A056240 -id:A056240 - cp's OEIS frontend

A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2*n).

2, 4, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753, 771, 789, 807, 813, 831, 843

Offset: 1

David James Sycamore, Jun 07 2017

nonn

This is an ascending subsequence of A056240 with even argument terms.
After the first three (even) terms, a(1) = b(2) = 2, a(2) = b(4) = 4, a(3) = b(6) = 8 respectively, all subsequent terms are odd (semiprime) numbers of the form 3*r, for r = primes 5, 7, 11, 13, .... The graph of all odd-valued terms a(n) for n >= 4 is a straight line (y = 3*x - 9), corresponding to b(2*n) = 3*(2*n) - 9 = 3*(2*n - 3) = 3*r, where r = 2*n - 3 is prime, and n is in sequence A098090. The sequence a(n) for n >= 4 is identical term for term to A001748(n) for n >= 3. In other words, for n >= 4, a(n) = 3*A000040(n-1).
If, for any even number n >= 6, n - 3 is prime, then A056240(n) belongs to this sequence.

			a(1) = 2 is included because for all n > 1, b(2n) > 2; likewise a(2) = b(4) = 4, and a(3) = b(6) = 8 are included. The first odd term, a(4) = b(8) = 15, is included since for all n > 4, b(2n) > 15. b(12) = 35 is not in this sequence because b(14) = 33 < 35, and only ascending terms are permitted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001748, A056240, A098090. Essentially the same as A063534.

Join[{2, 4, 8}, 3*Prime[Range[3, 100]]] (* Paolo Xausa, Apr 16 2024 *)

a(1) = 2, a(2) = 4, a(3) = 8, and for n >= 4, a(n) = 3*A000040(n-1).

Offset changed to 1 and entry edited to reflect this change by Michel Marcus, Jul 03 2017

A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).

Original entry on oeis.org

5, 211, 4327, 4547, 25523, 81611, 966109, 1654111, 3851587, 1895479, 66407189, 134965049, 129312889, 425845151, 35914507, 504365461, 2400397969, 8490141637, 8429770031, 20416021309, 23555107819, 23912414437

Offset: 1

David James Sycamore, Feb 06 2018

For a prime p >= 5 whose prime-index is m, the a056240-type of p is defined to be the unique integer k such that A288814(p) = prime(m-k)*A056240(prime(m)-prime(m-k)).
In other words, k is such that prime(n-k) is the greatest prime divisor of the smallest composite number whose sum of prime factors (taken with multiplicity) is prime(n).
The sequence lists the smallest prime of each successive a056240-type.
In the Examples section, the a056240-type k (=a(k)) of a prime p = prime(m) is indicated by p ~ k(g1,g2,...,gk) where gi = prime(m - i + 1) - prime(m - i). See also A295185.
For the values of the a056240-types of the primes 2, 3, 5, 7, ... see A299912. - N. J. A. Sloane, Mar 10 2018
a(20), a(21) > 14 * 10^9. Conjecture: a(k) > 14 * 10^9 for k > 22. - David A. Corneth, Mar 25 2018
a(20), a(21) computed on the basis of the above conjecture. Note that A321983 records the smallest composite number whose sum of prime divisors (with repetition) is a(n). - David James Sycamore, Nov 30 2018
a(23)..a(25) > 45.8 * 10^9. - David A. Corneth, Dec 02 2018

