A056240 -id:A056240 - cp's OEIS frontend

A354075 Lexicographically earliest infinite sequence of distinct positive integers such that A(a(n+1)) is prime to A(a(n)) but not to A(a(n-1)), where A is A001414.

2, 3, 4, 14, 15, 20, 16, 24, 18, 26, 33, 5, 7, 6, 10, 21, 12, 8, 94, 9, 124, 27, 25, 38, 30, 62, 32, 11, 35, 28, 36, 40, 39, 45, 42, 48, 44, 54, 46, 57, 86, 49, 74, 51, 13, 50, 22, 55, 56, 60, 63, 64, 75, 65, 80, 66, 90, 70, 96, 68, 69, 92, 84, 105, 85, 112, 87

Offset: 1

David James Sycamore, Jun 11 2022

nonn

2,3,4 is the earliest string of three consecutive numbers which satisfy the definition, therefore the sequence begins a(1)=2, a(2)=3, a(3)=4.
Sequence is infinite since there always exists a k which has not occurred before such that A(k) is prime to A(a(n)) but not to A(a(n-1)). Since A001414 covers N/{1} a number m can be found such that A(m)=k. Thus k can be chosen for a(n+1) unless there is a smaller number with the same property.
Similar to the Yellowstone sequence (A098550) in terms of coprime relations.
The first seven primes are in natural order but then we have ...,17,23,19,31,37,43,41,47,29,...
Conjectured to be a permutation of N/{0,1}.

			a(4)=14 because A(14)=9 is prime to A(a(3))=4 but not to A(a(2))=3, and is the smallest number not already seen in the sequence which has this property.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A001414, A056240, A098440, A277272.

f(n) = my(f=factor(n)); f[, 1]~*f[, 2]; \\ A001414
lista(nn) = {my(va = vector(nn)); va[1] = 2; va[2] = 3; for (n=3, nn, my(k=1); while ((gcd(f(va[n-1]), f(k)) != 1) || (gcd(f(va[n-2]), f(k)) == 1) || #select(x->(x==k), va), k++); va[n] = k;); va;} \\ Michel Marcus, Jun 12 2022

Corrected and extended by Michel Marcus, Jun 12 2022

A354547 Least number k <= n such that sopfr(k) = sopfr(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 8, 7, 11, 7, 13, 14, 15, 15, 17, 15, 19, 14, 21, 13, 23, 14, 21, 26, 14, 11, 29, 21, 31, 21, 33, 19, 35, 21, 37, 38, 39, 11, 41, 35, 43, 26, 11, 46, 47, 11, 33, 35, 51, 17, 53, 11, 39, 13, 57, 31, 59, 35, 61, 62, 13, 35, 65, 39, 67, 38

Offset: 1

Jean-Marc Rebert, Aug 15 2022

nonn

			For n = 16 = 2^4, sopfr(16) = 2*4 = 8 and 15 = 3 * 5, sopfr(15)= 3 + 5 = 8 and for k < 15, sopfr(k) != 8, hence a(16) = 15.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001414 (sopfr), A056240, A064364.

f[n_] := Plus @@ Times @@@ FactorInteger[n]; f[1] = 0; m = 100; With[{s = Array[f, m]}, Table[FirstPosition[s, s[[n]]][[1]], {n, 1, m}]] (* Amiram Eldar, Aug 15 2022 *)

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
a(n) = my(s=sopfr(n), k=1); while (sopfr(k) != s, k++); k; \\ Michel Marcus, Aug 16 2022

A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

Original entry on oeis.org

5, 9, 7, 15, 27, 21, 11, 35, 13, 33, 105, 39, 17, 65, 19, 51, 195, 57, 23, 95, 171, 69, 285, 115, 29, 161, 31, 87, 483, 93, 261, 155, 37, 217, 465, 111, 41, 185, 43, 123, 555, 129, 47, 215, 387, 141, 645, 235, 53, 329, 705, 159, 987, 265, 59, 371, 61, 177

Offset: 5

Gus Wiseman, Oct 27 2023

nonn

All terms are odd.

