A331379 -id:A331379 - cp's OEIS frontend

A000607 Number of partitions of n into prime parts.

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 30, 35, 40, 46, 52, 60, 67, 77, 87, 98, 111, 124, 140, 157, 175, 197, 219, 244, 272, 302, 336, 372, 413, 456, 504, 557, 614, 677, 744, 819, 899, 987, 1083, 1186, 1298, 1420, 1552, 1695, 1850, 2018, 2198, 2394, 2605, 2833, 3079, 3344

Offset: 0

N. J. A. Sloane

a(n) gives the number of values of k such that A001414(k) = n. - Howard A. Landman, Sep 25 2001
Let W(n) = {prime p: There is at least one number m whose spf is p, and sopfr(m) = n}. Let V(n,p) = {m: sopfr(m) = n, p belongs to W(n)}. Then a(n) = sigma(|V(n,p)|). E.g.: W(10) = {2,3,5}, V(10,2) = {30,32,36}, V(10,3) = {21}, V(10,5) = {25}, so a(10) = 3+1+1 = 5. - David James Sycamore, Apr 14 2018
From Gus Wiseman, Jan 18 2020: (Start)
Also the number of integer partitions such that the sum of primes indexed by the parts is n. For example, the sum of primes indexed by the parts of the partition (3,2,1,1) is prime(3)+prime(2)+prime(1)+prime(1) = 12, so (3,2,1,1) is counted under a(12). The a(2) = 1 through a(14) = 10 partitions are:
  1  2  11  3   22   4    32    41    33     5      43      6       44
            21  111  31   221   222   42     322    331     51      52
                     211  1111  311   321    411    421     332     431
                                2111  2211   2221   2222    422     3222
                                      11111  3111   3211    3221    3311
                                             21111  22111   4111    4211
                                                    111111  22211   22221
                                                            31111   32111
                                                            211111  221111
                                                                    1111111
(End)

			n = 10 has a(10) = 5 partitions into prime parts: 10 = 2 + 2 + 2 + 2 + 2 = 2 + 2 + 3 + 3 = 2 + 3 + 5 = 3 + 7 = 5 + 5.
n = 15 has a(15) = 12 partitions into prime parts: 15 = 2 + 2 + 2 + 2 + 2 + 2 + 3 = 2 + 2 + 2 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 + 5 = 2 + 2 + 2 + 2 + 7 = 2 + 2 + 3 + 3 + 5 = 2 + 3 + 5 + 5 = 2 + 3 + 3 + 7 = 2 + 2 + 11 = 2 + 13 = 3 + 3 + 3 + 3 + 3 = 3 + 5 + 7 = 5 + 5 + 5.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 203.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
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).

