A002098 -id:A002098 - cp's OEIS frontend

A064364 Positive integers sorted by A001414(n), the sum of their prime divisors, as the major key and n as the minor key.

1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 14, 20, 24, 27, 21, 25, 30, 32, 36, 11, 28, 40, 45, 48, 54, 35, 42, 50, 60, 64, 72, 81, 13, 22, 56, 63, 75, 80, 90, 96, 108, 33, 49, 70, 84, 100, 120, 128, 135, 144, 162, 26, 44, 105, 112, 125, 126, 150, 160, 180, 192, 216, 243

Offset: 1

Howard A. Landman, Sep 25 2001

This is a permutation of the positive integers.
a(1) could be taken as 0 because 1 is not a member of A001414 and one could start with a(0)=1 (see the W. Lang link).
The row length sequence of this array is A000607(n), n>=2.
If the array is [1,0,2,3,4,5,6,6,...] with offset 0 then the row length sequence is A000607(n), n>=0.
From David James Sycamore, May 11 2018: (Start)
For n > 1, a(n) is the smallest number not yet seen such that sopfr(a(n)) is the least possible integer. The sequence lists in increasing order elements of the finite sets S(k) = {x: sopfr(x)=k}, k >= 0, where sopfr(x) = 0 iff x = 1. When a(n) = A056240(k) for some k >= 2, then sopfr(a(n)) = k and a(n) is the first of A000607(k) terms, all of which have sopfr = k. (A000607(k) is the number of partitions of k into prime parts.) Consequently the sequence follows a sawtooth profile, rising from a(n) = A056240(k) to A000792(k), the greatest number with sopfr = k, then starting over with A056240(k+1) for the next larger value of sopfr. (End) [Edited by M. F. Hasler, Jan 19 2019]

			The triangle reads:
1,
(0,) (see comment in link to "first 16 rows" by W. Lang)
2,
3,
4,
5,  6,
8,  9,
7,  10, 12,
15, 16, 18,
14, 20, 24, 27,
21, 25, 30, 32, 36,
11, 28, 40, 45, 48, 54,
35, 42, 50, 60, 64, 72, 81,
13, 22, 56, 63, 75, 80, 90, 96, 108,
...

Alois P. Heinz, Rows n = 1..60, flattened (first 32 rows from Reinhard Zumkeller)
H. Havermann: The first 100 sums (complete, a 6 MB file)
H. Havermann: Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums)
Wolfdieter Lang, First 16 rows.
Index entries for sequences that are permutations of the natural numbers

Cf. A001414.
Cf. A000607 (row lengths), A002098 (row sums), A056240 (least = first term in the n-th row), A000792 (greatest term in the n-th row).
Cf. A257815 (inverse).

import Data.List (partition, union)
a064364 n k = a064364_tabf !! (n-1) !! (k-1)
a064364_row n = a064364_tabf !! (n-1)
a064364_tabf = [1] : tail (f 1 [] 1 (map a000792 [2..])) where
   f k pqs v (w:ws) = (map snd pqs') :
     f (k + 1) (union pqs'' (zip (map a001414 us) us )) w ws where
       us = [v + 1 .. w]
       (pqs', pqs'') = partition ((== k) . fst) pqs
a064364_list = concat a064364_tabf
-- Reinhard Zumkeller, Jun 11 2015

terms = 1000; nmax0 = 100000 (* a rough estimate of max sopfr *);
sopfr[n_] := sopfr[n] = Total[Times @@@ FactorInteger[n]];
f[n1_, n2_] := Which[t1 = sopfr[n1]; t2 = sopfr[n2]; t1 < t2, True, t1 == t2, n1 <= n2, True, False];
Clear[g];
g[nmax_] := g[nmax] = Sort[Range[nmax], f][[1 ;; terms]];
g[nmax = nmax0];
g[nmax += nmax0];
While[g[nmax] != g[nmax - nmax0], Print[nmax]; nmax += nmax0];
A064364 = g[nmax] (* Jean-François Alcover, Mar 13 2019 *)

lista(nn) = {nmax = A000792(nn); v = vector(nmax, k, A001414(k)); for (n=1, nn, vn = select(x->x==n, v, 1); for (k = 1, #vn, print1(vn[k], ", ")))} \\ Michel Marcus, May 01 2018