It seems that all composite terms not divisible by 3 form a supersequence of A292081. - Ivan N. Ianakiev, Oct 30 2023

			The terms together with their prime factors (which sum to n) begin:
    5 = 5
    9 = 3*3
    7 = 7
   15 = 3*5
   27 = 3*3*3
   21 = 3*7
   11 = 11
   35 = 5*7
   13 = 13
   33 = 3*11
  105 = 3*5*7

Michel Marcus, Table of n, a(n) for n = 5..1000

This is the odd case of A056240.
Positions of first appearances in A366840 (sum of odd prime factors).
The partition triangle for this statistic is A366851, even A116598.
A001414 adds up prime factors, triangle A331416.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A027746 lists prime factors, length A001222.
A087436 counts odd prime factors, even A007814.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A031368, A036349, A056239, A066208, A100005, A112798, A239261, A257991, A335657, A366839.

nn=1000;
w=Table[Total[Times@@@DeleteCases[If[n==1,{},FactorInteger[n]],{2,_}]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[i]]];
Table[Position[w,k][[1,1]],{k,5,spnm[Join[{1,2,3,4},Take[w,nn]/.(0->1)]]}]

f(n) = my(f=factor(n), j=if (n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ A366840
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Nov 02 2023

A376511 a(1) = 2. For n > 1, if a(n-1) is a novel term, a(n)=A001414(a(n-1)), else if there are k (>1) occurrences of a(j) = a(n-1) (1<=j<=n-1), a(n) is the smallest novel m such that A001414(m) = k*a(n-1).

Original entry on oeis.org

2, 2, 4, 4, 15, 8, 6, 5, 5, 21, 10, 7, 7, 33, 14, 9, 6, 35, 12, 7, 38, 21, 185, 42, 12, 95, 24, 9, 65, 18, 8, 39, 16, 8, 114, 24, 215, 48, 11, 11, 57, 22, 13, 13, 69, 26, 15, 161, 30, 10, 51, 20, 9, 92, 27, 9, 155, 36, 10, 209, 30, 371, 60, 12, 186, 36, 335, 72

Offset: 1

David James Sycamore, Sep 25 2024

nonn

Sequence inspired by a revisit to A353125. a(n) is a novel prime p iff a(n-1) is a term in A046363, following which a(n+1) is also = p. The first occurrences of 4 or p are followed by 4 or p respectively (4 being the only composite m such that Sopfr(m)=m), and these are the only terms repeated contiguously in this sequence. 3 cannot be a term because it is not given, and there is no composite g such that Sopfr(g)=3. A string of descending composite terms follows primes p,p until reaching (i) a repeat of an earlier term, or (ii) a term in A046363 (which produces a new prime pair q,q). If (i) the sequence resets immediately to a new string of descending composite terms, and if (ii) the reset occurs after the next pair q,q of primes. Every positive integer m (other than 3) occurs a maximum of A000607(m) times, this being the number of numbers k such that Sopfr(k)=m.

Row n of table T(n,k) in A064364 lists numbers j such that A001414(j) = n, with T(n,1) = A056240(n), and every term in this sequence is taken from the appropriate row of A064364. When a(n-1) is a novel term a(n) = A001414(a(n-1)), which is defined. Otherwise a(n) = smallest m such that A001414(m) = k*a(n-1), a number which is also defined since it is the smallest unused term in T(k*a(n-1),k) of A064364. Therefore the sequence is well defined and infinite. Conjecture: For any n > 1 every term in T(n,k) of A064364 appears eventually.

			a(1) = 2 is given, then since 2 is a novel term, a(2) = A001414(2) = 2. 2 has now been seen k = 2 times so a(3) is the smallest novel m such that A001414(m) = 2*2 = 4, so a(3) = 4, a novel term meaning that a(4) = A001414(4) = 4. now 4 has been seen twice so a(5) is the smallest novel m such that A001414(m) = 2*4 = 8, so a(5) = 15.

Michael De Vlieger, Table of n, a(n) for n = 1..5000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..5000, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers. Primes first appear in pairs.

Cf. A000040, A000607, A000792, A001414, A046363, A056240, A064364, A353125, A376147.

nn = 120; c[_] := 0; j = a[1] = 2; u = 2;
f[x_] := f[x] = Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]];
s = PositionIndex@ Array[f, 2^16];
Do[If[Set[m, c[j]] == 0, Set[k, f[j]],
  Set[{k, t, w}, {1, #, Length[#]}] &@Lookup[s, (m + 1)*j];
  While[c[t[[k]]] > 0, k++]; k = t[[k]] ]; c[j]++;
  Set[{a[i], j}, {k, k}], {i, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 27 2024 *)

If a(k) = first occasion of prime p, a(k+1) = p, a(k+2) = A056240(2*p), a(k+3) = 2*p.

A386524 a(n+1) is the smallest product of two or more primes which sum to a(n), starting from a(1) = 5.

Original entry on oeis.org

5, 6, 8, 15, 26, 69, 134, 393, 1556, 4659, 9314, 27933, 921327, 85680249, 171360494, 2227686253, 17821489976, 124750429783, 17589810592353, 70359242369396, 11046401051970523, 22092802103941042, 1634867355691634222, 238690633930978585462, 45589911080816909786761, 2963344220253099136138295, 2474392423911337778675332705

Offset: 1

Marc Morgenegg, Jul 24 2025

nonn

a(1) = 5 is a prime and is not taken as a sum of just itself, but thereafter terms are never prime.

