A286561 -id:A286561 - cp's OEIS frontend

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

2, 6, 2, 20, 2, 18, 2, 28, 2, 12, 2, 120, 2, 6, 2, 88, 2, 60, 2, 60, 2, 12, 2, 168, 2, 6, 2, 40, 2, 72, 2, 104, 2, 6, 2, 800, 2, 6, 2, 168, 2, 54, 2, 40, 2, 12, 2, 528, 2, 12, 2, 40, 2, 84, 2, 56, 2, 12, 2, 1440, 2, 6, 2, 136, 2, 72, 2, 20, 2, 24, 2, 2240, 2, 6, 2, 20, 2, 36, 2, 528, 2, 12, 2, 720, 2, 6, 2, 112, 2

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A067513, A185633, A286561, A322313 (rgs-transform), A322314.

Cf. also A293514, A322310.

A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };

a(n) = Product_{d|n} A000040(1+A286561(n,1+d))^A010051(1+d).
a(n) = A181819(A185633(n)).
For all n, A001222(a(n)) = A067513(n).

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

A323155 a(n) = Product_{d|n, d-1 is prime} (d-1)^(1+A286561(n,d-1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

Original entry on oeis.org

1, 1, 2, 3, 1, 20, 1, 21, 2, 1, 1, 3960, 1, 13, 2, 21, 1, 340, 1, 57, 2, 1, 1, 1275120, 1, 1, 2, 39, 1, 2900, 1, 651, 2, 1, 1, 201960, 1, 37, 2, 399, 1, 10660, 1, 129, 2, 1, 1, 119861280, 1, 1, 2, 3, 1, 18020, 1, 1911, 2, 1, 1, 643678200, 1, 61, 2, 651, 1, 20, 1, 201, 2, 13, 1, 4617209520, 1, 73, 2, 111, 1, 20, 1, 31521, 2, 1, 1, 175186440, 1, 1, 2, 903, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A010051, A286561, A323156, A323157.

Cf. also A185633.

A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); }; \\ Antti Karttunen, Jan 09 2019

a(n) = Product_{d|n, d>2} [(d-1)^(1+A286561(n,d-1))]^A010051(d-1).

A293515 a(n) = Product_{d^k|n, d>1, k>1} prime(A286561(n,d)-1), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 10, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 14, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 10, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 66, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 10, 10, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 14, 1, 2, 2, 8, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 10, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

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

Cf. A000040, A008578, A286561.
Cf. A293514, A294875.
Cf. A046951.

A293515(n) = { my(m=1,v); fordiv(n,d,if(d>1, v = valuation(n,d); if(v>1, m *= prime(v-1)))); m; };

a(n) = Product_{d|n, d>1} A008578(A286561(n,d)).
a(n) = A064989(A293514(n)).
Other identities. For all n >= 1:
1 + A001222(a(n)) = A046951(n).

A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

Original entry on oeis.org

3, 6, 1, 10, 1, 12, 1, 14, 1, 4, 1, 28, 1, 1, 1, 22, 1, 12, 1, 20, 1, 4, 1, 102, 1, 1, 1, 4, 1, 4, 1, 26, 1, 1, 1, 66, 1, 1, 1, 104, 1, 12, 1, 6, 1, 4, 1, 92, 1, 1, 1, 4, 1, 4, 1, 6, 1, 4, 1, 132, 1, 1, 1, 34, 1, 4, 1, 1, 1, 4, 1, 1240, 1, 1, 1, 1, 1, 4, 1, 57, 1, 4, 1, 21, 1, 1, 1, 28, 1, 1, 1, 6, 1, 1, 1, 492, 1, 1, 1, 12, 1, 4, 1, 6, 1

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A014197, A320000, A322311 (rgs-transform).

Cf. also A322312.

A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ From A320000
A322310(n) = if(1==n,3,my(m=1); fordiv(n,d, my(s, p=d+1, q=n/d); if(isprime(p) && (s = sum(i=0,valuation(n, p), A320000sq(q/(p^i),p))), m *= prime(s))); (m));

a(n) = Product_{d|n} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)])^A010051(1+d).

For all n, A056239(a(n)) = A014197(n).

A327167 a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 8, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 12, 1, 1, 1, 3, 6, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 8, 1, 1, 1, 1, 48, 1, 2, 1, 1, 2, 7, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 128

Offset: 1

Antti Karttunen, Sep 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A008578, A276086, A286561, A327168.

Cf. also A293514, A322312, A323155, A327154, A327155, A327156.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327167(n) = { my(m=1,v); fordiv(A276086(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)).
Other identities. For all n >= 1:
1+A001222(a(n)) = A327168(n).

A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 6, 1, 5, 2, 1, 1, 2, 1, 2, 1, 3, 1, 48, 1, 2, 1, 80, 1, 45, 1, 1, 2, 2, 1, 1, 1, 3, 1, 8, 1, 44, 1, 6, 2, 5, 1, 6, 1, 1, 3, 2, 1, 20, 1, 20, 1, 2, 1, 80, 1, 11, 1, 1, 1, 63, 1, 2, 2, 7, 1, 2, 1, 2, 1, 6, 1, 20, 1, 2, 1, 2, 1, 264, 1, 3, 2, 6, 1, 48, 2, 10, 1, 7, 2, 108, 1, 1, 2, 1, 1, 125, 1, 2, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327155, A327156.

A327154(n) = { my(m=1,s=sigma(n),v); fordiv(n,d,if((d>1) && ((v = valuation(s,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 6, 1, 2, 2, 1, 1, 3, 1, 3, 1, 2, 1, 80, 1, 2, 1, 48, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 1, 20, 1, 8, 1, 6, 3, 2, 1, 21, 1, 1, 2, 3, 1, 20, 1, 20, 1, 2, 1, 48, 1, 2, 1, 1, 1, 8, 1, 3, 2, 2, 1, 3, 1, 2, 1, 6, 1, 8, 1, 7, 1, 2, 1, 48, 1, 2, 2, 10, 1, 48, 2, 6, 1, 2, 2, 264, 1, 1, 3, 1, 1, 8, 1, 5, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327154, A327156.

A327155(n) = { my(m=1,v); fordiv(sigma(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 5, 1, 8, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A173441.

Cf. also A293514, A322312, A323155, A327154, A327155.

A327156(n) = { my(m=1,v); fordiv(n,d,if((d>1) && ((v = valuation(n,sigma(d)))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A173441(n).

A329037 a(n) = Product_{d|n, d>1} A008578(1+A286561(A276086(n),d)), where A286561(x,d) gives the exponent of the highest power of d dividing x.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 12, 1, 1, 1, 1, 5, 2, 1, 1, 1, 21, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 12, 1, 1, 1, 2, 10, 2, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 12, 1, 1, 1, 1, 48, 1, 3, 1, 1, 5, 2, 1, 1, 3, 7, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 10, 2, 2, 1, 1, 1, 1, 720

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base

Cf. A008578, A276086, A286561, A327168.

Cf. also A327167.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329037(n) = { my(m=1,x=A276086(n),v); fordiv(n,d,if((d>1) && ((v = valuation(x,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(A276086(n),d)).

1+A001222(a(n)) = A327168(n).

cp's OEIS Frontend

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

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A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A323155 a(n) = Product_{d|n, d-1 is prime} (d-1)^(1+A286561(n,d-1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

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A293515 a(n) = Product_{d^k|n, d>1, k>1} prime(A286561(n,d)-1), where A286561(n,d) gives the highest exponent of d dividing n.

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A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

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A327167 a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

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Formula

A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

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