			a(1) = 5 since 6 = 3 * 2, the smallest composite number whose prime divisors add to 5, is a multiple of 3, the greatest prime < 5, so k=1; 5 ~ 1(2).
a(2) = 211 since 6501 = 3 * 11 * 197, the smallest composite whose prime divisors add to 211, and 197 < 199 < 211 is the second prime below 211, so k=2, and 211 ~ 2(12,2), and since no smaller prime has this property, a(2)=211.
a(3) = 4327 since 526809 = 3 * 41 * 4283, the smallest composite whose prime divisors add to 4327, and 4283 < 4289 < 4297 < 4327 is the third prime below 4327, so k=3, 4327 ~ 3(30,8,6) and since no smaller prime has this property, a(3)=4327. Likewise,
a(4) = 4547 ~ 4(24, 4, 2, 4),
a(5) = 25523 ~ 5(52, 2, 6, 6, 4),
a(6) = 81611 ~ 6(42, 6, 4, 6, 2, 4),
a(7) = 966109 ~ 7(68, 12, 16, 2, 22, 6, 14),
a(8) = 1654111 ~ 8(54, 14, 4, 6, 2, 4, 6, 2),
a(9) = 3851587 ~ 9(128, 16, 12, 2, 6, 10, 14, 10, 2),
a(10) = 1895479 ~ 10(120, 2, 6, 30, 4, 30, 14, 10, 2, 12),
a(11) = 66407189 ~ 11(120, 6, 6, 16, 14, 6, 4, 8, 10, 2, 4),
a(12) = 134965049 ~ 12(138, 10, 2, 22, 18, 20, 6, 12, 18, 16, 8, 10),
a(13) = 129312889 ~ 13(98, 60, 22, 18, 8, 4, 18, 12, 38, 24, 6, 4, 8),
a(14) = 425845151 ~ 14(148, 2, 42, 16, 50, 24, 12, 6, 4, 20, 6, 48, 10, 12),
a(15) = 35914859 ~ 15(126, 82, 8, 4, 18, 12, 8, 4, 14, 6, 16, 8, 6, 30, 10),
a(16) = 504365461 ~ 16(122, 42, 10, 14, 36, 4, 6, 6, 12, 48, 2, 6, 10, 20, 6, 6),
a(17) = 2400397969 ~ 17(122, 58, 8, 4, 18, 36, 2, 4, 6, 32, 10, 2, 16,12,18,32,12),
a(18) = 8490141637 ~ 18(126, 2, 82, 8, 52, 20, 34, 2, 10, 24, 8, 6,34,2,6,28,24,2),
a(19) = 8429770031 ~ 19(148, 26, 16, 18, 12, 2, 18, 18, 10,20,4,2,6,18,6,4,2,18,4),
a(20) = 20416021309 ~ 20(122, 4, 2, 64, 20, 40, 6, 12, 12, 20, 10, 6, 8, 10, 30, 2, 10, 38, 22, 140,
a(21) = 23555107819 ~ 21(192, 20, 156, 30, 18, 10, 2, 12, 58, 12, 12, 26, 28, 32, 4, 6, 12, 2, 6, 22, 2),
a(22) = 23912414437 ~ 22(344, 4, 12, 14, 40, 2, 4, 18, 2, 36, 10, 12, 2, 10, 26, 10, 24, 14, 40, 30, 14, 12).

David A. Corneth, PARI program

Cf. A056240, A288814, A289993, A295185, A299912, A300334, A300359, A321983.

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
snumbr(n) = my(k=2); while(!isok(k, n), k++); k; /* A056240 */
scompo(n) = forcomposite(k=4, , if (isok(k, n), return(k))); /* A288814 */
a(n) = {forprime(p=5,,ip = primepi(p); if (ip > n, x = scompo(p); fmax = vecmax(factor(x)[,1]); ifmax = primepi(fmax); if (ip - ifmax == n, y = fmax*snumbr(p - fmax;); if (y == x, return (p);););););} \\ Michel Marcus, Feb 17 2018

PARI
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\\ see Corneth link
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a(7)-a(10) from Michel Marcus, Feb 23 2018
Name changed by N. J. A. Sloane, Mar 10 2018
a(11)-a(19) from David A. Corneth, Mar 24 2018, Mar 25 2018
a(20)-a(21) from David James Sycamore, Nov 30 2018
a(22) from David A. Corneth, Dec 02 2018

A292081 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that for some m > n, b(2m) < b(2n).

Original entry on oeis.org

35, 65, 95, 115, 161, 155, 217, 185, 215, 235, 329, 265, 371, 305, 427, 335, 365, 511, 395, 415, 581, 445, 623, 1246, 485, 515, 545, 565, 791, 1417, 1243, 1469, 2071, 635, 655, 917, 695, 973, 1507, 1529, 755, 1057, 785, 1099, 815, 835, 1169, 865, 1211, 905, 1267

Offset: 1

David James Sycamore, Sep 12 2017

nonn

Subsequence of even argument terms b(2n) of A056240 (listed in order), which do not appear in A288313. - David James Sycamore, Sep 13 2017

			b(12) = 35 is included since b(14) = 33 < 35.
b(28) = 115 is in since b(32) = 87, b(34) = 93, and b(40) = 111 are all less than 115.