			For n=2, a(2) = 8 and the possible sums of primes are 2+2+2+2 = 8 or 3+5 = 8 and the smallest product of those terms is 3*5=15 (versus 2*2*2*2=16) so that a(3) = 15.

David A. Corneth, PARI program

Cf. A056240.

PARI
```
\\ See Corneth link
```

More terms from David A. Corneth, Jul 28 2025

A067566 Smallest k such that k = nA001414(k) (or 0 if no such k exists), where A001414(k) is the integer log of k, i.e., Sum p_ie_i if the prime factorization of k is Product p_i^e_i.

Original entry on oeis.org

2, 16, 27, 0, 60, 72, 105, 0, 0, 150, 220, 180, 0, 308, 240, 256, 476, 288, 627, 440, 378, 0, 805, 528, 450, 624, 594, 560, 1160, 540, 1581, 576, 0, 1122, 0, 648, 1776, 1748, 0, 800, 3526, 0, 2322, 1056, 900, 0, 3055, 960, 0, 0, 1530, 1456, 4028

Offset: 1

Naohiro Nomoto, Jan 30 2002

			a(3)=27 since 27 = 3^3 and 27 = 3*(3*3), but not 30 even though 30 = 2^1*3^1*5^1 and 30 = 3*(2*1+3*1+5*1) because 27 is less than 30.

Cf. A001414, A056240.

Edited by Henry Bottomley, Jul 07 2002

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.

Original entry on oeis.org

3, 7, 13, 43, 67, 109, 127, 151, 163, 211, 277, 307, 373, 457, 463, 601, 613, 673, 727, 853, 919, 967, 1021, 1117, 1171, 1231, 1399, 1471, 1483, 1747, 1789, 1933, 2029, 2251, 2311, 2389, 2503, 2521, 2557, 2659, 2851, 2857, 3019, 3067, 3121, 3229, 3583, 3613, 3637, 3691, 3697

Offset: 1

David James Sycamore, Aug 02 2017

nonn

The terms of this sequence belong to two disjoint subsequences, namely those for which |A(5*p) - A(4*p)| = 9; (3,7,13,43,67,127,163,211,277,307,457,...), and those for which 5*A(4*p) - 3*A(5*p) = 3, (109,151,373,673,919,...), where A = A288814.

Note: A288814(n) = A056240(n) for all composite n.

			P=7: 5*7 - 4 = 31, 4*7 - 5 = 23, both prime so 7 is in this sequence, and belongs to the subsequence of terms satisfying A(4*p) - A(3*p) = 9.
P=109: 5*109 - 4 = 541, 4*109 - 5 = 431, both prime so 109 is in this sequence, and belongs to the subsequence of terms satisfying 5*A(4*p) - 3*A(5*p) = 3.

Cf. A259730, A288814, A290163, A290164, A056240.

Intersection of A136051 and A156300. - Michel Marcus, Aug 04 2017

Select[Prime@ Range@ 516, Times @@ Boole@ Map[PrimeQ, {5 # - 4, 4 # - 5}] > 0 &] (* Michael De Vlieger, Aug 02 2017 *)

More terms from Altug Alkan, Aug 02 2017

cp's OEIS Frontend

A354075 Lexicographically earliest infinite sequence of distinct positive integers such that A(a(n+1)) is prime to A(a(n)) but not to A(a(n-1)), where A is A001414.

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A354547 Least number k <= n such that sopfr(k) = sopfr(n).

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A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

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A376511 a(1) = 2. For n > 1, if a(n-1) is a novel term, a(n)=A001414(a(n-1)), else if there are k (>1) occurrences of a(j) = a(n-1) (1<=j<=n-1), a(n) is the smallest novel m such that A001414(m) = k*a(n-1).

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A386524 a(n+1) is the smallest product of two or more primes which sum to a(n), starting from a(1) = 5.

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A067566 Smallest k such that k = n*A001414(k) (or 0 if no such k exists), where A001414(k) is the integer log of k, i.e., Sum p_i*e_i if the prime factorization of k is Product p_i^e_i.

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A289556 Primes p such that both 5*p - 4 and 4*p - 5 are prime.

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A067566 Smallest k such that k = nA001414(k) (or 0 if no such k exists), where A001414(k) is the integer log of k, i.e., Sum p_ie_i if the prime factorization of k is Product p_i^e_i.

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.