A337205 -id:A337205 - cp's OEIS frontend

A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

1, 6, 120, 19399380, 195534950863140268380, 538938984694949877040715541221415046162838700, 216487559804430601784907786655491617909711008142914104790481010259258659171900

Offset: 0

Antti Karttunen, Aug 07 2020

For n > 0, the least k such that for at least n-1 iterations of map x -> A003961(x), starting from x=k, x stays nondeficient. In other words, from each a(n) starts a chain of at least n nondeficient numbers (A023196) obtained by successive prime shifts, e.g, for a(3) we have: 19399380 -> 334639305 -> 5391411025, where -> stands for applying A003961, the prime shift towards larger primes.
After 1 all other terms here are even, because if an odd number k is nondeficient, then A064989(k) is nondeficient also, where A064989 is the prime shift towards smaller primes. Moreover, because A047802 is defined for every n >= 0, also this sequence is.
From Peter Munn, Aug 13 2020 (Start)
Upper bounds for a(4) and a(5) are:
a(4) <= 195534950863140268380 = A064989(A064989(A064989(20169691981106018776756331))) = A337202(3).
a(5) <= 538938984694949877040715541221415046162838700 = A064989^4((A047802(4)*17*19)/137).
(End)
From David A. Corneth, Aug 21 2020: (Start)
Subsequence of A025487.
Let prime(n)# be the n-th primorial number, A002110(n) = A034386(prime(n)). Then:
a(6) <= 191# * 7#;
a(7) <= 311# * 5#;
a(8) <= 457# * 5#.
(End)
That each term occurs in A025487 follows because (1), the abundancy index of prime(i)^e is larger than that of prime(i+1)^e, that is, sigma(prime(i)^e)/prime(i)^e > sigma(prime(i+1)^e)/prime(i+1)^e, and (2) because the abundancy index of p^(e+d) * q^e is larger than that of p^e * q^(e+d), where p and q are distinct primes, p < q, and e, d > 0. Thus, for any n, we can first find a "prime-factorization compressed version" of it, A071364(n), and then sort the exponents to the non-ascending order with A046523 (and actually, A046523(A071364(n)) = A046523(n), so we need to apply just A046523), to get a term x of A025487, that certainly have the abundancy index >= n [and this inequivalence stays same for their successive prime shifts as well, the abundancy index of A003961(x) being at least that of A003961(n), etc.], and as A046523(n) <= n for all n, it is guaranteed that the least k for which A336835(k) >= n are found from A025487, which is the range of A046523.

Cf. A003961, A005100, A023196, A025487, A047802, A108951, A246277, A329900, A336834, A336835, A337202, A337205, A337477, A337478.

From term a(2) = 120 onward a subsequence of A337386.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
A336389(n) = for(i=1,oo,if(A336835(i)>=n,return(i)));

For n >= 0, A336835(a(n)) >= n.
For all n >= 1, a(n) <= A337202(n-1) [= 2*A246277(A047802(n-1))].
a(n) = A025487(A337477(n)).
a(n) = A108951(A337478(n)).

a(4) - a(6) from combined work of David A. Corneth and Peter Munn Aug 13-26 2020

A337203 Sum of the divisors of the primorial inflation of n.

Original entry on oeis.org

1, 3, 12, 7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, 31, 1741824, 195, 34836480, 360, 4368, 16128, 836075520, 124, 2821, 225792, 600, 2880, 25082265600, 1170, 802632499200, 63, 52416, 4064256, 22568, 403, 30500034969600, 81285120, 733824, 744, 1281001468723200, 9360, 56364064623820800, 34560, 3600, 1950842880

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 0 of A337205, and of A337472.
Cf. A000203, A034386, A108951, A337204.
Cf. also A323173.

Array[DivisorSigma[1, Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e]] &, 46] (* Michael De Vlieger, Aug 27 2020 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A337203(n) = sigma(A108951(n));

A337203(n) = if(1==n,n, my(f=factor(n), 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); s *= ((p^(1+e)-1)/(p-1))); if(!pid,return(s)); prevpid = pid; e += f[i,2]); (s));

a(n) = A000203(A108951(n)).

