A001414 -id:A001414 - cp's OEIS frontend

A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 70, 71, 72, 73, 79, 83, 84, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 180, 181, 191, 193, 197, 199, 211, 220, 223

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002

nonn

Union of A046346 and the primes. - T. D. Noe, Feb 20 2007
These are the Heinz numbers of the partitions counted by A330953. - Gus Wiseman, Jan 17 2020
Alladi and Erdős (1977) noted that sopfr(k) = k if k is a prime or k = 4. They called the terms for which k/sopfr(k) > 1 "special numbers", and proved that there are infinitely many such terms that are squarefree. - Amiram Eldar, Nov 02 2020

			a(12) = 27 because sopfr(27) = 3 + 3 + 3 = 9 and 27 is divisible by 9.

Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.

sopfr(n) is defined in A001414.
The version for prime indices instead of prime factors is A324851.
Partitions whose Heinz number is divisible by their sum: A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions whose product is divisible by their sum of primes: A330954.
Partitions whose product divides their sum of primes: A331381.
Product of prime indices is divisible by sum of prime factors: A331378.
Sum of prime factors is divisible by sum of prime indices: A331380.
Product of prime indices equals sum of prime factors: A331384.
Cf. A056239, A112798, A120383, A238525, A331379, A331382, A331383.

a036844 n = a036844_list !! (n-1)
a036844_list = filter ((== 0). a238525) [2..]
-- Reinhard Zumkeller, Jul 21 2014

Select[Range[2, 224], Divisible[#, Plus @@ Times @@@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

is_A036844(n)=n>1 && !(n%A001414(n)) \\ M. F. Hasler, Mar 01 2014

A238525(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2014

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

Original entry on oeis.org

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

A373364 a(n) = gcd(A001414(n), A003415(n)), where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 1, 6, 6, 7, 1, 1, 1, 9, 8, 8, 1, 1, 1, 3, 10, 13, 1, 1, 10, 15, 9, 1, 1, 1, 1, 10, 14, 19, 12, 10, 1, 21, 16, 1, 1, 1, 1, 3, 1, 25, 1, 1, 14, 3, 20, 1, 1, 1, 16, 1, 22, 31, 1, 4, 1, 33, 1, 12, 18, 1, 1, 3, 26, 1, 1, 12, 1, 39, 1, 1, 18, 1, 1, 1, 12, 43, 1, 2, 22, 45, 32, 1, 1, 1, 20, 3, 34, 49, 24

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

For n >= 1, a(n) is a multiple of A373363(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A003415, A373375 (positions of even terms), A373376 (of odd terms).

Cf. also A082299, A373362, A373363.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373364(n) = gcd(A001414(n), A003415(n));

A373362 a(n) = gcd(A001414(n), A276085(n)), where A001414 is the sum of prime factors with repetition, and A276085 is the primorial base log-function.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 7, 1, 1, 1, 1, 8, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 12, 2, 1, 1, 8, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 4, 17, 1, 1, 8, 1, 2, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 9, 1, 1, 4, 1, 1, 2, 2, 1, 8, 1, 1, 1, 20, 1, 2, 1, 12, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

As A001414 and A276085 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication: For example, A373373, which gives the indices of multiples of 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A002110, A276085.
Cf. A345452 (positions of even terms), A373373 (positions of multiples of 3).
Cf. also A082299, A373361, A373363, A373367.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373362(n) = gcd(A001414(n), A276085(n));

A373363 a(n) = gcd(A001414(n), A083345(n)), where A001414 is the sum of prime factors with repetition, and A083345 is the numerator of the sum of the inverses of prime factors with repetition.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 1, 1, 9, 8, 2, 1, 1, 1, 3, 10, 13, 1, 1, 2, 15, 1, 1, 1, 1, 1, 5, 14, 19, 12, 5, 1, 21, 16, 1, 1, 1, 1, 3, 1, 25, 1, 1, 2, 3, 20, 1, 1, 1, 16, 1, 22, 31, 1, 1, 1, 33, 1, 3, 18, 1, 1, 3, 26, 1, 1, 1, 1, 39, 1, 1, 18, 1, 1, 1, 4, 43, 1, 1, 22, 45, 32, 1, 1, 1, 20, 3, 34, 49, 24, 1, 1, 1, 1, 7

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001414, A083345.
Cf. A345452 (positions of even terms), A353374 (their characteristic function).
Cf. also A082299, A373362.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373363(n) = gcd(A001414(n), A083345(n));

A373478 Numbers k such that A001414(k) and A003415(k) are both multiples of 3, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

Original entry on oeis.org

1, 8, 9, 14, 20, 26, 27, 35, 38, 44, 50, 62, 64, 65, 68, 72, 74, 77, 81, 86, 92, 95, 110, 112, 116, 119, 122, 125, 126, 134, 143, 146, 155, 158, 160, 161, 164, 170, 180, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 216, 218, 221, 230, 234, 236, 242, 243, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305

Offset: 1

Antti Karttunen, Jun 07 2024

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001414, A003415, A373477 (characteristic function).
Positions of multiples of 3 in A373364.
Intersection of A289142 and A327863.
Disjoint union of A373475 and A373479.

