A065769 -id:A065769 - cp's OEIS frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

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

Offset: 1

Henry Bottomley, Nov 19 2001

nonn

It seems that a(n) <= A297113(n) for all n. Of the first 10000 positive natural numbers, 6454 are such that a(n) = A297113(n). - Antti Karttunen, Dec 31 2017

Also one plus the maximum number of unit steps East or South in the Young diagram of the integer partition with Heinz number n > 1, starting from the upper-left square and ending in a boundary square in the lower-right quadrant. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 06 2019

			a(50) = 4 since the cascade goes from 50 = 2^1 * 5^2 to 15 = 3^1 * 5^1 to 6 = 2^1 * 3^1 to 2 = 2^1 to 1.
From _Gus Wiseman_, Apr 06 2019: (Start)
The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has longest path from (1,1) to (5,3) of length 6, so a(7865) = 7.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
FindStat, St000384: The maximal part of the shifted composition of an integer partition

Cf. A065769.
Differs from A297113 for the first time at n=20, where a(20) = 3, while A297113(20) = 4.
Cf. A001221, A001222, A052126, A056239, A064989, A112798, A174090, A257990, A325166, A325167, A325169.

Table[If[n==1,0,Max@@Total/@Position[PadRight[ConstantArray[1,#]&/@Sort[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],Greater]],1]-1],{n,100}] (* Gus Wiseman, Apr 06 2019 *)

(definec (A065770 n) (if (= 1 n) 0 (+ 1 (A065770 (A065769 n))))) ;; Antti Karttunen, Dec 31 2017

Inverse of primes, powers of 2 and primorials in sense that a(A000040(n))=n; a(A000079(n))=n; a(A002110(n))=n. If n>0: a(3^n)=n+1; a(2^n*3^k)=n+k; a(p(k)^n)=n+k-1; a(n!)=A022559(n).

a(1) = 0; and for n > 1, a(n) = 1 + A065769(n). - Antti Karttunen, Dec 31 2017

A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

Original entry on oeis.org

1, 1, 1, 7, 3, 1, 5, 25, 5, 3, 7, 7, 11, 5, 3, 363, 13, 5, 17, 21, 5, 7, 19, 25, 51, 11, 13, 35, 23, 3, 29, 1335, 7, 13, 15, 35, 31, 17, 11, 75, 37, 5, 41, 49, 15, 19, 43, 363, 115, 51, 13, 77, 47, 13, 21, 125, 17, 23, 53, 21, 59, 29, 25, 9923, 33, 7, 61, 91, 19, 15, 67, 125, 71, 31, 51, 119, 35, 11, 73, 1089, 139, 37, 79, 35, 39, 41, 23

Offset: 1

Antti Karttunen, Sep 03 2018

Multiplicative because A065769 and A317932 are.

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

Cf. A065769, A317932 (denominators).

up_to = 1+(2^16);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A065769(n) = { my(f=factor(n>>valuation(n,2))[, 1]~); (A003557(n) * factorback(vector(#f,i,precprime(f[i]-1)))); }; \\ Antti Karttunen, Sep 03 2018
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA065769(n)));
A318669(n) = numerator(v318669_aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A065769(n) - Sum_{d|n, d>1, d 1.

A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

Original entry on oeis.org

1, 2, 1, 8, 2, 2, 2, 16, 2, 4, 2, 8, 2, 4, 2, 128, 2, 4, 2, 16, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 2, 256, 2, 4, 4, 16, 2, 4, 2, 32, 2, 4, 2, 16, 4, 4, 2, 128, 8, 16, 2, 16, 2, 4, 4, 32, 2, 4, 2, 16, 2, 4, 4, 1024, 4, 4, 2, 16, 2, 8, 2, 32, 2, 4, 8, 16, 4, 4, 2, 256, 8, 4, 2, 16, 4, 4, 2, 32, 2, 8, 4, 16, 2, 4, 4, 256, 2, 16, 4, 64, 2, 4, 2, 32, 4

Offset: 1

Antti Karttunen, Aug 11 2018

These are denominators for rational valued sequences that are obtained as "Dirichlet Square Roots" of sequences b that satisfy the condition b(3) = 2, and b(p) = odd number for any other primes p. For example, A064989, A065769 and A234840. - Antti Karttunen, Aug 31 2018

The original definition was: Denominators of the rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence. However, this definition depends on the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A317930, A318319, A318669 (some of the numerator sequences), A317931 (conjectured, for A002487).
Cf. A305439 (the 2-adic valuation), A318666.
Cf. also A046644, A261179, A299150, A237649, A299150, A317928, A317839, A317843, A317934, A318319.

\\ Original program, based on conjectural formula:
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2);
A317932(n) = denominator(A317931perA317932(n));

\\ New fast program implementing the new definition:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
A317932(n) = (A046644(n)/2^valuation(n,3)); \\ Antti Karttunen, Aug 31 2018

A011371(n) = (A005187(n)-n);
A317932(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^(A011371(f[i,2])), m *= 2^A005187(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

a(n) = A046644(n)/A318666(n) = 2^A305439(n).
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d 1, where b can be A064989, A065769 or A234840 for example, conjecturally also A002487.
Multiplicative with a(3^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for any other primes. - Antti Karttunen, Sep 03 2018

Definition changed, the original (now conjectured alternative definition) moved to the comments section by Antti Karttunen, Aug 31 2018

Keyword:mult added by Antti Karttunen, Sep 03 2018

cp's OEIS Frontend

A065770 Number of prime cascades to reach 1, where a prime cascade (A065769) is multiplicative with a(p(m)^k) = p(m-1) * p(m)^(k-1).

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A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

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A317932 Denominators of certain "Dirichlet Square Root" sequences: a(n) = A046644(n)/(2^A007949(n)).

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