A212171 -id:A212171 - cp's OEIS frontend

A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

0, 1, 1, 1, 1, 0, 1, 4, 1, 5, 4, 1, 6, 5, 1, 7, 6, 2, 1, 8, 5, 7, 2, 1, 9, 6, 8, 2, 1, 10, 7, 1, 9, 2, 6, 1, 11, 8, 0, 10, 2, 7, 1, 12, 9, 18, 0, 11, 2, 8, 15, 1, 13, 10, 22, 0, 7, 14, 12, 2, 9, 20, 1, 14, 11, 26, 7, 8, 18, 13, 2, 10, 25, 1, 15, 15, 12, 30, 9

Offset: 1

Matthew Vandermast, Jun 05 2012

Significance of the sequence: Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. For all integers n with second signature {S}, A212180(n) = k and A085082(n) is congruent to j modulo k; see examples.

Note: b(n) = A212642(n); c(n) = A212644(n).

			4 is the smallest integer with second signature {2}, and its divisors represent 3 distinct prime signatures and 2 distinct second signatures. 1 = 3 mod 2. Since 4 = A181800(2), a(2) = 1. For all integers m with second signature {2}, A085082(m) is congruent to 1 modulo 2.
10800 is the smallest integer with second signature {4,3,2}, and its divisors represent 28 distinct prime signatures and 14 distinct second signatures. 0 = 28 mod 14.  Since 10800 = A181800(39), a(39) = 0. For all integers m with second signature {4,3,2}, A085082(m) is congruent to 0 modulo 14.

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

Cf. A085082, A212171, A212172, A212180, A212642, A212644.

a(n) = A212642(n)-A212644(n), reduced modulo A212644(n).

A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 4, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 3, 6, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A212177(n).

			First rows of table read: 2; 3; 2; 2; 4; 2; 2; 3;...
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, but only exponents that are 2 or greater appear in a number's second signature. Hence, 12's second signature is {2}. Since 12 = A013929(4), row 4 of the table represents the second signature {2}.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Jason Kimberley, Table of i, a(i) for i = 1..4492 (n = 1..3917)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

Cf. A013929, A212172, A212175, A212176, A212177.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)|pe[2]gt 1])):n in[1..247]]; // Jason Kimberley, Jun 13 2012

a(n) = A212172(A013929(n)).

This sequence is both the subsequence of A212171 formed by omitting all 1s and the subsequence of A212172 formed by omitting all 0's. - Jason Kimberley, Jun 13 2012

A320390 Prime signature of n (sorted in decreasing order), concatenated.

Original entry on oeis.org

0, 1, 1, 2, 1, 11, 1, 3, 2, 11, 1, 21, 1, 11, 11, 4, 1, 21, 1, 21, 11, 11, 1, 31, 2, 11, 3, 21, 1, 111, 1, 5, 11, 11, 11, 22, 1, 11, 11, 31, 1, 111, 1, 21, 21, 11, 1, 41, 2, 21, 11, 21, 1, 31, 11, 31, 11, 11, 1, 211, 1, 11, 21, 6, 11, 111, 1, 21, 11, 111, 1

Offset: 1

M. F. Hasler, Oct 12 2018

In the variant A037916, the exponents of the prime factorization are concatenated without being sorted first (i.e., rows of A124010).

			For n = 1, the prime signature is the empty sequence, so the concatenation of its terms yields 0 by convention.
For n = 2 = 2^1, n = 3 = 3^1 and any prime p = p^1, the prime signature is (1), and concatenation yields a(n) = 1.
For n = 4 = 2^2, the prime signature is (2), and concatenation yields a(n) = 2.
For n = 6 = 2^1 * 3^1, the prime signature is (1,1), and concatenation yields a(n) = 11.
For n = 12 = 2^2 * 3^1 but also n = 18 = 2^1 * 3^2, the prime signature is (2,1) since exponents are sorted in decreasing order; concatenation yields a(n) = 21.
For n = 30 = 2^1 * 3^1 * 5^1, the prime signature is (1,1,1), and concatenation yields a(n) = 111.
For n = 3072 = 2^10 * 3^1, the prime signature is (10,1), and concatenation yields a(n) = 101. This is the first term with nondecreasing digits.

Cf. A037916, A118914, A124010, A212171.

{0}~Join~Array[FromDigits@ Flatten[IntegerDigits /@ FactorInteger[#][[All, -1]] ] &, 78, 2] (* Michael De Vlieger, Oct 13 2018 *)

a(n)=fromdigits(vecsort(factor(n)[,2]~,,4)) \\ Except for multiples of 2^10, 3^10, etc.

a(n)=eval(concat(apply(t->Str(t),vecsort(factor(n)[,2]~,,4)))) \\ Slower but correct for all n.

a(n) = concatenation of row n of A212171.

a(n) = a(A046523(n)). - David A. Corneth, Oct 13 2018

A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 2, 2, 2, 3, 6, 4, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 6, 3, 6, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 6, 4, 1, 1, 2, 2, 2, 6, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 1, 2, 2, 2, 3, 2, 6, 6, 4, 12, 5

Offset: 1

Thomas Anton, Dec 09 2018

The method of indexing the unordered factorizations of n in this array is as follows: take all unordered factorizations of n and write them with their factors in nonincreasing order (e.g., 2*4*5*3 becomes 5*4*3*2), and order these reverse-lexicographically (e.g., for 12: 12, 6*2, 4*3, 3*2*2), then assign the index k to the k-th factorization in this ordering.
For a sequence f with Dirichlet inverse f^(-1), f^(-1)(n) is the sum over all multisets M of integers > 1 with product n, of the product of the terms f(m) with indices m in M (counted with multiplicity) multiplied by T(n,k)*(-1)^c/f(1)^(c+1) where c = |M| and T(n,k) corresponds to M.
The multiset of entries in the n-th row is determined by the prime signature of n.
For the p^j-th row with p a prime, the entries give the number of compositions of j corresponding to each partition of j, indexed by k in an analogous manner, given by the j-th row of A048996.