A064364_vec(N, k=6, L=9)={vector(N, i, if(i<7, N=i, until(A001414(N+=1)==k, ); NA056240(k)-1))} \\ To compute terms up to a given value of k=sopfr(n) and/or for large N >> 1000, it is more efficient to use code similar to lista() above, with "for(k...)" replaced by "a=concat(a, vn)". - M. F. Hasler, Jan 19 2019

If a(n) = A056240(k) for some k then a(n+A000607(k)-1) = A000792(k). - David James Sycamore, May 11 2018

More terms from Vladeta Jovovic, Sep 27 2005

A002099 G.f.: -1 + Product_{k>=1} (1 + prime(k)*x^prime(k)).

Original entry on oeis.org

0, 0, 2, 3, 0, 11, 0, 17, 15, 14, 51, 11, 77, 35, 103, 131, 160, 227, 330, 218, 426, 794, 426, 1533, 641, 2072, 2071, 2491, 4381, 3880, 3925, 6079, 5446, 7914, 14017, 10421, 19768, 15486, 30589, 31901, 46119, 50917, 65574, 48002, 85155, 112648, 115127, 150995

Offset: 0

N. J. A. Sloane

nonn

S. M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500

Cf. A002098.

Better description and more terms from Vladeta Jovovic, May 09 2003

A116864 Array of product of parts of the partitions of n with only prime parts.

Original entry on oeis.org

0, 2, 0, 3, 0, 0, 0, 0, 4, 0, 0, 5, 0, 6, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 8, 0, 0, 0, 0, 7, 0, 10, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 20, 0, 27

Offset: 1

Wolfdieter Lang, Mar 24 2006

The inverse of sequence A001414 (sopfr(n)=sum of prime factors of n). See the examples and the W. Lang link.
The row length sequence of this array is p(n)=A000041(n) (number of partitions).
The partitions of n are ordered according to Abramowitz-Stegun (A-St), pp. 831-2.
Row n gives the values k for which A001414(k)=n>=2. E.g. n=10 appears 5 times in A001414, namely for the k values 21, 25, 30, 36 and 32.

			[0];
[2, 0];
[3, 0, 0];
[0, 0, 4, 0, 0];
[5, 0, 6, 0, 0, 0, 0];
...
a(4,3)=4 because the third partition of 4 is, in A-St order, (2,2)
with product 4. There is only this partition of 4 with only prime parts.
Row n=5 shows: n=5 appears twice in A001414(k), namely for k= 5 and
6. This is related to the two partitions (5) and (3,2) with only prime parts.

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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
W. Lang: First 10 rows.

Row sums give A002098(n), n>=1.

Row sums (with nonzero numbers replaced by 1) give A000607(n), n>=1. See the array A116865.

a(n,k)=product(part(i),i=1..m(n,k)) if the k-th partition of n in the A-St order has only prime parts. Here m(n,k) is the number of parts of this partition. Otherwise a(n,k)=0. See A000040 for the prime numbers.

A298159 G.f.: Product_{k>=1} (1-prime(k)*x^prime(k)).

Original entry on oeis.org

1, 0, -2, -3, 0, 1, 0, 3, 15, 14, -9, -11, -7, 9, -37, -79, 28, 193, -46, -150, -142, -58, -142, 171, 73, 652, 643, 349, -1047, 404, -2743, 873, 3082, 2498, -3039, 411, -2188, 4534, -4571, -9997, -5917, 32879, -14830, -11926, -16333, 7208, -22517, 59885, -27601

Offset: 0

Seiichi Manyama, Jan 14 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A002098, A046675, A298160.

Convolution inverse of A002098.