Ivan N. Ianakiev, Table of n, a(n) for n = 1..1000

Cf. A056240, A288813.

a056240=Cases[Import["https://oeis.org/A056240/b056240.txt","Table"],{,}][[All,2]]; even=Take[a056240,{1,Length[a056240],2}];leven=Length[even]; a292081= Select[even,#>Min[Part[even,1;;Position[even,#][[1,1]]]]&&#>Min[Part[even,Position[even,#][[1,1]];;leven]]&] (* Ivan N. Ianakiev, Nov 02 2023 *)

A299912 a(n) = a056240-type of n-th prime, or -1 if the a056240-type is not defined.

Original entry on oeis.org

-1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Mar 10 2018

sign

For the definition of the a056240-type of a prime, see A293652, which also gives the smallest prime of a056240-type k for k = 1, 2, 3, ....

Michel Marcus, Table of n, a(n) for n = 1..5001
Michel Marcus, PARI/GP program

Cf. A056240, A293652.

For primes of a056240-types 2 and 3 see A300334 and A300359.

PARI
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\\ See Marcus link.
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A298615 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that there is at least one m > n for which b(2m) < b(2n) belongs to A297150.

Original entry on oeis.org

161, 217, 329, 371, 427, 511, 581, 623, 1246, 791, 1417, 1243, 1469, 2071, 917, 973, 1507, 1529, 1057, 1099, 1169, 1211, 1267, 1969, 1991, 1393, 2167, 2189, 2587, 1477, 2954, 2321, 2743, 1631, 1687, 2629, 2651, 1757, 1799, 1841, 1897, 1981, 3091, 3113, 2051, 4102

Offset: 1

David James Sycamore, Jan 26 2018

nonn

For even number n, if n-5 and n-3 are both composite then A056240(n) belongs to this sequence. The union of terms in this sequence together with those in A288313 and A297150 combine to make A056240(2n), for n >= 3. A288313(n) = A056240(A298252(n)), A297150(n) = A056240(A297925(n)), and the terms of this sequence correspond to A056240(A298366). Distinct sequences A298252, A297925 and A298366 form a partition of the nonnegative even integers (A005843) >= 6. These partitions holds because any even integer n >= 6 is such that, either n-3 is prime (A298252), or n-5 is prime but n-3 is composite (A297925), or both n-5 and n-3 are composite (A298366).

			n=1, a(1) = A056240(A298366(1)) = A056240(30) = 161;
n=24, a(24) = A056240(A298366(24)) = A056240(190) = 1969.

Michel Marcus, Table of n, a(n) for n = 1..2000

Cf. A056240, A288313, A297150, A298252, A298366, A005843.

A056240(n, p=n-1, m=oo)=if(n<6 || isprime(n), n, n==6, 8, until(p<3 || (n-p=precprime(p-1))*p >= m=min(m, A056240(n-p)*p), ); m);
is_A298366(n)= !isprime(n-5) && !isprime(n-3) && !(n%2) && (n>5);
lista(nn) = {for (n=0, nn, if (is_A298366(n), print1(A056240(n), ", ")););} \\ Michel Marcus, Apr 03 2020

a(n) = A056240(A298366(n)).

A300334 Primes of a056240-type 2.

Original entry on oeis.org

211, 541, 631, 673, 1693, 1801, 2879, 3181, 3271, 3299, 3343, 3571, 3943, 4177, 4441, 4561, 4751, 4783, 4813, 4861, 5147, 5381, 5431, 5501, 5779, 6029, 6197, 6421, 6469, 6521, 6599, 6637, 6883, 7103, 7321, 7369, 7477, 7573, 7603, 7789, 7901, 7963, 8419, 8443, 8641, 8923, 9091, 9587, 9643, 9733, 9781, 9871, 10513

Offset: 1

David James Sycamore, Mar 03 2018

nonn

Prime(r) has a056240-type k if A295185(prime(r))=prime(r-k)*A056240(prime(r)-prime(r-k)).