Hans Havermann, Table of n, a(n) for n = 0..50000 (first 1000 terms from T. D. Noe, terms 1001-20000 from Vaclav Kotesovec, terms 20001-50000 extracted from files by Hans Havermann)
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
George E. Andrews, Arnold Knopfmacher, and John Knopfmacher, Engel expansions and the Rogers-Ramanujan identities, J. Number Theory 80 (2000), 273-290. See Eq. 2.1.
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the Number of Distinct Multinomial Coefficients, Journal of Number Theory, Volume 118, Issue 1, May 2006, pp. 15-30.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Johann Bartel, R. K. Bhaduri, Matthias Brack, and M. V. N. Murthy, Asymptotic prime partitions of integers, Phys. Rev. E, 95 (2017), 052108, arXiv:1609.06497 [math-ph], 2016-2017.
P. T. Bateman and P. Erdős, Partitions into primes, Publ. Math. Debrecen 4 (1956), 198-200.
J. Browkin, Sur les décompositions des nombres naturels en sommes de nombres premiers, Colloquium Mathematicum 5 (1958), 205-207.
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
L. M. Chawla and S. A. Shad, Review of "On a trio-set of partition functions and their tables", Mathematics of Computation, Vol. 24, No. 110 (Apr., 1970), pp. 490-491.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see Section VIII.6, pp. 576ff.
H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187.
O. P. Gupta and S. Luthra, Partitions into primes, Proc. Nat. Inst. Sci. India. Part A. 21 (1955), 181-184.
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12. (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
H. Havermann, Tables of sum-of-prime-factors sequences (including sequence lengths, i.e., number of prime-parts partitions, for the first 50000).
BongJu Kim, Partition number identities which are true for all set of parts, arXiv:1803.08095 [math.CO], 2018.
Vaclav Kotesovec, Graph - asymptotic ratio for log(a(n)), without minor asymptotic term.
Vaclav Kotesovec, Wrong asymptotics of OEIS A000607? (MathOverflow). Includes discussion of the contradiction between the results for the next-to-leading term in the asymptotic formulas by Vaughan and by Bartel et al.
John F. Loase (splurge(AT)aol.com), David Lansing, Cassie Hryczaniuk and Jamie Cahoon, A Variant of the Partition Function, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), pp. 320-321.
Ljuben Mutafchiev, A Note on Goldbach Partitions of Large Even Integers, arXiv:1407.4688 [math.NT], 2014-2015.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
N. J. A. Sloane, Transforms.
R. C. Vaughan, On the number of partitions into primes, Ramanujan J. vol. 15, no. 1 (2008) 109-121.
Eric Weisstein's World of Mathematics, Prime Partition.
Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.
Index entries for sequences related to Goldbach conjecture

G.f. = 1 / g.f. for A046675. See A046113 for the ordered (compositions) version.
Cf. A046676, A048165, A004526, A051034, A000040, A001414, A000586 (distinct parts), A000041, A070214, A192541, A112021, A056768, A128515, A319265, A319266.
Row sums of array A116865 and of triangle A261013.
Column sums of A331416.
Partitions whose Heinz number is divisible by their sum of primes are A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Cf. A000720, A014689, A331385, A331387, A331415, A331417.

a000607 = p a000040_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 05 2012

[1] cat [#RestrictedPartitions(n,{p:p in PrimesUpTo(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019

with(gfun):
t1:=mul(1/(1-q^ithprime(n)),n=1..51):
t2:=series(t1,q,50):
t3:=seriestolist(t2); # fixed by Vaclav Kotesovec, Sep 14 2014

CoefficientList[ Series[1/Product[1 - x^Prime[i], {i, 1, 50}], {x, 0, 50}], x]
f[n_] := Length@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]; Array[f, 57] (* Robert G. Wilson v, Jul 23 2010 *)
Table[Length[Select[IntegerPartitions[n],And@@PrimeQ/@#&]],{n,0,60}] (* Harvey P. Dale, Apr 22 2012 *)
a[n_] := a[n] = If[PrimeQ[n], 1, 0]; c[n_] := c[n] = Plus @@ Map[# a[#] &, Divisors[n]]; b[n_] := b[n] = (c[n] + Sum[c[k] b[n - k], {k, 1, n - 1}])/n; Table[b[n], {n, 1, 20}] (* Thomas Vogler, Dec 10 2015: Uses Euler transform, caches computed values, faster than IntegerPartitions[] function. *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j + 1]] -= poly[[j + 1 - p]], {j, nmax, p, -1}];, {k, 2, pmax}]; s = Sum[poly[[k + 1]]*x^k, {k, 0, Length[poly] - 1}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2021 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,1-x^prime(k))) \\ Joerg Arndt, Sep 04 2014

from sympy import primefactors
l = [1, 0]
for n in range(2, 101):
    l.append(sum(sum(primefactors(k)) * l[n - k] for k in range(1, n + 1)) // n)
l  # Indranil Ghosh, Jul 13 2017

[Partitions(n, parts_in=prime_range(n + 1)).cardinality() for n in range(100)]  # Giuseppe Coppoletta, Jul 11 2016