			  1;
  1;
  1;
  1, 1;
  1;
  1, 2;
  1;
  1, 2, 1;
  1, 1;
  1, 2;
  1;
  1, 2, 2, 3;
  etc.
The 12th row is 1,2,2,3, because 12 can be factored as 12, 6*2, 3*4 or 3*2*2 with respective sets of ordered factorizations {12}, {6*2, 2*6}, {4*3, 3*4} and {3*2*2, 2*3*2, 2*2*3}, with respective cardinalities 1, 2, 2 and 3.

Cf. A048996, A002033 (row sums), A212171, A251683, A001055 (row lengths).

A353248 Irregular table, read by rows, where row n is the concatenation of all prime signatures leading to n divisors, in reverse lexicographic order.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Apr 08 2022

The number-of-divisors function d = A000005 is multiplicative with d(p^e) = e+1. Therefore, a prime signature (e1, e2, ..., eN) which yields a given number of divisors corresponds to a factorization (e1+1)*...*(eN+1) of that number. Following the usual convention, we only consider prime signatures with decreasing exponents (e1 >= e2 >= ... >= eN). Furthermore, we list them in reverse lexicographic order, i.e., largest e1 first, etc.

The sequence starts with row 1 which has length 0 (the only number having only 1 divisor is 1 which has the empty product as prime factorization), so the first term a(1) = 1 is the first element of row 2. Here and thereafter, the first element of row n is easily recognized as the first occurrence of n-1, which is the only element of the row (and therefore followed by n) iff n is prime.

			Table begins:
row n | prime signatures
   1  | ()
   2  | (1)
   3  | (2)
   4  | (3), (1,1)
   5  | (4)
   6  | (5), (2,1)
   7  | (6)
   8  | (7), (3,1), (1,1,1)
   9  | (8), (2,2)
  10  | (9), (4,1)
  11  | (10)
  12  | (11), (5,1), (3,2), (2,1,1)

Cf. A000005 (d = tau = number-of-divisors function).

Cf. A025487 (products of primorial numbers, representatives of prime signatures), A046523 (representative of prime signature of n), A118914, A212171 and A124010 (prime signature of n), A124832 (prime signatures listed in order of representatives A025487), A080577 (partitions in grad.rev.lex order), A036036 (partitions in rev.lex order).

A353248_row(n, M=n)={if(n>1, my(f=factor(n)~, m=f[1,#f], L=List()); fordiv(n, d, n < m*d && break; n > M*d || foreach(self()(d, n/d), S, listput(L,concat(n/d-1,S)))); Vec(L), [[]])}

A364814 Numbers k whose largest divisor <= sqrt(k) is a power of 2, listing only the first such number with any given prime signature.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 20, 24, 32, 64, 72, 80, 96, 128, 256, 288, 320, 336, 384, 512, 1024, 1056, 1152, 1280, 1344, 1536, 2048, 4096, 4224, 4608, 4800, 5120, 5376, 6144, 8192, 16384, 16896, 17280, 18432, 18816, 19200, 20480, 21504, 24576, 32768, 65536, 67584, 69120, 69888

Offset: 1

David A. Corneth, Oct 21 2023

nonn

This sequence is a primitive sequence related to A365406 in the sense that it can be used to find the smallest term k in A365406 such that tau(k), omega(k) or bigomega(k) has some particular value.

Not every prime signature produces a term. For example no term has prime signature (3, 2, 1). Proof: any number with prime signature (3, 2, 1) has 24 divisors. Hence the 12th divisor must be a power of 2. But the largest power of 2 such number can have as a divisor is 8. 8 can never be the 12th divisor of a number. Therefore (3, 2, 1) can never be the prime signature of a term.

			k = 20 = 2^2 * 5 is in the sequence as it has prime signature (2, 1) and its largest divisor <= sqrt(k) is 4, a power of 2. It is the smallest such number since smaller numbers with prime signature (2, 1), namely 12 and 18, do not have the relevant divisor being a power of 2.

Cf. A025487, A212171, A365406.

upto(n) = {
	my(res = List([1]), m = Map());
	forstep(i = 2, n, 2,
		if(isok(i),
			s = sig(i);
			sb = sigback(s);
			if(!mapisdefined(m, sb),
				listput(res, i);	
				mapput(m, sb, i)
			)
		)
	);
	res
}
sig(n) = {
	vecsort(factor(n)[,2],,4)
}
sigback(v) = {
	my(pr = primes(#v));
	prod(i = 1, #v, pr[i]^v[i])
}
isok(n) = my(d = divisors(n)); hammingweight(d[(#d + 1)\2]) == 1

Edited by Peter Munn, Oct 26 2023

cp's OEIS Frontend

A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

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A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

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A320390 Prime signature of n (sorted in decreasing order), concatenated.

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A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

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A353248 Irregular table, read by rows, where row n is the concatenation of all prime signatures leading to n divisors, in reverse lexicographic order.

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A364814 Numbers k whose largest divisor <= sqrt(k) is a power of 2, listing only the first such number with any given prime signature.

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