This sequence lists primes having a056240-type k=2, each having form ~2(g1,g2) where g1 is the first gap below prime(r), and g2 is the second (notation explained in A295185). The majority of primes appear to be of a056240-type 1.

			211 is included because the smallest composite number whose sum of prime factors (with repetition)=211 is 6501=197*33, a multiple of the second prime below 211, not the first. 211~2(12,2) is the smallest prime to have this property. Likewise 541~2(18,2), 1693~2(24,2), 2879~2(18,4), etc.

Cf. A056240, A295185, A299110, A293652, A299912, A300359.

Edited by N. J. A. Sloane, Mar 10 2018

A300359 Primes of a056240-type 3.

Original entry on oeis.org

4327, 6947, 7283, 7993, 11863, 12143, 14591, 15859, 18583, 18617, 18839, 18947, 19661, 20593, 21059, 21313, 22853, 29567, 29917, 31307, 32531, 32831, 35023, 35491, 35671, 39499, 39659, 40343, 41579, 42821, 43853, 46807, 48221, 49253, 49363, 49597, 50411, 51109, 52667, 54401

Offset: 1

David James Sycamore, Mar 03 2018

nonn

			4327 is included because the smallest composite number whose sum of prime factors (with repetition)=4327 is 526809=4283*123, a multiple of the third prime below 4327 (not the first or second). 4327~3(30,8,6) is the smallest prime to have this property. Likewise 6947~3(30,6,4), and 7993~3(30,12,2).

Cf. A056240, A295185, A299704, A293652, A299912, A300334.

Edited by N. J. A. Sloane, Mar 10 2018

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 *)

A069968 Duplicate of A056240.

Original entry on oeis.org

2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13, 33, 26, 55, 17, 65, 19, 51, 38, 53, 23, 95, 46, 69

Offset: 2

dead

A318387 Starting with a(1) = 6, a(n) is the smallest number whose sum of prime divisors (taken with multiplicity) is a(n-1). In other words, a(n) = A056240(a(n-1)).

Original entry on oeis.org

6, 8, 15, 26, 69, 134, 393, 1556, 4659, 9314, 27933, 921327, 85680249, 171360494, 2227686253, 17821489976, 124750429783, 19336316610785, 4544034403522255, 3567067006764843005, 203322819385596050031, 25008706784428314148401, 825287323886134366896771, 91606892951360914725537141, 1923744751978579209236279751

Offset: 1

David James Sycamore, Aug 25 2018

nonn

Any nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a nontrivial sequence (because A056240(r)=r for r=4 or a prime). In this sequence a(1) is set to 6 because it is the smallest composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non-stationary sequence of this kind.

Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).

			a(2) = 8, the smallest number whose sopfr is 6: A056240(8) = 6;
a(3) = 15, the smallest number whose sopfr is 8: A056240(8) = 15; etc.

Cf. A001414, A048133, A002217, A056240.

With[{s = Array[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &,10^6]}, Nest[Append[#, First@ FirstPosition[s, #[[-1]] ]] &, {6}, 11]] (* Michael De Vlieger, Aug 25 2018 *)

a(n) = A056240(a(n-1)); A002217(a(n)) = 1 + A002217(a(n - 1))

a(13)-a(17) from Giovanni Resta, Aug 28 2018

Terms a(18) onward from Max Alekseyev, Sep 20 2024

cp's OEIS Frontend

A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2*n) where for all m > n, b(2*m) > b(2*n).

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A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).

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A292081 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that for some m > n, b(2m) < b(2n).

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A299912 a(n) = a056240-type of n-th prime, or -1 if the a056240-type is not defined.

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A298615 Let b(k) be A056240(k); this sequence lists numbers b(2n) such that there is at least one m > n for which b(2m) < b(2n) belongs to A297150.

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A300334 Primes of a056240-type 2.

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A300359 Primes of a056240-type 3.

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2*n).