PARI
```
isA373478 = A373477;
```

A025281 a(n) = sopfr(n!), where sopfr = A001414 is the integer log.

Original entry on oeis.org

0, 0, 2, 5, 9, 14, 19, 26, 32, 38, 45, 56, 63, 76, 85, 93, 101, 118, 126, 145, 154, 164, 177, 200, 209, 219, 234, 243, 254, 283, 293, 324, 334, 348, 367, 379, 389, 426, 447, 463, 474, 515, 527, 570, 585, 596, 621, 668, 679, 693, 705, 725, 742, 795, 806, 822, 835, 857, 888

Offset: 0

Wouter Meeussen

nonn

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms n = 0..1000 from T. D. Noe)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294; alternative link.

Partial sums of A001414.

Cf. A000142, A072691.

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)+add(i[1]*i[2], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 09 2021

sopfr[n_] := Plus @@ Times @@@ FactorInteger@ n; a[n_] := a[n] = a[n - 1] + sopfr[n]; a[0] = a[1] = 0; Array[a, 59, 0] (* Robert G. Wilson v, May 18 2015 *)

for(n=1,100,print1(sum(k=1,n,sum(i=1,omega(k), component(component(factor(k),1),i)*component(component(factor(k),2),i))),","))

from sympy import factorial, factorint
def A025281(n): return sum(p*e for p, e in factorint(factorial(n)).items()) # Chai Wah Wu, Apr 09 2021

a(n) = A001414(A000142(n)).
From Benoit Cloitre, Apr 14 2002: (Start)
a(0)=0; for n>0, a(n) = Sum_{k=1..n} A001414(k).
Asymptotic formula: a(n) ~ (Pi^2/12)*n^2/log(n). [Proven by Alladi and Erdős (1977). - Amiram Eldar, Mar 04 2021]
(End)

A086711 Primes p such that A001414(p-1) = A001414(p+1), where A001414 = sum of primes dividing n (with repetition).

Original entry on oeis.org

11, 17, 31, 251, 1429, 3041, 16561, 16927, 53299, 56897, 89783, 95089, 213599, 282977, 345547, 432587, 592223, 763457, 906949, 915799, 1050449, 1058389, 1485017, 1577341, 1678399, 1780253, 1855549, 2131687, 2374289, 2658259

Offset: 1

Jason Earls, Jul 28 2003

nonn

Conjecture: sequence is infinite.

			a(3)=31 because it is prime and 30=2*3*5, 32=2^5 and 2+3+5 = 2+2+2+2+2 = 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1280

Cf. A001414, A190680.

A373475 Numbers k such that A001414(k) and A083345(k) are both multiples of 3, where A001414 is fully additive with a(p) = p, and A083345 is the numerator of the fully additive function with a(p) = 1/p.

Original entry on oeis.org

1, 8, 14, 20, 26, 35, 38, 44, 50, 62, 64, 65, 68, 74, 77, 86, 92, 95, 110, 112, 116, 119, 122, 125, 134, 143, 146, 155, 158, 160, 161, 164, 170, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 218, 221, 230, 236, 242, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305, 314, 323, 326, 329, 332, 335, 341, 343

Offset: 1

Antti Karttunen, Jun 06 2024

nonn

If k is a term, then 3^9 * k is also a term. See A373476.
A369659 is a subsequence of this sequence, giving the terms that are not multiples of 3. This follows because A083345(n) = n' / gcd(n',n) and from the following lemma: When k is not a multiple of 3, then either sopfr(k) [= A001414(k)] and k' [= A003415(k)] are both multiples of 3, or both are non-multiples of 3.
Proof of the lemma: As k is not a multiple of 3, all its prime factors p, q, r, s, t, u, v, w, ... (not necessarily all distinct) are either of the form 3m+1 or 3m-1. Let's first eliminate from k all triplets of primes that are of the same type modulo 3, either -1 or +1, (marked now as p, q, r) as they do not affect the divisibility by 3 of either the sopfr(k) or k'. In the case of the arithmetic derivative this is because we have k' = (pqr)' * (k/pqr) + (k/pqr)' * pqr, and as we know that the first summand is a multiple of 3 (because (pqr)' is), therefore the divisibility of the whole expression by 3 depends only on whether (k/pqr)' is a multiple of 3, as certainly pqr is not a multiple of 3.
  What will remain after such elimination process has been completed as far as possible, must be either 1, or of the form p*q (p and q of different types), or p*q*r*s (with two primes of one type, and two primes of the other type), in which cases both sopfr(k) and k' are multiples of 3, or then alternatively, what remains must be of the form p*q (p and q of the same type), or p*q*r (with two primes of one type and the third of the other type), both cases which indicate that both sopfr(k) and k' are non-multiples of 3.

