A337476 -id:A337476 - cp's OEIS frontend

A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 0, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 3, 2, 3, 0, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 2, 4, 3, 2, 0, 2, 2, 4, 2, 3, 2, 4, 1, 4, 3, 2, 3, 2, 2, 4, 2, 1, 3, 4, 2, 2, 3, 3, 2, 4, 2, 2, 3, 3, 3, 3, 1, 4, 2, 2, 2, 4, 2, 4, 2, 2

Offset: 1

Antti Karttunen, Aug 28 2020

nonn

a(n) is the least k for which A337473(k, n) = 1.

Cf. A003961, A108951, A181815, A336835, A337473, A337475.

Cf. A337476 (position of the first occurrence of each n), A337478.

A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337474(n) = for(i=0,oo,if(1==A337473sq(i,n),return(i)));

\\ This version uses binary search, which is faster in certain cases:
isA337473sq1(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(s<2)); prevpid = pid; e += f[i,2]); (s<2));
A337474(n) = if(!bitand(n,n-1),0,my(imin=0,imax=n,imid); for(i=0,oo, imid=(imax+imin)\2; if(1!=isA337473sq1(imid,n), imin = imid+1, if(1!=isA337473sq1(imid-1,n),return(imid), imax = imid-1))));

a(n) = A336835(A108951(n)).
a(A181815(n)) = A337475(n).
For all n >= 0, a(A337476(n)) = n.
For all n >= 0, a(A337478(n)) >= n.

A337473 Square array read by falling antidiagonals, where A(n,k) = floor(A337472(n, k)/A337470(n, k)); Abundancy index of A337470(n, k) floored down.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 28 2020

Array is read by descending antidiagonals with n >= 0 and k >= 1 ranging as: (0, 1), (0, 2), (1, 1), (0, 3), (1, 2), (2, 1), (0, 4), (1, 3), (2, 2), (3, 1), ...

			The top left corner of the array begins as:
n/k | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
----|----------------------------------------------------------------------
  0 | 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 3, 3, 1, 3, 2, 3, 3, 3, 3, 3,
  1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2,
  2 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
  3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  4 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
  5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
etc.
A337474 gives the distance in each column to the first 1 in that column, being 0 for columns 1, 2, 4, 8, 16, ..., where 1 is already in the top row.

Cf. A003961, A108951, A337470, A337472, A337474, A337476.

up_to = 105858-1; \\ Or 105-1.
A337473sq(n, k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j+n); s *= ((p^(1+e)-1)/((p-1)*(p^e)))); if(!pid,return(floor(s))); prevpid = pid; e += f[i,2]); floor(s));
A337473list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337473sq(b-1, (a-(b-1))))); (v); };
v337473 = A337473list(up_to);
A337473(n) = v337473[1+n];

A(n,k) = floor(A337472(n, k)/A337470(n, k)).

cp's OEIS Frontend

A337474 Number of prime shifts (x -> A003961(x)) needed before the result is deficient, when starting from x = A108951(n), the primorial inflation of n.

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A337478 Primorial deflation of A336389.

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A337477 Position of the first term >= n in A337475.

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A337473 Square array read by falling antidiagonals, where A(n,k) = floor(A337472(n, k)/A337470(n, k)); Abundancy index of A337470(n, k) floored down.

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