A100118 -id:A100118 - cp's OEIS frontend

A316520 Heinz numbers of integer partitions whose average is a prime number.

3, 5, 9, 10, 11, 17, 21, 22, 25, 27, 28, 30, 31, 41, 46, 57, 59, 67, 68, 78, 81, 82, 83, 84, 85, 88, 90, 91, 98, 99, 100, 105, 109, 110, 111, 121, 125, 127, 146, 155, 157, 164, 179, 184, 191, 194, 203, 208, 211, 213, 222, 228, 241, 243, 247, 252, 253, 264, 267

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			60 = 2*2*3*5 has prime indices {1,1,2,3} with average 7/4, so 60 does not belong to the sequence.
90 = 2*3*3*5 has prime indices {1,2,2,3} with average 8/2 = 4, so 90 belongs to the sequence.

Cf. A000040, A000041, A000607, A056239, A067538, A078175, A100118, A237984, A296150, A316313, A316413.

Select[Range[100],PrimeQ[Mean[If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]&]

A358490 Composite Fibonacci numbers whose sum of prime factors (with multiplicity) is a prime.

Original entry on oeis.org

34, 75025, 196418, 701408733, 225851433717, 591286729879, 23416728348467685, 420196140727489673, 927372692193078999176, 16641027750620563662096, 114059301025943970552219, 1264937032042997393488322, 5358359254990966640871840, 2353412818241252672952597492098, 3807901929474025356630904134051

Offset: 1

Marc Kouyoumdjian, Nov 18 2022

nonn

			75025 is a term because it is a composite Fibonacci number whose sum of prime factors 5, 5 and 3001 is 3011, a prime number.

Shyam Sunder Gupta, Prime Curios! 34

Intersection of A000045 and A046363.
Intersection of A090206 and A100118.
Cf. A001414.

A385724 The least integer of n consecutive numbers where each has its sum of prime factors, with multiplicity, being a prime.

Original entry on oeis.org

17, 2, 5, 10, 1547, 8837, 1293224, 52445796, 3267037, 896531141, 183208285259

Offset: 1

Zhining Yang, Jul 08 2025

According to Primepuzzles, puzzle 445:
a(10) = 896531141 given by Carlos Rivera.
a(11) = 183208285259 given by Jens Kruse Andersen.

			a(4) = 10 because 10=2*5, 11=11, 12 = 2^2*3, 13 = 13 and sum of prime factors of the four consecutive numbers are all primes : 2 + 5 = 7, 11, 2 + 2 + 3 = 7, 13.

Fang, Can we find that the sum of prime factors of each of twelve consecutive positive integers is a prime number?
G. L. Honaker, Jr., Prime Curio 183208285259.
Carlos Rivera, Puzzle 445. Consecutive integers with prime SPF, The Prime Puzzles & Problems Connection.

Cf. A100118, A337310.

f[n_] := PrimeQ@Total[Times @@@ FactorInteger[n]]; a = Table[0, 6];
c = 0; Do[If[c == 0, s = n]; If[f[n] == False, If[a[[c]] == 0, a[[c]] = s]; c = 0, c++], {n, 2, 10000}]; a

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A316520 Heinz numbers of integer partitions whose average is a prime number.

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A358490 Composite Fibonacci numbers whose sum of prime factors (with multiplicity) is a prime.

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A385724 The least integer of n consecutive numbers where each has its sum of prime factors, with multiplicity, being a prime.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica