A342010 -id:A342010 - cp's OEIS frontend

A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

2, 3, 6, 10, 20, 30, 42, 84, 132, 156, 312, 468, 780, 1020, 1140, 1380, 2760, 3480, 3720, 5208, 7812, 9324, 10332, 10836, 21672, 23688, 26712, 29736, 49560, 51240, 56280, 59640, 61320, 96360, 104280, 208560, 219120, 328680, 352440, 384120, 453960, 472680, 482040, 500760, 510120, 528840, 594360, 613080, 641160, 650520, 1301040

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

In contrast to A329902, this sequence is monotonic, because each term is obtained from the previous, either by multiplying it by 2, or by "bumping" one [or hypothetically: two] of its prime factors one step up (i.e., replacing it with the next larger prime), and both operations are guaranteed to make the number larger.

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

Cf. A004490, A005940, A073751, A108951, A319626, A329900, A342010 [= A001222(a(n))], A342011, A342013.

Cf. also A217867, A329902.

v073751 = readvec("b073751_to.txt");
A073751(n) = v073751[n];
A004490list(v073751) = { my(v=vector(#v073751)); v[1] = 2; for(n=2,#v,v[n] = v073751[n]*v[n-1]); (v); };
v004490 = A004490list(v073751);
A004490(n) = v004490[n];
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319626(n) = (n / gcd(n, A064989(n)));
A342012(n) = A319626(A004490(n));

a(n) = A319626(A004490(n)) = A329900(A004490(n)).

a(n) = A005940(1+A342013(n)).

A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 2, 5, 9, 19, 21, 37, 75, 139, 267, 535, 539, 555, 1067, 2091, 4139, 8279, 16471, 32855, 32919, 32923, 65691, 131227, 262299, 524599, 1048887, 2097463, 4194615, 4194647, 8388951, 16777559, 33554775, 67109207, 67109463, 134218327, 268436655, 536872111, 536872119, 1073743031, 2147484855, 2147485879, 4294969527, 8589936823

Offset: 1

Antti Karttunen and Michael De Vlieger, Mar 08 2021

nonn

a(n) is the unique integer k such that A329886(k) = A004490(n).
Like A342012, also this sequence is monotonic. Proof: the doubling step corresponds here to step *2 + 1, and "bumping up" some of the prime factors likewise results a larger A156552-code, thus both steps keep the result growing.
The binary length of these numbers (A070939, = 1+A000523) grows by 0 or 1 at each step, thus the next colossally abundant number is always found on either on the same row (right of the current CA-number), or the next row of A329886, the row immediately below. The next CA-number will be on the same row only when its factorization contains neither a new prime nor yet another instance of prime 2.

Cf. A004490, A005940, A073751, A108951, A156552, A319626, A329886, A329900, A342000, A342010 (the binary weight), A342011, A342012.

A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
A342013(n) = A156552(A342012(n)); \\ Uses also code from A342012.

a(n) = A156552(A342012(n)) = A156552(A319626(A004490(n))).

A342011 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, with f(1) = 2 and f(n) = A004490(n)/A004490(n-1) when n > 1, where A004490(n) is the n-th colossally abundant number.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 1, 5, 6, 1, 2, 3, 7, 8, 9, 1, 10, 11, 4, 2, 12, 13, 14, 1, 15, 16, 17, 3, 18, 19, 20, 21, 5, 22, 1, 23, 2, 24, 25, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 1, 35, 36, 4, 37, 38, 39, 7, 40, 41, 42, 43, 44, 45, 46, 8, 47, 2, 48, 49, 50, 3, 51, 52, 53, 54, 1, 55, 56, 57, 58, 59, 60, 61, 62, 9, 63, 64

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

This is also the restricted growth sequence transform of A073751, provided that quotient A004490(1+n)/A004490(n) is always prime, which is implied by a conjecture mentioned in Lagarias' paper. Note that the b-file of A073751 is computed based on the knowledge that the conjecture holds at least for the first 10^7 quotients.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A073751 provided by T. D. Noe)
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A000720, A004490, A073751, A342010, A342012, A342013.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v073751 = readvec("b073751_to.txt"); \\ Prepared with gawk '{ print $2 }' < b073751.txt > b073751_to.txt
v342011 = rgs_transform(v073751);
A342011(n) = v342011[n];
for(n=1,#v342011,write("b342011.txt", n, " ", A342011(n)));

a(n) = A000720(A073751(n)), up to the first n where A004490(n)/A004490(n-1) is not a prime.

cp's OEIS Frontend

A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

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A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.

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A342011 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, with f(1) = 2 and f(n) = A004490(n)/A004490(n-1) when n > 1, where A004490(n) is the n-th colossally abundant number.

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