A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 4, 15, 5, 1, 30, 9, 35, 7, 1, 12, 105, 25, 77, 11, 1, 210, 45, 385, 49, 143, 13, 1, 8, 1155, 175, 1001, 121, 221, 17, 1, 36, 27, 5005, 539, 2431, 169, 323, 19, 1, 60, 225, 125, 17017, 1573, 4199, 289, 437, 23, 1, 2310, 315, 1225, 343, 46189, 2873, 7429, 361, 667, 29, 1, 24, 15015, 1925, 5929, 1331, 96577, 5491, 12673, 529, 899, 31

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
----|------------------------------------------------------------
  0 | 1,  2,   6,   4,   30,   12,   210,    8,    36,    60, ...
  1 | 1,  3,  15,   9,  105,   45,  1155,   27,   225,   315, ...
  2 | 1,  5,  35,  25,  385,  175,  5005,  125,  1225,  1925, ...
  3 | 1,  7,  77,  49, 1001,  539, 17017,  343,  5929,  7007, ...
  4 | 1, 11, 143, 121, 2431, 1573, 46189, 1331, 20449, 26741, ...
  5 | 1, 13, 221, 169, 4199, 2873, 96577, 2197, 48841, 54587, ...
etc.

Cf. A108951 (row 0), A337471 (row 1).
Cf. A003961, A242378.
Cf. also A337205, A337472.

up_to = 105-1; \\ Or 78-1.
Ashifted_primorial(n,d) = prod(i=1, primepi(n), prime(i+d));
A337470sq(n, k) = { my(f=factor(k)); prod(i=1, #f~, Ashifted_primorial(f[i, 1], n)^f[i, 2]); };
A337470list(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] = A337470sq(b-1, (a-(b-1))))); (v); };
v337470 = A337470list(up_to);
A337470(n) = v337470[1+n];

A(n,k) = A242378(n,A108951(k)).

A337472 Square array read by falling antidiagonals, where A(n,k) = sigma(A337470(n, k)).

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 7, 24, 6, 1, 72, 13, 48, 8, 1, 28, 192, 31, 96, 12, 1, 576, 78, 576, 57, 168, 14, 1, 15, 2304, 248, 1344, 133, 252, 18, 1, 91, 40, 8064, 684, 3024, 183, 360, 20, 1, 168, 403, 156, 24192, 1862, 5040, 307, 480, 24, 1, 6912, 624, 1767, 400, 60480, 3294, 8640, 381, 720, 30, 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
----|--------------------------------------------------------------
  0 | 1,  3,  12,   7,   72,   28,    576,   15,    91,   168, ...
  1 | 1,  4,  24,  13,  192,   78,   2304,   40,   403,   624, ...
  2 | 1,  6,  48,  31,  576,  248,   8064,  156,  1767,  2976, ...
  3 | 1,  8,  96,  57, 1344,  684,  24192,  400,  7581,  9576, ...
  4 | 1, 12, 168, 133, 3024, 1862,  60480, 1464, 24339, 33516, ...
  5 | 1, 14, 252, 183, 5040, 3294, 120960, 2380, 56181, 65880, ...
etc.

Cf. A000203, A003961.
Cf. A337203 (row 0).
Cf. also arrays A337205, A337470.

up_to = 105-1; \\ Or 78-1.
Ashifted_primorial(n,d) = prod(i=1, primepi(n), prime(i+d));
A337470sq(n, k) = { my(f=factor(k)); prod(i=1, #f~, Ashifted_primorial(f[i, 1], n)^f[i, 2]); };
A337472sq(n, k) = sigma(A337470sq(n, k));
A337472list(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] = A337472sq(b-1, (a-(b-1))))); (v); };
v337472 = A337472list(up_to);
A337472(n) = v337472[1+n];

A(n,k) = A000203(A337470(n, k)).

A337204 Sum of the odd divisors of the primorial inflation of n.

Original entry on oeis.org

1, 1, 4, 1, 24, 4, 192, 1, 13, 24, 2304, 4, 32256, 192, 78, 1, 580608, 13, 11612160, 24, 624, 2304, 278691840, 4, 403, 32256, 40, 192, 8360755200, 78, 267544166400, 1, 7488, 580608, 3224, 13, 10166678323200, 11612160, 104832, 24, 427000489574400, 624, 18788021541273600, 2304, 240, 278691840, 901825033981132800, 4, 22971, 403

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 1 of A337205.
Cf. A000593, A034386, A108951, A337203.
Cf. also A322819.

Array[DivisorSum[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e], # &, OddQ] &, 50] (* Michael De Vlieger, Aug 27 2020 *)

A000593(n) = sigma(n>>valuation(n, 2));
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A337204(n) = A000593(A108951(n));

a(n) = A000593(A108951(n)).

cp's OEIS Frontend

A336389 The least positive integer k for which A336835(k) >= n, where A336835(k) is the number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A337203 Sum of the divisors of the primorial inflation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A337472 Square array read by falling antidiagonals, where A(n,k) = sigma(A337470(n, k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A337204 Sum of the odd divisors of the primorial inflation of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula