (mod 10).
For n = 12, we have S(2,12) = {1, 2, 4, 8} and S(3,12) = {1, 3, 9}. Therefore we have the following distinct products:
   1  2  4  8
   3  6  0
   9
Thus T(12) = {0, 1, 2, 3, 4, 6, 8, 9}, terms in A003586 belong to these residues (mod 12).
For n = 30, we have {1, 2, 4, 8, 16}, {1, 3, 9, 21, 27}, and {1, 5, 25}. Therefore we have the following distinct products:
   1  2  4  8  16         5  10  20         25
   3  6 12 24            15   0
   9 18
  27
Thus T(30) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}; terms in A051037 belong to these residues (mod 30).

Michael De Vlieger, Table of n, a(n) for n = 1..22906 (rows n = 1..500, flattened)
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..36, showing reduced residues mod n in gray and labeling terms in row n. The number n appears on the left in red italic, and row length A381800(n) in blue at right.
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..5000.

Cf. A007947, A038566, A121998, A162306, A381799, A381800.

Table[Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n], {n, 30}]

Row 1 is {0} since 1 is the empty product and the only number that has zero prime factors is 1, congruent to 0 (mod 1).
Row n begins with {0,1} for n > 1.
For prime p, row p = {0,1}.
For prime power p^m, m > 0, row p = union of {0} and {p^i, i < m}.
Row n is a subset of row n of A121998, considering n in A121998 instead as n mod n = 0.
Row n is a superset of row n of A162306, considering n in A162306 instead as n mod n = 0.

A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 49, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69, 70, 71, 36, 73

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) is a first power if and only if n is not a powerful number (A001694, A052485).

			n = 1800: squarefree kernel is 2*3*5 = 30 and a(1800) = 900 = 30^2 divides n, exponent of 30 is the smallest prime exponent of 1800 = 2*2*2*3*3*5*5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001694, A003557, A007947, A051904, A052485, A062759, A065463.

a062759 n = a007947 n ^ a051904 n  -- Reinhard Zumkeller, Jul 15 2012

{1}~Join~Table[#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 73}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] = e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmin(f[,2])); prod(i = 1, #f~, f[i,1]^e));} \\ Amiram Eldar, Feb 12 2023

a(n) = A007947(n)^A051904(n).
From Amiram Eldar, Feb 12 2023: (Start)
a(n) = n/A062759(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 / 2 = 0.352221... . (End)

A066087 a(n) = gcd(sigma(n), phi(n)) - gcd(sigma(rad(n)), phi(rad(n))); rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 1, -1, 0, -4, 0, 4, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -4, -2, 0, 0, 0, 0, -1, 0, 0, -4, 0, 0, 0, 18, 0, -2, 0, 2, 0, 0, 0, 2, 0, -3, 8, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 12, -6

Offset: 1

Labos Elemer, Dec 04 2001

sign

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048250, A023900, A000203, A007947, A000010, A009223, A066086.

Table[GCD[DivisorSigma[1, n], EulerPhi@ n] - GCD[DivisorSigma[1, #], EulerPhi@ #] &[Times @@ FactorInteger[n][[All, 1]]], {n, 120}] (* Michael De Vlieger, Feb 19 2017 *)

rad(f)=for(i=1,#f~,f[i,2]=1); f
g(f)=gcd(sigma(f),eulerphi(f))
a(n)=my(f=factor(n),k=rad(f)); g(f)-g(k) \\ Charles R Greathouse IV, Dec 09 2013

A009223(n) - A066086(n) = gcd(sigma(n), phi(n)) - gcd(sigma(A007947(n)), phi(A007947(n))).

A075425 Number of steps to reach 1 starting with n and iterating the map n ->rad(n)-1, where rad(n) is the squarefree kernel of n (A007947).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

Sequence is defined for all n, as A075423(n) < n.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

a075425 n = snd $ until ((== 1) . fst)
                        (\(x, i) -> (a075423 x, i + 1)) (n, 0)
-- Reinhard Zumkeller, Aug 14 2013

rad(n)=vecprod(factor(n)[,1])
a(n)=my(k);while(n>1,n=rad(n)-1;k++); k \\ Charles R Greathouse IV, Jul 09 2013

A078313 Number of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 3, 2, 2, 4, 1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 4, 1, 2, 2, 3, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 1, 2, 1, 3, 1, 3, 3, 2, 2, 3, 2, 3

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

a(n)=A001221(A078310(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A078314, A078315.

a078313 = a001221 . a078310  -- Reinhard Zumkeller, Jul 23 2013

A078313[n_] := PrimeNu[n*Times @@ FactorInteger[n][[All, 1]] + 1]; Array[A078313, 50] (* G. C. Greubel, Apr 25 2017 *)

a(n)=omega(n*vecprod(factor(n)[,1])+1) \\ Charles R Greathouse IV, Jul 09 2013

A078314 Total number of prime factors of n*rad(n)+1 counted with multiplicity, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A007947, A078310, A078313, A078315.

a078314 = a001222 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := PrimeOmega[1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n)=bigomega(n*vecprod(factor(n)[,1])+1) \\ Charles R Greathouse IV, Jul 09 2013

a(n) = A001222(A078310(n)).

A078315 Minimum exponent in prime factorization of n*rad(n)+1, where rad = A007947 (the radical or squarefree kernel).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

2 = a(4) = a(45) = a(48) = a(140) = a(529) = a(682) = a(3264) = a(3564) = a(4680) = a(4756) = a(166320) = a(194873) = a(330096) = a(364905) = a(2100332) = a(4160200) with all terms in between equal to 1. Is there an n with a(n) > 2? - Charles R Greathouse IV, May 20 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A051904, A078310, A078314, A078316.

a078315 = a051904 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Min[FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]][[;;, 2]]]; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n)=my(f=factor(n));f[,2]=apply(n->n+1,f[,2]);vecmin(factor(factorback(f)+1)[,2]) \\ Charles R Greathouse IV, May 20 2013

a(n) = A051904(A078310(n)).

cp's OEIS Frontend

A347960 Numbers k for which A348036(k) > A007947(k).

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A360767 Numbers k that are neither prime power nor squarefree, such that k/rad(k) < q, where rad(k) = A007947(k) and prime q = A119288(k).

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A363061 Number of k <= P(n) such that rad(k) | P(n), where rad(n) = A007947(n) and P(n) = A002110(n).

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A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.

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A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

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A066087 a(n) = gcd(sigma(n), phi(n)) - gcd(sigma(rad(n)), phi(rad(n))); rad = A007947.

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A075425 Number of steps to reach 1 starting with n and iterating the map n ->rad(n)-1, where rad(n) is the squarefree kernel of n (A007947).

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A078313 Number of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).

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