A279513 -id:A279513 - cp's OEIS frontend

A284761 a(n) = gcd(A279513(n), A279513(n+1)).

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

Offset: 1

Rémy Sigrist, Apr 02 2017

nonn

Two consecutive numbers, say n and n+1, cannot share a prime factor (gcd(n, n+1)=1). However, their prime tower factorizations can share some prime numbers; this is the case iff a(n)>1 (see A182318 for the definition of the prime tower factorization of a number).
If p is prime, then a(p-1) = a(p) = 1.
If p is an odd prime, then a(p^2) = 2.
This sequence contains a multiple of p for any prime p:
- let m = A074792(p)^p-1,
- m is a multiple of p, hence p divides A279513(m),
- m+1 = A074792(p)^p, hence p divides A279513(m+1),
- hence p divides gcd(A279513(m), A279513(m+1)) = a(m).
This sequence contains infinitely many distinct values; see A284821 for these distinct values in order of appearance, and A284822 for the corresponding indexes.

			a(8) = gcd(A279513(8), A279513(9)) = gcd(A279513(2^3), A279513(3^2)) = gcd(2*3, 3*2) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of the first terms

Cf. A074792, A182318, A284821, A284822.

A284763 Numbers n such that A279513(n) is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Rémy Sigrist, Apr 02 2017

nonn

Also numbers with no duplicate prime number in their prime tower factorization (see A182318 for the definition of the prime tower factorization of a number).
This sequence contains the squarefree numbers (A005117); 8 = 2^3 is the first term in this sequence that is not squarefree.
All terms belong to A144146; 81 = 3^2^2 is the first term of A144146 that is not in this sequence.

			8 = 2^3 belongs to this sequence.
24 = 3*2^3 does not belong to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A005117, A144146, A182318.

a279513(n) =  my (f=factor(n)); prod(i=1, #f~, f[i, 1]*a279513(f[i, 2]));
isok(n) = issquarefree(a279513(n)); \\ Michel Marcus, Apr 08 2017

A284889 Numbers n such that A279513(n) is a primorial number (A002110).

Original entry on oeis.org

1, 2, 6, 8, 9, 30, 40, 45, 75, 96, 210, 250, 280, 315, 486, 525, 672, 735, 1750, 1920, 2310, 3080, 3402, 3430, 3465, 5775, 6125, 7392, 8085, 8575, 10976, 11907, 12705, 15625, 16000, 19250, 21120, 21870, 30030, 31104, 32768, 37422, 37730, 40040, 45045, 54675

Offset: 1

Rémy Sigrist, Apr 05 2017

nonn

Also numbers with the k first prime numbers in their prime tower factorization, without duplicate, for some k (see A182318 for the definition of the prime tower factorization of a number).
This sequence contains the primorial numbers (A002110); 8 = 2^3 is the first term in this sequence that is not a primorial number.
This sequence contains A260548.
All terms belong to A284763.
If a(n) <= p# for some prime p, then a(n) is p-smooth (p# denotes the product of the primes <= p, see A002110).
There are A000272(k+1) terms with k prime numbers in their prime tower factorization:
- for k=0: 1,
- for k=1: 2,
- for k=2: 2*3, 2^3, 3^2,
- for k=3: 2*3*5, 2^3*5, 2^5*3, 3^2*5, 3^5*2, 5^2*3, 5^3*2, 2^(3*5), 3^(2*5), 5^(2*3), 2^3^5, 2^5^3, 3^2^5, 3^5^2, 5^2^3, 5^3^2.

			1626625 = 5^3*7*11*13^2 appears in this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..500
Rémy Sigrist, Illustration of the first terms

Cf. A000272, A002110, A182318, A260548, A279513, A284763.

isprimorial(n) = if (n==1, 1, my (f=factor(n)); (#f~ == primepi(vecmax(f[,1]))) && (vecmax(f[,2]) == 1));
a279513(n) =  my (f=factor(n)); prod(i=1, #f~, f[i, 1]*a279513(f[i, 2]));
isok(n) = isprimorial(a279513(n)); \\ Michel Marcus, Apr 08 2017

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A284695 Greatest prime number appearing in the prime tower factorization of n (when n > 1); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 3, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 5, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 3, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5

Offset: 1

Rémy Sigrist, Apr 01 2017

nonn

See A182318 for the definition of the prime tower factorization of a number.

a(n) >= A006530(n) for any n>0.

			8 = 2^3, hence a(8) = max(2, 3) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A006530, A182318, A279513, A284694.

a(n) = A006530(A279513(n)) for any n>0.