Asymptotically a(n) ~ exp(2 Pi sqrt(n/log n) / sqrt(3)) (Ayoub).
a(n) = (1/n)*Sum_{k=1..n} A008472(k)*a(n-k). - Vladeta Jovovic, Aug 27 2002
G.f.: 1/Product_{k>=1} (1-x^prime(k)).
See the partition arrays A116864 and A116865.
From Vaclav Kotesovec, Sep 15 2014 [Corrected by Andrey Zabolotskiy, May 26 2017]: (Start)
It is surprising that the ratio of the formula for log(a(n)) to the approximation 2 * Pi * sqrt(n/(3*log(n))) exceeds 1. For n=20000 the ratio is 1.00953, and for n=50000 (using the value from Havermann's tables) the ratio is 1.02458, so the ratio is increasing. See graph above.
A more refined asymptotic formula is found by Vaughan in Ramanujan J. 15 (2008), pp. 109-121, and corrected by Bartel et al. (2017): log(a(n)) = 2*Pi*sqrt(n/(3*log(n))) * (1 - log(log(n))/(2*log(n)) + O(1/log(n))).
See Bartel, Bhaduri, Brack, Murthy (2017) for a more complete asymptotic expansion. (End)
G.f.: 1 + Sum_{i>=1} x^prime(i) / Product_{j=1..i} (1 - x^prime(j)). - Ilya Gutkovskiy, May 07 2017
a(n) = A184198(n) + A184199(n). - Vaclav Kotesovec, Jan 11 2021

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

Original entry on oeis.org

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

A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 22, 15, 30, 42, 77, 42, 101, 56, 176, 176, 231, 135, 490, 490, 490, 792, 1002, 490, 1575, 627, 3010, 2436, 2436, 3718, 5604, 1958, 4565, 6842, 12310, 3718, 14883, 4565, 21637, 26015, 17977, 8349, 53174, 44583, 63261

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(10) = 11 partitions:
  1  11  21  211   32   321    43    5111      522      631
             1111  311  2211   421   32111     3222     3331
                        21111  4111  41111     4221     4321
                                     221111    22221    5311
                                     311111    32211    32221
                                     2111111   222111   33211
                                     11111111  2211111  43111
                                                        322111
                                                        331111
                                                        3211111
                                                        31111111
For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).

The Heinz numbers of these partitions are given by A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,n]&]],{n,20}]

A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 3, 4, 6, 3, 12, 10, 12, 14, 27, 38, 44, 52, 48, 77, 101, 106, 127, 206, 268, 377, 392, 496, 602, 671, 821, 1090, 1318, 1568, 1926, 2260, 2703, 3258, 3942, 4858, 5923, 6891, 8286, 9728, 11676, 13775, 16314, 19749, 23474, 27793, 32989, 38775

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
  1  2   3  4     5  6    7      8         9        A         B
     11     1111     222  3211   431       432      5311      542
                     321  22111  4211      3321     22111111  5411
                                 11111111  32211              33221
                                           321111             42221
                                           2211111            53111
                                                              322211
                                                              431111
                                                              521111
                                                              2222111
                                                              3311111
                                                              32111111
For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.

The Heinz numbers of these partitions are given by A036844.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product is divisible by their sum of primes are A330954.
Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,Plus@@Prime/@#]&]],{n,30}]

A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 3, 4, 1, 3, 4, 5, 0, 3, 3, 1, 6, 2, 1, 5, 4, 2, 3, 4, 2, 2, 3, 1, 5, 2, 3, 4, 6, 5, 2, 7, 1, 3, 5, 3, 4, 2, 5, 5, 4, 7, 3, 6, 4, 4, 2, 4, 4, 3, 9, 4, 3, 5, 3, 5, 4, 4, 4, 3, 7, 4, 2, 8, 2, 3

Offset: 1

Gus Wiseman, Jan 16 2020

nonn

			The a(n) partitions for n = 7, 9, 18, 24:
  (4,3)  (6,3)    (12,4,1,1)                 (19,4,1)
         (4,4,1)  (11,4,1,1,1)               (18,4,1,1)
                  (8,5,1,1,1,1,1)            (9,6,1,1,1,1,1,1,1,1,1)
                  (4,2,2,2,1,1,1,1,1,1,1,1)
For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).