A319112 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k)/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 6, 44, 170, 1644, 7448, 72624, 653112, 8510160, 62704752, 1324662624, 10772812752, 167386388064, 2413326453120, 52610523489024, 597065112874368, 18066985168806144, 212119023906342144, 4734822914239173120, 100734270778298352384, 2818116390408742291968, 48201015565806837709824

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Cf. A002098, A007841, A319113.

seq(n!*coeff(series(mul(1/(1-x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

nmax = 24; CoefficientList[Series[Product[1/(1 - x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] d^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^(1-k/p) ) * x^k/k).

A376147 a(1)=1, followed by array T(n,k), n>=1, k>=2 read by antidiagonals (downwards) wherein the first row is A056240, and the k-th column records in ascending order the numbers m such that A001414(m) = k.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 7, 9, 15, 10, 14, 16, 12, 21, 20, 18, 11, 25, 24, 35, 28, 30, 27, 13, 42, 40, 32, 33, 22, 50, 45, 36, 26, 49, 56, 60, 48, 39, 44, 70, 63, 64, 54, 17, 55, 105, 84, 75, 72, 65, 52, 66, 112, 100, 80, 81, 19, 77, 88, 98, 125, 120, 90, 51, 34, 78

Offset: 1

David James Sycamore, Sep 12 2024

nonn

The columns of T(n,k) are of finite length, corresponding to A000607(k), whereas the rows are of infinite length. This is a permutation of A064364 (which reads 1 plus the consecutive columns of T(n,k)), and hence of the positive integers A000027.

			Construct the irregular table T(n,k) as follows:
The first row T(1,k) is A056240, smallest number whose sum of prime divisors (with multiplicity) is k (k>=2).  The second row T(2,k) is the second smallest number (if it exists) whose sum of prime divisors is k, and so on. The k-th column is then the ordered list of the A000607(k) numbers (k>=2) whose prime divisors sum to k, the final term of which is A000792(k), after which the k-th column contains no further terms. The sum of the terms in the k-th column (k>=2) is A002098(k).
Read the table T(n,k) by antidiagonals downwards to obtain the data:
  2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13.. (A056240)
           6, 9,10, 16, 20, 25, 28, 42, 22..
                12, 18, 24, 30, 40, 50, 56..
                        27, 32, 45, 60, 63..
                            36, 48, 64, 75..
                                54, 72, 80..
                                    81, 90..
And so on…

Michael De Vlieger, Table of n, a(n) for n = 1..12011 (rows n = 1..180, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..22572.

Cf. A000027, A000607, A000792, A001414, A002098, A056240, A064364.

kk = 30;
MapIndexed[Set[t[First[#2]], #1] &,
  Rest@ CoefficientList[
    Series[(1 + x + 2 x^2 + x^4)/(1 - 3 x^3), {x, 0, kk}], x] ];
Array[Set[r[#],
  Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, t[kk]];
s = Table[
  Select[Range[Prime@ PrimePi[k], t[k]], r[#] == k &], {k, 2, kk}];
Join[{1}, s[[1]],
  Table[i = 1; m = n;
    Reap[While[And[m > 1, Length@ s[[m]] >= i], Sow[s[[m, i]] ]; m--;
i++]][[-1, 1]], {n, 2, kk - 1}] ] // Flatten (* Michael De Vlieger, Sep 18 2024 *)

cp's OEIS Frontend

A064364 Positive integers sorted by A001414(n), the sum of their prime divisors, as the major key and n as the minor key.

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A002099 G.f.: -1 + Product_{k>=1} (1 + prime(k)*x^prime(k)).

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A116864 Array of product of parts of the partitions of n with only prime parts.

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A298159 G.f.: Product_{k>=1} (1-prime(k)*x^prime(k)).

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A319112 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k)/prime(k)).

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A376147 a(1)=1, followed by array T(n,k), n>=1, k>=2 read by antidiagonals (downwards) wherein the first row is A056240, and the k-th column records in ascending order the numbers m such that A001414(m) = k.

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A302837 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^prime(k)).

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