A284694 Least prime number appearing in the prime tower factorization of n (when n > 1); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 2, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 2, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 2, 2, 47, 2, 2, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 2, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 2, 2, 7, 2, 79, 2, 2

Offset: 1

Rémy Sigrist, Apr 01 2017

nonn

See A182318 for the definition of the prime tower factorization of a number.

a(n) <= A020639(n) for any n>0.

			9 = 3^2, hence a(9) = min(3, 2) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A020639, A182318, A279513, A284695.

a(n) = A020639(A279513(n)) for any n>0.

A336965 a(n) is the product of the distinct prime numbers appearing in the prime tower factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 6, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 6, 14, 10, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70

Offset: 1

Rémy Sigrist, Aug 09 2020

nonn

The prime tower factorization of a number is defined in A182318.

For any n > 0, a(n) is the product of the terms in n-th row of A336964.

			A001221(a(n)) = A115588(n) for any n > 1.
a(n) = A007947(A279513(n)).
a(n) = n iff n is squarefree (A005117).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A001221, A005117, A007947, A115588, A182318, A336964.

a(n) = { my (f=factor(n), v=vecprod(f[,1]~)); for (k=1, #f~, v=lcm(v, a(f[k,2]))); v }

A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 10, 18, 16, 67, 21, 26, 14, 71, 12, 73

Offset: 1

Ferdinand Rönngren and Lars Kevin Haagensen Strömberg, Aug 22 2020

nonn

a(n) <= A001414(n) for n > 1, with equality if and only if all the exponents in the prime factorization of n are either less than 6 or prime themselves. - Mital Ashok, Jun 22 2025

			a(100) = a(2^2*5^2) = 2*a(2) + 5*a(2) = 2*2 + 5*2 = 14.
a(192) = a(2^6*3^1) = 2*a(6) + 3*a(1) = 2*a(2^1*3^1) + 3*1 = 2*(2*a(1) + 3*a(1)) + 3 = 2*(2*1 + 3*1) + 3 = 13.

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

Cf. A000040, A001414, A064372, A279513.

a:= proc(n) option remember; `if`(n=1, 1,
      add(i[1]*a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2020

f[p_, e_] := p * a[e]; a[1] = 1; a[n_] := a[n] = Plus @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 22 2020 *)

a(n)={my(f=factor(n)); if(n==1, 1, sum(i=1, #f~, my([p,e]=f[i,]); p*a(e)))} \\ Andrew Howroyd, Aug 22 2020

a(1)=1, a(p_1^b_1*p_2^b_2*...*p_n^b_n)=p_1*a(b_1)+p_2*a(b_2)+...+p_n*a(b_n) where p_i is the i-th prime number.

A284696 Numbers of the form p^^k, with p prime and k>=0, where ^^ denotes tetration.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Rémy Sigrist, Apr 01 2017

nonn

Also numbers n such that A284694(n)=A284695(n).
Also numbers with no distinct prime numbers in their prime tower factorization (see A182318 for the definition of the prime tower factorization of a number).
Also numbers n such that A279513(n) is a power of prime (A000961).
This sequence is the union of 1, the prime numbers (A000040), and A275211.

			16 = 2^^3 is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Wikipedia, Tetration

Cf. A000040, A000961, A182318, A275211, A279513, A284694, A284695.

A343068 Multiplicative with a(p^e) = e*a(p-1).

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Apr 04 2021

			a(2) = a(2-1) = 1; a(3) = a(3-1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A279513, A191614.

a:= proc(n) option remember; `if`(n=1, 1,
      mul(i[2]*a(i[1]-1), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 04 2021

a[1] = 1; a[p_,s_]:= a[p, s]=s a[p-1];
a[n_]:=a[n]= Module[{aux = FactorInteger[n]},Product[a[aux[[i, 1]],aux[[i, 2]]], {i, Length[aux]}]]; Table[a[n],{n,100}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,2]*a(f[k,1]-1); f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 23 2021

a(2^n) = n*a(2-1) = n.

cp's OEIS Frontend

A284761 a(n) = gcd(A279513(n), A279513(n+1)).

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A284763 Numbers n such that A279513(n) is squarefree.

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A284889 Numbers n such that A279513(n) is a primorial number (A002110).

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A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

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Maple

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A284695 Greatest prime number appearing in the prime tower factorization of n (when n > 1); a(1) = 1.

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A284694 Least prime number appearing in the prime tower factorization of n (when n > 1); a(1) = 1.

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A336965 a(n) is the product of the distinct prime numbers appearing in the prime tower factorization of n.

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PARI

A337310 Additive function with a(p) = p, a(p^e) = p*a(e) for prime p and e > 1, with a(1) = 1.

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Maple