The Heinz numbers of these partitions are given by A331384.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Times@@#==Plus@@Prime/@#&]],{n,30}]

a(n) = my(c=0); forpart(v=n, if(vecprod(Vec(v))==sum(i=1, #v, prime(v[i])), c++)); c; \\ Jinyuan Wang, Feb 14 2025

a(71)-a(87) from Robert Price, Apr 10 2020

A331416 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 3, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 6, 3, 4, 2

Offset: 0

Gus Wiseman, Jan 17 2020

			Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 0 0 2 1
  0 0 0 0 0 0 1 3 1
  0 0 0 0 0 0 0 0 2 3 1 1
  0 0 0 0 0 0 0 0 0 1 4 3 1 2
  0 0 0 0 0 0 0 0 0 0 0 2 5 3 2 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1 4 6 3 4 2 0 2
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 6 4 6 2 1 2 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 8 6 6 7 2 4 2 0 1 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 6 9 7 9 7 3 7 2 1 1 0 0 0 2
Row n = 8 counts the following partitions (empty column not shown):
  (2222)  (332)    (44)      (41111)    (53)        (611)   (8)
          (422)    (431)     (311111)   (62)        (5111)  (71)
          (3221)   (3311)    (2111111)  (521)
          (22211)  (4211)               (11111111)
                   (32111)
                   (221111)
Column k = 19 counts the following partitions:
  (8)   (6111)   (532)        (443)       (33222)
  (71)  (51111)  (622)        (4331)      (42222)
                 (5221)       (4421)      (322221)
                 (4111111)    (33311)     (2222211)
                 (31111111)   (43211)
                 (211111111)  (332111)
                              (422111)
                              (3221111)
                              (22211111)

Row lengths are A331417.
Row sums are A000041.
Column sums are A000607.
Shifting row n to the left n times gives A331385.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions of whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A014689, A056239, A330950, A330954, A331378, A331387, A331415, A331418.

maxm[n_]:=Max@@Table[Total[Prime/@y],{y,IntegerPartitions[n]}];
Table[Length[Select[IntegerPartitions[n],Total[Prime/@#]==k&]],{n,0,10},{k,0,maxm[n]}]

A330954 Number of integer partitions of n whose product is divisible by the sum of primes of their parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 4, 2, 3, 9, 8, 18, 15, 25, 35, 44, 50, 70, 71, 93, 141, 158, 226, 286, 337, 439, 532, 648, 789, 1013, 1261, 1454, 1776, 2176, 2701, 3258, 3823, 4606, 5521, 6613, 7810, 9202, 11074, 13145, 15498, 18413, 21818, 25774, 30481, 35718

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

			The a(7) = 1 through a(15) = 8 partitions (empty column not shown):
  43  63   541     83     552   6322   4433       5532
      441  4222    3332   6411  7411   7322       6522
           222211  5222         62221  44321      84111
                   33221               63311      333222
                                       65111      432222
                                       72221      3322221
                                       433211     32222211
                                       4322111    333111111
                                       322211111
For example, the partition (3,3,2,2,1) has product 3 * 3 * 2 * 2 * 1 = 36 and sum of primes 5 + 5 + 3 + 3 + 2 = 18, and 36 is divisible by 18, so (3,3,2,2,1) is counted under a(11).

The Heinz numbers of these partitions are given by A331378.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose sum of primes divides their product of primes are A330953.
Partitions whose sum of primes divides of their product are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A036844, A056239, A324850, A326149, A330950, A331379, A331382, A331384, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,Plus@@Prime/@#]&]],{n,30}]

A331381 Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jan 16 2020

nonn

			The a(n) partitions for n = 1, 5, 7, 8, 9, 13, 14:
  1  221    43       311111    63         7411           65111
     311    511      11111111  441        721111         322211111
     11111  3211               711        43111111       311111111111
            22111              42111      421111111      11111111111111
            1111111            2211111    3211111111
                               111111111  22111111111
                                          1111111111111

The Heinz numbers of these partitions are given by A331382.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379, A331415, A331416.