			110 = 2*5*11 is a term of this sequence because 2+5+11 = 18 is a multiple of 3, and also 2*5 + 2*11 + 5*11 = 87 is a multiple of 3.
54 (= A369644(10)) is NOT a term of this sequence, because A001414(54) = 11 is not a multiple of 3, although A083345(54) = 3 is.
19683 = 3^9 is a term of this sequence, because both A001414(19683) = 9*3 = 27 and A083345(19683) = A003415(3^9)/gcd(3^9, A003415(3^9)) = 3, are multiples of 3.

Antti Karttunen, Table of n, a(n) for n = 1..33911
Index entries for sequences which agree for a long time but are different

Cf. A001414, A003415, A083345, A373474 (characteristic function).
Positions of multiples of 3 in A373363.
Intersection of A289142 and A369644.
Subsequence of A373478.
Disjoint union of A369659 and A373476.
Differs from A369659 for the first time at n=4186, where a(4186) = A373476(1) = 19683, a term not present in A369659, as it is the first multiple of 3 in this sequence.

PARI
```
isA373475 = A373474;
```

a(n) = A373476(n) / 3^9.

A036288 a(n) = 1 + integer log of n: if the prime factorization of n is n = Product (p_j^k_j) then a(n) = 1 + Sum (p_j * k_j) (cf. A001414).

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 8, 7, 7, 8, 12, 8, 14, 10, 9, 9, 18, 9, 20, 10, 11, 14, 24, 10, 11, 16, 10, 12, 30, 11, 32, 11, 15, 20, 13, 11, 38, 22, 17, 12, 42, 13, 44, 16, 12, 26, 48, 12, 15, 13, 21, 18, 54, 12, 17, 14, 23, 32, 60, 13, 62, 34, 14, 13, 19, 17, 68, 22

Offset: 1

N. J. A. Sloane

nonn

If this function is iterated then, starting at any number n >= 7, we will always reach an 8 - see A212813, A212814, A212815. - N. J. A. Sloane, May 30 2012

a(n) = 1 + Sum_{k=1..A001221(n)} A027748(k) * A124010(k). - Reinhard Zumkeller, May 30 2012

			12 = 2^2 * 3 so a(12) = 1 + 2^2 + 3 = 8.

Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006) - From N. J. A. Sloane, May 30 2012
R. Honsberger, Problem 89, Another Curious Sequence, Mathematical Morsels, MAA, 1978, pp. 223-227.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. B. Roberts, Problem E2356, Amer. Math. Monthly, 79 (1972); solution by H. Kappus, loc. cit., 80 (1973), p. 810.

Equals A001414 + 1.

Cf. A212813, A212814, A212815, A212816, A212908, A212909.

a036288 n = 1 + sum (zipWith (*)
            (a027748_row n) (map fromIntegral $ a124010_row n))
-- Reinhard Zumkeller, May 30 2012

f:=proc(n) local i,t1; t1:=ifactors(n)[2]; 1+add( t1[i][1]*t1[i][2], i=1..nops(t1)); end; # N. J. A. Sloane, May 30 2012

f[1]=1;f[n_]:=Total[Apply[Times,FactorInteger[n],1]]+1;f/@Range@68 (* Ivan N. Ianakiev, Apr 18 2016 *)

A036288(n)=1+(n=factor(n))[,1]~*n[,2]  \\ M. F. Hasler, May 30 2012

Edited by N. J. A. Sloane, Jun 01 2012

cp's OEIS Frontend

A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A373364 a(n) = gcd(A001414(n), A003415(n)), where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A373362 a(n) = gcd(A001414(n), A276085(n)), where A001414 is the sum of prime factors with repetition, and A276085 is the primorial base log-function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A373363 a(n) = gcd(A001414(n), A083345(n)), where A001414 is the sum of prime factors with repetition, and A083345 is the numerator of the sum of the inverses of prime factors with repetition.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A373478 Numbers k such that A001414(k) and A003415(k) are both multiples of 3, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A025281 a(n) = sopfr(n!), where sopfr = A001414 is the integer log.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A086711 Primes p such that A001414(p-1) = A001414(p+1), where A001414 = sum of primes dividing n (with repetition).

Views