Table[Length[Select[IntegerPartitions[n],Divisible[Plus@@Prime/@#,Times@@#]&]],{n,0,30}]

A331378 Numbers whose product of prime indices is divisible by their sum of prime factors.

Original entry on oeis.org

35, 65, 95, 98, 154, 189, 297, 324, 363, 364, 375, 450, 476, 585, 623, 702, 763, 765, 791, 812, 826, 918, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1197, 1225, 1287, 1288, 1300, 1305, 1309, 1449, 1470, 1484, 1517, 1566, 1593, 1665, 1708, 1710, 1736, 1769

Offset: 1

Gus Wiseman, Jan 15 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    35: {3,4}
    65: {3,6}
    95: {3,8}
    98: {1,4,4}
   154: {1,4,5}
   189: {2,2,2,4}
   297: {2,2,2,5}
   324: {1,1,2,2,2,2}
   363: {2,5,5}
   364: {1,1,4,6}
   375: {2,3,3,3}
   450: {1,2,2,3,3}
   476: {1,1,4,7}
   585: {2,2,3,6}
   623: {4,24}
   702: {1,2,2,2,6}
   763: {4,29}
   765: {2,2,3,7}
   791: {4,30}
   812: {1,1,4,10}
For example, 450 = prime(1)*prime(2)*prime(2)*prime(3)*prime(3) has prime indices {1,2,2,3,3} and prime factors {2,3,3,5,5}, and since 36 is divisible by 18, 450 is in the sequence.

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

These are the Heinz numbers of the partitions counted by A330954.
Numbers divisible by the sum of their prime factors are A036844.
Numbers divisible by the sum of their prime indices are A324851.
Sum of prime indices divides product of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product divides their sum of primes are A331381.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A057568, A324850, A330953, A331379, A331381, A331382, A331383.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@primeMS[#],Total[Prime/@primeMS[#]]]&]

A331384 Numbers whose sum of prime factors is equal to their product of prime indices.

Original entry on oeis.org

35, 65, 95, 98, 154, 324, 364, 476, 623, 763, 791, 812, 826, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1288, 1484, 1708, 1736, 2044, 2408, 2632, 4320, 5408, 6688, 6974, 8000, 10208, 12623, 12701, 12779, 14144, 19624, 23144, 25784, 26048, 44176, 47696

Offset: 1

Gus Wiseman, Jan 16 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Numbers k such that A001414(k) = A003963(k). - Jason Yuen, Dec 19 2024

			The sequence of terms together with their prime indices begins:
     35: {3,4}
     65: {3,6}
     95: {3,8}
     98: {1,4,4}
    154: {1,4,5}
    324: {1,1,2,2,2,2}
    364: {1,1,4,6}
    476: {1,1,4,7}
    623: {4,24}
    763: {4,29}
    791: {4,30}
    812: {1,1,4,10}
    826: {1,4,17}
    938: {1,4,19}
    994: {1,4,20}
   1036: {1,1,4,12}
   1064: {1,1,1,4,8}
   1106: {1,4,22}
   1144: {1,1,1,5,6}
   1148: {1,1,4,13}
For example, 476 has prime factors {2,2,7,17} and prime indices {1,1,4,7}, and 2+2+7+17 = 28 = 1*1*4*7, so 476 is in the sequence.

These are the Heinz numbers of the partitions counted by A331383.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
Cf. A000040, A001414, A003963, A324850, A330954, A331378, A331379, A331382, A331415, A331416.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==Plus@@Prime/@primeMS[#]&]

cp's OEIS Frontend

A000607 Number of partitions of n into prime parts.

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

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A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

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A331383 Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.

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A331416 Irregular triangle read by rows where T(n,k) is the number of integer partitions y of n such that Sum_i prime(y_i) = k.

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A330954 Number of integer partitions of n whose product is divisible by the sum of primes of their parts.

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A331381 Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.

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